Require terminaison.

Require Relations.

Require term.

Require List.

Require equational_theory.

Require rpo_extension.

Require equational_extension.

Require closure_extension.

Require term_extension.

Require dp.

Require Inclusion.

Require or_ext_generated.

Require ZArith.

Require ring_extention.

Require Zwf.

Require Inverse_Image.

Require matrix.

Require more_list_extention.

Import List.

Import ZArith.

Set Implicit Arguments.

Module algebra.
 Module F
  <:term.Signature.
  Inductive symb  :
   Set := 
     (* id_a____ *)
    | id_a____ : symb
     (* id_i *)
    | id_i : symb
     (* id_isList *)
    | id_isList : symb
     (* id_a__and *)
    | id_a__and : symb
     (* id_isQid *)
    | id_isQid : symb
     (* id_isPal *)
    | id_isPal : symb
     (* id_mark *)
    | id_mark : symb
     (* id_u *)
    | id_u : symb
     (* id_isNeList *)
    | id_isNeList : symb
     (* id_a__isList *)
    | id_a__isList : symb
     (* id_a *)
    | id_a : symb
     (* id___ *)
    | id___ : symb
     (* id_o *)
    | id_o : symb
     (* id_a__isQid *)
    | id_a__isQid : symb
     (* id_tt *)
    | id_tt : symb
     (* id_isNePal *)
    | id_isNePal : symb
     (* id_a__isPal *)
    | id_a__isPal : symb
     (* id_nil *)
    | id_nil : symb
     (* id_and *)
    | id_and : symb
     (* id_a__isNePal *)
    | id_a__isNePal : symb
     (* id_a__isNeList *)
    | id_a__isNeList : symb
     (* id_e *)
    | id_e : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_a____,id_a____ => true
      | id_i,id_i => true
      | id_isList,id_isList => true
      | id_a__and,id_a__and => true
      | id_isQid,id_isQid => true
      | id_isPal,id_isPal => true
      | id_mark,id_mark => true
      | id_u,id_u => true
      | id_isNeList,id_isNeList => true
      | id_a__isList,id_a__isList => true
      | id_a,id_a => true
      | id___,id___ => true
      | id_o,id_o => true
      | id_a__isQid,id_a__isQid => true
      | id_tt,id_tt => true
      | id_isNePal,id_isNePal => true
      | id_a__isPal,id_a__isPal => true
      | id_nil,id_nil => true
      | id_and,id_and => true
      | id_a__isNePal,id_a__isNePal => true
      | id_a__isNeList,id_a__isNeList => true
      | id_e,id_e => true
      | _,_ => false
      end.
  
  
   (* Proof of decidability of equality over symb *)
  Definition symb_eq_bool_ok(f1 f2:symb) :
   match symb_eq_bool f1 f2 with
     | true => f1 = f2
     | false => f1 <> f2
     end.
  Proof.
    intros f1 f2.
    
    refine match f1 as u1,f2 as u2 return 
             match symb_eq_bool u1 u2 return 
               Prop with
               | true => u1 = u2
               | false => u1 <> u2
               end with
             | id_a____,id_a____ => refl_equal _
             | id_i,id_i => refl_equal _
             | id_isList,id_isList => refl_equal _
             | id_a__and,id_a__and => refl_equal _
             | id_isQid,id_isQid => refl_equal _
             | id_isPal,id_isPal => refl_equal _
             | id_mark,id_mark => refl_equal _
             | id_u,id_u => refl_equal _
             | id_isNeList,id_isNeList => refl_equal _
             | id_a__isList,id_a__isList => refl_equal _
             | id_a,id_a => refl_equal _
             | id___,id___ => refl_equal _
             | id_o,id_o => refl_equal _
             | id_a__isQid,id_a__isQid => refl_equal _
             | id_tt,id_tt => refl_equal _
             | id_isNePal,id_isNePal => refl_equal _
             | id_a__isPal,id_a__isPal => refl_equal _
             | id_nil,id_nil => refl_equal _
             | id_and,id_and => refl_equal _
             | id_a__isNePal,id_a__isNePal => refl_equal _
             | id_a__isNeList,id_a__isNeList => refl_equal _
             | id_e,id_e => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_a____ => term.Free 2
      | id_i => term.Free 0
      | id_isList => term.Free 1
      | id_a__and => term.Free 2
      | id_isQid => term.Free 1
      | id_isPal => term.Free 1
      | id_mark => term.Free 1
      | id_u => term.Free 0
      | id_isNeList => term.Free 1
      | id_a__isList => term.Free 1
      | id_a => term.Free 0
      | id___ => term.Free 2
      | id_o => term.Free 0
      | id_a__isQid => term.Free 1
      | id_tt => term.Free 0
      | id_isNePal => term.Free 1
      | id_a__isPal => term.Free 1
      | id_nil => term.Free 0
      | id_and => term.Free 2
      | id_a__isNePal => term.Free 1
      | id_a__isNeList => term.Free 1
      | id_e => term.Free 0
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_a____,id_a____ => true
      | id_a____,id_i => false
      | id_a____,id_isList => false
      | id_a____,id_a__and => false
      | id_a____,id_isQid => false
      | id_a____,id_isPal => false
      | id_a____,id_mark => false
      | id_a____,id_u => false
      | id_a____,id_isNeList => false
      | id_a____,id_a__isList => false
      | id_a____,id_a => false
      | id_a____,id___ => false
      | id_a____,id_o => false
      | id_a____,id_a__isQid => false
      | id_a____,id_tt => false
      | id_a____,id_isNePal => false
      | id_a____,id_a__isPal => false
      | id_a____,id_nil => false
      | id_a____,id_and => false
      | id_a____,id_a__isNePal => false
      | id_a____,id_a__isNeList => false
      | id_a____,id_e => false
      | id_i,id_a____ => true
      | id_i,id_i => true
      | id_i,id_isList => false
      | id_i,id_a__and => false
      | id_i,id_isQid => false
      | id_i,id_isPal => false
      | id_i,id_mark => false
      | id_i,id_u => false
      | id_i,id_isNeList => false
      | id_i,id_a__isList => false
      | id_i,id_a => false
      | id_i,id___ => false
      | id_i,id_o => false
      | id_i,id_a__isQid => false
      | id_i,id_tt => false
      | id_i,id_isNePal => false
      | id_i,id_a__isPal => false
      | id_i,id_nil => false
      | id_i,id_and => false
      | id_i,id_a__isNePal => false
      | id_i,id_a__isNeList => false
      | id_i,id_e => false
      | id_isList,id_a____ => true
      | id_isList,id_i => true
      | id_isList,id_isList => true
      | id_isList,id_a__and => false
      | id_isList,id_isQid => false
      | id_isList,id_isPal => false
      | id_isList,id_mark => false
      | id_isList,id_u => false
      | id_isList,id_isNeList => false
      | id_isList,id_a__isList => false
      | id_isList,id_a => false
      | id_isList,id___ => false
      | id_isList,id_o => false
      | id_isList,id_a__isQid => false
      | id_isList,id_tt => false
      | id_isList,id_isNePal => false
      | id_isList,id_a__isPal => false
      | id_isList,id_nil => false
      | id_isList,id_and => false
      | id_isList,id_a__isNePal => false
      | id_isList,id_a__isNeList => false
      | id_isList,id_e => false
      | id_a__and,id_a____ => true
      | id_a__and,id_i => true
      | id_a__and,id_isList => true
      | id_a__and,id_a__and => true
      | id_a__and,id_isQid => false
      | id_a__and,id_isPal => false
      | id_a__and,id_mark => false
      | id_a__and,id_u => false
      | id_a__and,id_isNeList => false
      | id_a__and,id_a__isList => false
      | id_a__and,id_a => false
      | id_a__and,id___ => false
      | id_a__and,id_o => false
      | id_a__and,id_a__isQid => false
      | id_a__and,id_tt => false
      | id_a__and,id_isNePal => false
      | id_a__and,id_a__isPal => false
      | id_a__and,id_nil => false
      | id_a__and,id_and => false
      | id_a__and,id_a__isNePal => false
      | id_a__and,id_a__isNeList => false
      | id_a__and,id_e => false
      | id_isQid,id_a____ => true
      | id_isQid,id_i => true
      | id_isQid,id_isList => true
      | id_isQid,id_a__and => true
      | id_isQid,id_isQid => true
      | id_isQid,id_isPal => false
      | id_isQid,id_mark => false
      | id_isQid,id_u => false
      | id_isQid,id_isNeList => false
      | id_isQid,id_a__isList => false
      | id_isQid,id_a => false
      | id_isQid,id___ => false
      | id_isQid,id_o => false
      | id_isQid,id_a__isQid => false
      | id_isQid,id_tt => false
      | id_isQid,id_isNePal => false
      | id_isQid,id_a__isPal => false
      | id_isQid,id_nil => false
      | id_isQid,id_and => false
      | id_isQid,id_a__isNePal => false
      | id_isQid,id_a__isNeList => false
      | id_isQid,id_e => false
      | id_isPal,id_a____ => true
      | id_isPal,id_i => true
      | id_isPal,id_isList => true
      | id_isPal,id_a__and => true
      | id_isPal,id_isQid => true
      | id_isPal,id_isPal => true
      | id_isPal,id_mark => false
      | id_isPal,id_u => false
      | id_isPal,id_isNeList => false
      | id_isPal,id_a__isList => false
      | id_isPal,id_a => false
      | id_isPal,id___ => false
      | id_isPal,id_o => false
      | id_isPal,id_a__isQid => false
      | id_isPal,id_tt => false
      | id_isPal,id_isNePal => false
      | id_isPal,id_a__isPal => false
      | id_isPal,id_nil => false
      | id_isPal,id_and => false
      | id_isPal,id_a__isNePal => false
      | id_isPal,id_a__isNeList => false
      | id_isPal,id_e => false
      | id_mark,id_a____ => true
      | id_mark,id_i => true
      | id_mark,id_isList => true
      | id_mark,id_a__and => true
      | id_mark,id_isQid => true
      | id_mark,id_isPal => true
      | id_mark,id_mark => true
      | id_mark,id_u => false
      | id_mark,id_isNeList => false
      | id_mark,id_a__isList => false
      | id_mark,id_a => false
      | id_mark,id___ => false
      | id_mark,id_o => false
      | id_mark,id_a__isQid => false
      | id_mark,id_tt => false
      | id_mark,id_isNePal => false
      | id_mark,id_a__isPal => false
      | id_mark,id_nil => false
      | id_mark,id_and => false
      | id_mark,id_a__isNePal => false
      | id_mark,id_a__isNeList => false
      | id_mark,id_e => false
      | id_u,id_a____ => true
      | id_u,id_i => true
      | id_u,id_isList => true
      | id_u,id_a__and => true
      | id_u,id_isQid => true
      | id_u,id_isPal => true
      | id_u,id_mark => true
      | id_u,id_u => true
      | id_u,id_isNeList => false
      | id_u,id_a__isList => false
      | id_u,id_a => false
      | id_u,id___ => false
      | id_u,id_o => false
      | id_u,id_a__isQid => false
      | id_u,id_tt => false
      | id_u,id_isNePal => false
      | id_u,id_a__isPal => false
      | id_u,id_nil => false
      | id_u,id_and => false
      | id_u,id_a__isNePal => false
      | id_u,id_a__isNeList => false
      | id_u,id_e => false
      | id_isNeList,id_a____ => true
      | id_isNeList,id_i => true
      | id_isNeList,id_isList => true
      | id_isNeList,id_a__and => true
      | id_isNeList,id_isQid => true
      | id_isNeList,id_isPal => true
      | id_isNeList,id_mark => true
      | id_isNeList,id_u => true
      | id_isNeList,id_isNeList => true
      | id_isNeList,id_a__isList => false
      | id_isNeList,id_a => false
      | id_isNeList,id___ => false
      | id_isNeList,id_o => false
      | id_isNeList,id_a__isQid => false
      | id_isNeList,id_tt => false
      | id_isNeList,id_isNePal => false
      | id_isNeList,id_a__isPal => false
      | id_isNeList,id_nil => false
      | id_isNeList,id_and => false
      | id_isNeList,id_a__isNePal => false
      | id_isNeList,id_a__isNeList => false
      | id_isNeList,id_e => false
      | id_a__isList,id_a____ => true
      | id_a__isList,id_i => true
      | id_a__isList,id_isList => true
      | id_a__isList,id_a__and => true
      | id_a__isList,id_isQid => true
      | id_a__isList,id_isPal => true
      | id_a__isList,id_mark => true
      | id_a__isList,id_u => true
      | id_a__isList,id_isNeList => true
      | id_a__isList,id_a__isList => true
      | id_a__isList,id_a => false
      | id_a__isList,id___ => false
      | id_a__isList,id_o => false
      | id_a__isList,id_a__isQid => false
      | id_a__isList,id_tt => false
      | id_a__isList,id_isNePal => false
      | id_a__isList,id_a__isPal => false
      | id_a__isList,id_nil => false
      | id_a__isList,id_and => false
      | id_a__isList,id_a__isNePal => false
      | id_a__isList,id_a__isNeList => false
      | id_a__isList,id_e => false
      | id_a,id_a____ => true
      | id_a,id_i => true
      | id_a,id_isList => true
      | id_a,id_a__and => true
      | id_a,id_isQid => true
      | id_a,id_isPal => true
      | id_a,id_mark => true
      | id_a,id_u => true
      | id_a,id_isNeList => true
      | id_a,id_a__isList => true
      | id_a,id_a => true
      | id_a,id___ => false
      | id_a,id_o => false
      | id_a,id_a__isQid => false
      | id_a,id_tt => false
      | id_a,id_isNePal => false
      | id_a,id_a__isPal => false
      | id_a,id_nil => false
      | id_a,id_and => false
      | id_a,id_a__isNePal => false
      | id_a,id_a__isNeList => false
      | id_a,id_e => false
      | id___,id_a____ => true
      | id___,id_i => true
      | id___,id_isList => true
      | id___,id_a__and => true
      | id___,id_isQid => true
      | id___,id_isPal => true
      | id___,id_mark => true
      | id___,id_u => true
      | id___,id_isNeList => true
      | id___,id_a__isList => true
      | id___,id_a => true
      | id___,id___ => true
      | id___,id_o => false
      | id___,id_a__isQid => false
      | id___,id_tt => false
      | id___,id_isNePal => false
      | id___,id_a__isPal => false
      | id___,id_nil => false
      | id___,id_and => false
      | id___,id_a__isNePal => false
      | id___,id_a__isNeList => false
      | id___,id_e => false
      | id_o,id_a____ => true
      | id_o,id_i => true
      | id_o,id_isList => true
      | id_o,id_a__and => true
      | id_o,id_isQid => true
      | id_o,id_isPal => true
      | id_o,id_mark => true
      | id_o,id_u => true
      | id_o,id_isNeList => true
      | id_o,id_a__isList => true
      | id_o,id_a => true
      | id_o,id___ => true
      | id_o,id_o => true
      | id_o,id_a__isQid => false
      | id_o,id_tt => false
      | id_o,id_isNePal => false
      | id_o,id_a__isPal => false
      | id_o,id_nil => false
      | id_o,id_and => false
      | id_o,id_a__isNePal => false
      | id_o,id_a__isNeList => false
      | id_o,id_e => false
      | id_a__isQid,id_a____ => true
      | id_a__isQid,id_i => true
      | id_a__isQid,id_isList => true
      | id_a__isQid,id_a__and => true
      | id_a__isQid,id_isQid => true
      | id_a__isQid,id_isPal => true
      | id_a__isQid,id_mark => true
      | id_a__isQid,id_u => true
      | id_a__isQid,id_isNeList => true
      | id_a__isQid,id_a__isList => true
      | id_a__isQid,id_a => true
      | id_a__isQid,id___ => true
      | id_a__isQid,id_o => true
      | id_a__isQid,id_a__isQid => true
      | id_a__isQid,id_tt => false
      | id_a__isQid,id_isNePal => false
      | id_a__isQid,id_a__isPal => false
      | id_a__isQid,id_nil => false
      | id_a__isQid,id_and => false
      | id_a__isQid,id_a__isNePal => false
      | id_a__isQid,id_a__isNeList => false
      | id_a__isQid,id_e => false
      | id_tt,id_a____ => true
      | id_tt,id_i => true
      | id_tt,id_isList => true
      | id_tt,id_a__and => true
      | id_tt,id_isQid => true
      | id_tt,id_isPal => true
      | id_tt,id_mark => true
      | id_tt,id_u => true
      | id_tt,id_isNeList => true
      | id_tt,id_a__isList => true
      | id_tt,id_a => true
      | id_tt,id___ => true
      | id_tt,id_o => true
      | id_tt,id_a__isQid => true
      | id_tt,id_tt => true
      | id_tt,id_isNePal => false
      | id_tt,id_a__isPal => false
      | id_tt,id_nil => false
      | id_tt,id_and => false
      | id_tt,id_a__isNePal => false
      | id_tt,id_a__isNeList => false
      | id_tt,id_e => false
      | id_isNePal,id_a____ => true
      | id_isNePal,id_i => true
      | id_isNePal,id_isList => true
      | id_isNePal,id_a__and => true
      | id_isNePal,id_isQid => true
      | id_isNePal,id_isPal => true
      | id_isNePal,id_mark => true
      | id_isNePal,id_u => true
      | id_isNePal,id_isNeList => true
      | id_isNePal,id_a__isList => true
      | id_isNePal,id_a => true
      | id_isNePal,id___ => true
      | id_isNePal,id_o => true
      | id_isNePal,id_a__isQid => true
      | id_isNePal,id_tt => true
      | id_isNePal,id_isNePal => true
      | id_isNePal,id_a__isPal => false
      | id_isNePal,id_nil => false
      | id_isNePal,id_and => false
      | id_isNePal,id_a__isNePal => false
      | id_isNePal,id_a__isNeList => false
      | id_isNePal,id_e => false
      | id_a__isPal,id_a____ => true
      | id_a__isPal,id_i => true
      | id_a__isPal,id_isList => true
      | id_a__isPal,id_a__and => true
      | id_a__isPal,id_isQid => true
      | id_a__isPal,id_isPal => true
      | id_a__isPal,id_mark => true
      | id_a__isPal,id_u => true
      | id_a__isPal,id_isNeList => true
      | id_a__isPal,id_a__isList => true
      | id_a__isPal,id_a => true
      | id_a__isPal,id___ => true
      | id_a__isPal,id_o => true
      | id_a__isPal,id_a__isQid => true
      | id_a__isPal,id_tt => true
      | id_a__isPal,id_isNePal => true
      | id_a__isPal,id_a__isPal => true
      | id_a__isPal,id_nil => false
      | id_a__isPal,id_and => false
      | id_a__isPal,id_a__isNePal => false
      | id_a__isPal,id_a__isNeList => false
      | id_a__isPal,id_e => false
      | id_nil,id_a____ => true
      | id_nil,id_i => true
      | id_nil,id_isList => true
      | id_nil,id_a__and => true
      | id_nil,id_isQid => true
      | id_nil,id_isPal => true
      | id_nil,id_mark => true
      | id_nil,id_u => true
      | id_nil,id_isNeList => true
      | id_nil,id_a__isList => true
      | id_nil,id_a => true
      | id_nil,id___ => true
      | id_nil,id_o => true
      | id_nil,id_a__isQid => true
      | id_nil,id_tt => true
      | id_nil,id_isNePal => true
      | id_nil,id_a__isPal => true
      | id_nil,id_nil => true
      | id_nil,id_and => false
      | id_nil,id_a__isNePal => false
      | id_nil,id_a__isNeList => false
      | id_nil,id_e => false
      | id_and,id_a____ => true
      | id_and,id_i => true
      | id_and,id_isList => true
      | id_and,id_a__and => true
      | id_and,id_isQid => true
      | id_and,id_isPal => true
      | id_and,id_mark => true
      | id_and,id_u => true
      | id_and,id_isNeList => true
      | id_and,id_a__isList => true
      | id_and,id_a => true
      | id_and,id___ => true
      | id_and,id_o => true
      | id_and,id_a__isQid => true
      | id_and,id_tt => true
      | id_and,id_isNePal => true
      | id_and,id_a__isPal => true
      | id_and,id_nil => true
      | id_and,id_and => true
      | id_and,id_a__isNePal => false
      | id_and,id_a__isNeList => false
      | id_and,id_e => false
      | id_a__isNePal,id_a____ => true
      | id_a__isNePal,id_i => true
      | id_a__isNePal,id_isList => true
      | id_a__isNePal,id_a__and => true
      | id_a__isNePal,id_isQid => true
      | id_a__isNePal,id_isPal => true
      | id_a__isNePal,id_mark => true
      | id_a__isNePal,id_u => true
      | id_a__isNePal,id_isNeList => true
      | id_a__isNePal,id_a__isList => true
      | id_a__isNePal,id_a => true
      | id_a__isNePal,id___ => true
      | id_a__isNePal,id_o => true
      | id_a__isNePal,id_a__isQid => true
      | id_a__isNePal,id_tt => true
      | id_a__isNePal,id_isNePal => true
      | id_a__isNePal,id_a__isPal => true
      | id_a__isNePal,id_nil => true
      | id_a__isNePal,id_and => true
      | id_a__isNePal,id_a__isNePal => true
      | id_a__isNePal,id_a__isNeList => false
      | id_a__isNePal,id_e => false
      | id_a__isNeList,id_a____ => true
      | id_a__isNeList,id_i => true
      | id_a__isNeList,id_isList => true
      | id_a__isNeList,id_a__and => true
      | id_a__isNeList,id_isQid => true
      | id_a__isNeList,id_isPal => true
      | id_a__isNeList,id_mark => true
      | id_a__isNeList,id_u => true
      | id_a__isNeList,id_isNeList => true
      | id_a__isNeList,id_a__isList => true
      | id_a__isNeList,id_a => true
      | id_a__isNeList,id___ => true
      | id_a__isNeList,id_o => true
      | id_a__isNeList,id_a__isQid => true
      | id_a__isNeList,id_tt => true
      | id_a__isNeList,id_isNePal => true
      | id_a__isNeList,id_a__isPal => true
      | id_a__isNeList,id_nil => true
      | id_a__isNeList,id_and => true
      | id_a__isNeList,id_a__isNePal => true
      | id_a__isNeList,id_a__isNeList => true
      | id_a__isNeList,id_e => false
      | id_e,id_a____ => true
      | id_e,id_i => true
      | id_e,id_isList => true
      | id_e,id_a__and => true
      | id_e,id_isQid => true
      | id_e,id_isPal => true
      | id_e,id_mark => true
      | id_e,id_u => true
      | id_e,id_isNeList => true
      | id_e,id_a__isList => true
      | id_e,id_a => true
      | id_e,id___ => true
      | id_e,id_o => true
      | id_e,id_a__isQid => true
      | id_e,id_tt => true
      | id_e,id_isNePal => true
      | id_e,id_a__isPal => true
      | id_e,id_nil => true
      | id_e,id_and => true
      | id_e,id_a__isNePal => true
      | id_e,id_a__isNeList => true
      | id_e,id_e => true
      end.
  
  
  Module Symb.
   Definition A  := symb.
   
   Definition eq_A  := @eq A.
   
   
   Definition eq_proof : equivalence A eq_A.
   Proof.
     constructor.
     red ;reflexivity .
     red ;intros ;transitivity y ;assumption.
     red ;intros ;symmetry ;assumption.
   Defined.
   
   
   Add Relation A eq_A 
  reflexivity proved by (@equiv_refl _ _ eq_proof)
    symmetry proved by (@equiv_sym _ _ eq_proof)
      transitivity proved by (@equiv_trans _ _ eq_proof) as EQA
.
   
   Definition eq_bool  := symb_eq_bool.
   
   Definition eq_bool_ok  := symb_eq_bool_ok.
  End Symb.
  
  Export Symb.
 End F.
 
 Module Alg := term.Make'(F)(term_extension.IntVars).
 
 Module Alg_ext := term_extension.Make(Alg).
 
 Module EQT := equational_theory.Make(Alg).
 
 Module EQT_ext := equational_extension.Make(EQT).
End algebra.

Module R_xml_0_deep_rew.
 Inductive R_xml_0_rules  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* a____(__(X_,Y_),Z_) -> a____(mark(X_),a____(mark(Y_),mark(Z_))) *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 2)::nil))::
                   (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 3)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
      (algebra.Alg.Var 3)::nil))
    (* a____(X_,nil) -> mark(X_) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id_nil nil)::nil))
    (* a____(nil,X_) -> mark(X_) *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
      algebra.F.id_nil nil)::(algebra.Alg.Var 1)::nil))
    (* a__and(tt,X_) -> mark(X_) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
      algebra.F.id_tt nil)::(algebra.Alg.Var 1)::nil))
    (* a__isList(V_) -> a__isNeList(V_) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNeList 
                   ((algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 4)::nil))
    (* a__isList(nil) -> tt *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))
   
    (* a__isList(__(V1_,V2_)) -> a__and(a__isList(V1_),isList(V2_)) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil))::
                   (algebra.Alg.Term algebra.F.id_isList 
                   ((algebra.Alg.Var 6)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term 
      algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil))
    (* a__isNeList(V_) -> a__isQid(V_) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isQid 
                   ((algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil))
   
    (* a__isNeList(__(V1_,V2_)) -> a__and(a__isList(V1_),isNeList(V2_)) *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil))::
                   (algebra.Alg.Term algebra.F.id_isNeList 
                   ((algebra.Alg.Var 6)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term 
      algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil))
   
    (* a__isNeList(__(V1_,V2_)) -> a__and(a__isNeList(V1_),isList(V2_)) *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isNeList ((algebra.Alg.Var 5)::nil))::
                   (algebra.Alg.Term algebra.F.id_isList 
                   ((algebra.Alg.Var 6)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term 
      algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil))
    (* a__isNePal(V_) -> a__isQid(V_) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isQid 
                   ((algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 4)::nil))
   
    (* a__isNePal(__(I_,__(P_,I_))) -> a__and(a__isQid(I_),isPal(P_)) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_a__isQid ((algebra.Alg.Var 7)::nil))::
                   (algebra.Alg.Term algebra.F.id_isPal 
                   ((algebra.Alg.Var 8)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term 
      algebra.F.id___ ((algebra.Alg.Var 7)::(algebra.Alg.Term 
      algebra.F.id___ ((algebra.Alg.Var 8)::
      (algebra.Alg.Var 7)::nil))::nil))::nil))
    (* a__isPal(V_) -> a__isNePal(V_) *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNePal 
                   ((algebra.Alg.Var 4)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 4)::nil))
    (* a__isPal(nil) -> tt *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))
    (* a__isQid(a) -> tt *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
      algebra.F.id_a nil)::nil))
    (* a__isQid(e) -> tt *)
   | R_xml_0_rule_15 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
      algebra.F.id_e nil)::nil))
    (* a__isQid(i) -> tt *)
   | R_xml_0_rule_16 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
      algebra.F.id_i nil)::nil))
    (* a__isQid(o) -> tt *)
   | R_xml_0_rule_17 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
      algebra.F.id_o nil)::nil))
    (* a__isQid(u) -> tt *)
   | R_xml_0_rule_18 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
      algebra.F.id_u nil)::nil))
    (* mark(__(X1_,X2_)) -> a____(mark(X1_),mark(X2_)) *)
   | R_xml_0_rule_19 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 10)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id___ 
      ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil))
    (* mark(and(X1_,X2_)) -> a__and(mark(X1_),X2_) *)
   | R_xml_0_rule_20 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::
                   (algebra.Alg.Var 10)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_and 
      ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil))
    (* mark(isList(X_)) -> a__isList(X_) *)
   | R_xml_0_rule_21 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isList 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isList ((algebra.Alg.Var 1)::nil))::nil))
    (* mark(isNeList(X_)) -> a__isNeList(X_) *)
   | R_xml_0_rule_22 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNeList 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNeList ((algebra.Alg.Var 1)::nil))::nil))
    (* mark(isQid(X_)) -> a__isQid(X_) *)
   | R_xml_0_rule_23 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isQid 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isQid ((algebra.Alg.Var 1)::nil))::nil))
    (* mark(isNePal(X_)) -> a__isNePal(X_) *)
   | R_xml_0_rule_24 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isNePal 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))::nil))
    (* mark(isPal(X_)) -> a__isPal(X_) *)
   | R_xml_0_rule_25 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__isPal 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
      algebra.F.id_isPal ((algebra.Alg.Var 1)::nil))::nil))
    (* mark(nil) -> nil *)
   | R_xml_0_rule_26 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_nil nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil 
      nil)::nil))
    (* mark(tt) -> tt *)
   | R_xml_0_rule_27 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt 
      nil)::nil))
    (* mark(a) -> a *)
   | R_xml_0_rule_28 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_a nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_a 
      nil)::nil))
    (* mark(e) -> e *)
   | R_xml_0_rule_29 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_e nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_e 
      nil)::nil))
    (* mark(i) -> i *)
   | R_xml_0_rule_30 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_i nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_i 
      nil)::nil))
    (* mark(o) -> o *)
   | R_xml_0_rule_31 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_o nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_o 
      nil)::nil))
    (* mark(u) -> u *)
   | R_xml_0_rule_32 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_u nil) 
     (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_u 
      nil)::nil))
    (* a____(X1_,X2_) -> __(X1_,X2_) *)
   | R_xml_0_rule_33 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 9)::
                   (algebra.Alg.Var 10)::nil)) 
     (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 9)::
      (algebra.Alg.Var 10)::nil))
    (* a__and(X1_,X2_) -> and(X1_,X2_) *)
   | R_xml_0_rule_34 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 9)::
                   (algebra.Alg.Var 10)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Var 9)::
      (algebra.Alg.Var 10)::nil))
    (* a__isList(X_) -> isList(X_) *)
   | R_xml_0_rule_35 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isList 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 1)::nil))
    (* a__isNeList(X_) -> isNeList(X_) *)
   | R_xml_0_rule_36 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNeList 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 1)::nil))
    (* a__isQid(X_) -> isQid(X_) *)
   | R_xml_0_rule_37 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isQid 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 1)::nil))
    (* a__isNePal(X_) -> isNePal(X_) *)
   | R_xml_0_rule_38 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNePal 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil))
    (* a__isPal(X_) -> isPal(X_) *)
   | R_xml_0_rule_39 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isPal 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
     algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 3)::nil))::nil))::nil)))::nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_nil nil)::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_nil 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil)))::
    R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term 
     algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Term 
     algebra.F.id_isList ((algebra.Alg.Var 6)::nil))::nil)))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 4)::nil)))::
    R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term 
     algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Term 
     algebra.F.id_isNeList ((algebra.Alg.Var 6)::nil))::nil)))::
    R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term 
     algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_a__isNeList ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Term 
     algebra.F.id_isList ((algebra.Alg.Var 6)::nil))::nil)))::
    R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 4)::nil)))::
    R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term 
     algebra.F.id___ ((algebra.Alg.Var 7)::(algebra.Alg.Term algebra.F.id___ 
     ((algebra.Alg.Var 8)::(algebra.Alg.Var 7)::nil))::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_a__isQid ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Term 
     algebra.F.id_isPal ((algebra.Alg.Var 8)::nil))::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 4)::nil)))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Term 
     algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
     algebra.F.id_a nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
     algebra.F.id_e nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
     algebra.F.id_i nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_15.
 
 
 Definition R_xml_0_rule_as_list_17  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
     algebra.F.id_o nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_16.
 
 
 Definition R_xml_0_rule_as_list_18  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term 
     algebra.F.id_u nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_17.
 
 
 Definition R_xml_0_rule_as_list_19  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id___ 
     ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 10)::nil))::nil)))::
    R_xml_0_rule_as_list_18.
 
 
 Definition R_xml_0_rule_as_list_20  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_and 
     ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
     algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::
     (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_19.
 
 
 Definition R_xml_0_rule_as_list_21  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isList ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_20.
 
 
 Definition R_xml_0_rule_as_list_22  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNeList ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_21.
 
 
 Definition R_xml_0_rule_as_list_23  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isQid ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_22.
 
 
 Definition R_xml_0_rule_as_list_24  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_23.
 
 
 Definition R_xml_0_rule_as_list_25  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
     algebra.F.id_isPal ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_24.
 
 
 Definition R_xml_0_rule_as_list_26  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_nil nil))::
    R_xml_0_rule_as_list_25.
 
 
 Definition R_xml_0_rule_as_list_27  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_26.
 
 
 Definition R_xml_0_rule_as_list_28  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_a 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_a nil))::
    R_xml_0_rule_as_list_27.
 
 
 Definition R_xml_0_rule_as_list_29  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_e 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_e nil))::
    R_xml_0_rule_as_list_28.
 
 
 Definition R_xml_0_rule_as_list_30  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_i 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_i nil))::
    R_xml_0_rule_as_list_29.
 
 
 Definition R_xml_0_rule_as_list_31  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_o 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_o nil))::
    R_xml_0_rule_as_list_30.
 
 
 Definition R_xml_0_rule_as_list_32  := 
   ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_u 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_u nil))::
    R_xml_0_rule_as_list_31.
 
 
 Definition R_xml_0_rule_as_list_33  := 
   ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 9)::
     (algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 9)::
     (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_32.
 
 
 Definition R_xml_0_rule_as_list_34  := 
   ((algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Var 9)::
     (algebra.Alg.Var 10)::nil)),
    (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 9)::
     (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_33.
 
 
 Definition R_xml_0_rule_as_list_35  := 
   ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_isList ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_34.
 
 
 Definition R_xml_0_rule_as_list_36  := 
   ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_isNeList ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_35.
 
 
 Definition R_xml_0_rule_as_list_37  := 
   ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_isQid ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_36.
 
 
 Definition R_xml_0_rule_as_list_38  := 
   ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_37.
 
 
 Definition R_xml_0_rule_as_list_39  := 
   ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_isPal ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_38.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_39.
 
 
 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 40.
   injection H;intros ;subst;constructor 39.
   injection H;intros ;subst;constructor 38.
   injection H;intros ;subst;constructor 37.
   injection H;intros ;subst;constructor 36.
   injection H;intros ;subst;constructor 35.
   injection H;intros ;subst;constructor 34.
   injection H;intros ;subst;constructor 33.
   injection H;intros ;subst;constructor 32.
   injection H;intros ;subst;constructor 31.
   injection H;intros ;subst;constructor 30.
   injection H;intros ;subst;constructor 29.
   injection H;intros ;subst;constructor 28.
   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.
   rewrite  <- or_ext_generated.or17_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 16.
   injection H;intros ;subst;constructor 15.
   injection H;intros ;subst;constructor 14.
   injection H;intros ;subst;constructor 13.
   injection H;intros ;subst;constructor 12.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   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_14 (x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 
 x25:Prop) :
  Prop := 
   | conj_14 :
    x12->x13->x14->x15->x16->x17->x18->x19->x20->x21->x22->x23->x24->
     x25->and_14 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25
 .
 
 
 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_14 (t = (algebra.Alg.Term algebra.F.id_i nil) ->
            t' = (algebra.Alg.Term algebra.F.id_i nil)) 
     (forall x13, 
      t = (algebra.Alg.Term algebra.F.id_isList (x13::nil)) ->
       exists x12,
         t' = (algebra.Alg.Term algebra.F.id_isList (x12::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x12 x13)) 
     (forall x13, 
      t = (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) ->
       exists x12,
         t' = (algebra.Alg.Term algebra.F.id_isQid (x12::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x12 x13)) 
     (forall x13, 
      t = (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) ->
       exists x12,
         t' = (algebra.Alg.Term algebra.F.id_isPal (x12::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x12 x13)) 
     (t = (algebra.Alg.Term algebra.F.id_u nil) ->
      t' = (algebra.Alg.Term algebra.F.id_u nil)) 
     (forall x13, 
      t = (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) ->
       exists x12,
         t' = (algebra.Alg.Term algebra.F.id_isNeList (x12::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x12 x13)) 
     (t = (algebra.Alg.Term algebra.F.id_a nil) ->
      t' = (algebra.Alg.Term algebra.F.id_a nil)) 
     (forall x13 x15, 
      t = (algebra.Alg.Term algebra.F.id___ (x13::x15::nil)) ->
       exists x12,
         exists x14,
           t' = (algebra.Alg.Term algebra.F.id___ (x12::x14::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x12 x13)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x14 x15)) 
     (t = (algebra.Alg.Term algebra.F.id_o nil) ->
      t' = (algebra.Alg.Term algebra.F.id_o nil)) 
     (t = (algebra.Alg.Term algebra.F.id_tt nil) ->
      t' = (algebra.Alg.Term algebra.F.id_tt nil)) 
     (forall x13, 
      t = (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) ->
       exists x12,
         t' = (algebra.Alg.Term algebra.F.id_isNePal (x12::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
           x12 x13)) 
     (t = (algebra.Alg.Term algebra.F.id_nil nil) ->
      t' = (algebra.Alg.Term algebra.F.id_nil nil)) 
     (forall x13 x15, 
      t = (algebra.Alg.Term algebra.F.id_and (x13::x15::nil)) ->
       exists x12,
         exists x14,
           t' = (algebra.Alg.Term algebra.F.id_and (x12::x14::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x12 x13)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x14 x15)) 
     (t = (algebra.Alg.Term algebra.F.id_e nil) ->
      t' = (algebra.Alg.Term algebra.F.id_e nil)).
 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 x13 H;exists x13;intuition;constructor 1.
   intros x13 H;exists x13;intuition;constructor 1.
   intros x13 H;exists x13;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x13 H;exists x13;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x13 x15 H;exists x13;exists x15;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x13 H;exists x13;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x13 x15 H;exists x13;exists x15;intuition;constructor 1.
   intros H;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList H_id_a 
    H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e].
   constructor.
   
   clear H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList H_id_a 
   H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;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_i H_id_isQid H_id_isPal H_id_u H_id_isNeList H_id_a H_id___ 
   H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;intros x13 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 x13 |- _ =>
      destruct (H_id_isList y (refl_equal _)) as [x12];intros ;intuition;
       exists x12;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isPal H_id_u H_id_isNeList H_id_a H_id___ 
   H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;intros x13 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 x13 |- _ =>
      destruct (H_id_isQid y (refl_equal _)) as [x12];intros ;intuition;
       exists x12;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_u H_id_isNeList H_id_a H_id___ 
   H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;intros x13 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 x13 |- _ =>
      destruct (H_id_isPal y (refl_equal _)) as [x12];intros ;intuition;
       exists x12;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_isPal H_id_isNeList H_id_a 
   H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_a H_id___ 
   H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;intros x13 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 x13 |- _ =>
      destruct (H_id_isNeList y (refl_equal _)) as [x12];intros ;intuition;
       exists x12;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;
    intros x13 x15 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 x13 |- _ =>
      destruct (H_id___ y x15 (refl_equal _)) as [x12 [x14]];intros ;
       intuition;exists x12;exists x14;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x15 |- _ =>
      destruct (H_id___ x13 y (refl_equal _)) as [x12 [x14]];intros ;
       intuition;exists x12;exists x14;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_tt H_id_isNePal H_id_nil H_id_and H_id_e;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_o H_id_isNePal H_id_nil H_id_and H_id_e;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_o H_id_tt H_id_nil H_id_and H_id_e;intros x13 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 x13 |- _ =>
      destruct (H_id_isNePal y (refl_equal _)) as [x12];intros ;intuition;
       exists x12;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_o H_id_tt H_id_isNePal H_id_and H_id_e;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_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_e;
    intros x13 x15 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 x13 |- _ =>
      destruct (H_id_and y x15 (refl_equal _)) as [x12 [x14]];intros ;
       intuition;exists x12;exists x14;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x15 |- _ =>
      destruct (H_id_and x13 y (refl_equal _)) as [x12 [x14]];intros ;
       intuition;exists x12;exists x14;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_i H_id_isList H_id_isQid H_id_isPal H_id_u H_id_isNeList 
   H_id_a H_id___ H_id_o H_id_tt H_id_isNePal H_id_nil H_id_and;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
    ).
 Qed.
 
 
 Lemma id_i_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_i nil) ->
     t' = (algebra.Alg.Term algebra.F.id_i nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isList_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13, 
     t = (algebra.Alg.Term algebra.F.id_isList (x13::nil)) ->
      exists x12,
        t' = (algebra.Alg.Term algebra.F.id_isList (x12::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x12 x13).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isQid_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13, 
     t = (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) ->
      exists x12,
        t' = (algebra.Alg.Term algebra.F.id_isQid (x12::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x12 x13).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isPal_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13, 
     t = (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) ->
      exists x12,
        t' = (algebra.Alg.Term algebra.F.id_isPal (x12::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x12 x13).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_u_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_u nil) ->
     t' = (algebra.Alg.Term algebra.F.id_u nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNeList_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13, 
     t = (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) ->
      exists x12,
        t' = (algebra.Alg.Term algebra.F.id_isNeList (x12::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x12 x13).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_a_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_a nil) ->
     t' = (algebra.Alg.Term algebra.F.id_a nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id____is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13 x15, 
     t = (algebra.Alg.Term algebra.F.id___ (x13::x15::nil)) ->
      exists x12,
        exists x14,
          t' = (algebra.Alg.Term algebra.F.id___ (x12::x14::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x12 x13)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x14 x15).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_o_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_o nil) ->
     t' = (algebra.Alg.Term algebra.F.id_o nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_tt_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_tt nil) ->
     t' = (algebra.Alg.Term algebra.F.id_tt nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_isNePal_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13, 
     t = (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) ->
      exists x12,
        t' = (algebra.Alg.Term algebra.F.id_isNePal (x12::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x12 x13).
 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_and_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x13 x15, 
     t = (algebra.Alg.Term algebra.F.id_and (x13::x15::nil)) ->
      exists x12,
        exists x14,
          t' = (algebra.Alg.Term algebra.F.id_and (x12::x14::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x12 x13)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x14 x15).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_e_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_e nil) ->
     t' = (algebra.Alg.Term algebra.F.id_e nil).
 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_i nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_i_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_isList (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isList_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_isQid (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isQid_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_isPal (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isPal_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_u nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_u_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_isNeList (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNeList_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_a nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_a_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___ (?x13::?x12::nil)) |- _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id____is_R_xml_0_constructor H (refl_equal _)) as 
               [x13 [x12 [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_o nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_o_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_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNePal (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_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_and (?x13::?x12::nil)) |- _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_and_is_R_xml_0_constructor H (refl_equal _)) as 
               [x13 [x12 [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_e nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_e_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 ))
    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_i nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_i_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_isList (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isList_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_isQid (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isQid_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_isPal (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isPal_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_u nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_u_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_isNeList (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNeList_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_a nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_a_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___ (?x13::?x12::nil)) |- _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id____is_R_xml_0_constructor H (refl_equal _)) as 
               [x13 [x12 [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_o nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_o_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_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_isNePal (?x12::nil)) |- _ =>
     let x12 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as 
           [x12 [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_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_and (?x13::?x12::nil)) |- _ =>
     let x13 := fresh "x" in 
      (let x12 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_and_is_R_xml_0_constructor H (refl_equal _)) as 
               [x13 [x12 [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_e nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_e_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 )))
    end
  .
End R_xml_0_deep_rew.

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

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

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

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

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

Module Interp.
 Section S.
   Require Import interp.
   
   Hypothesis A : Type.
   
   Hypothesis Ale Alt Aeq : A -> A -> Prop.
   
   Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
   
   Hypothesis A0 : A.
   
   Notation Local "a <= b" := (Ale a b).
   
   Hypothesis P_id_a____ : A ->A ->A.
   
   Hypothesis P_id_i : A.
   
   Hypothesis P_id_isList : A ->A.
   
   Hypothesis P_id_a__and : A ->A ->A.
   
   Hypothesis P_id_isQid : A ->A.
   
   Hypothesis P_id_isPal : A ->A.
   
   Hypothesis P_id_mark : A ->A.
   
   Hypothesis P_id_u : A.
   
   Hypothesis P_id_isNeList : A ->A.
   
   Hypothesis P_id_a__isList : A ->A.
   
   Hypothesis P_id_a : A.
   
   Hypothesis P_id___ : A ->A ->A.
   
   Hypothesis P_id_o : A.
   
   Hypothesis P_id_a__isQid : A ->A.
   
   Hypothesis P_id_tt : A.
   
   Hypothesis P_id_isNePal : A ->A.
   
   Hypothesis P_id_a__isPal : A ->A.
   
   Hypothesis P_id_nil : A.
   
   Hypothesis P_id_and : A ->A ->A.
   
   Hypothesis P_id_a__isNePal : A ->A.
   
   Hypothesis P_id_a__isNeList : A ->A.
   
   Hypothesis P_id_e : A.
   
   Hypothesis P_id_a_____monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id_a____ x13 x15 <= P_id_a____ x12 x14.
   
   Hypothesis P_id_isList_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
   
   Hypothesis P_id_a__and_monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id_a__and x13 x15 <= P_id_a__and x12 x14.
   
   Hypothesis P_id_isQid_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
   
   Hypothesis P_id_isPal_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
   
   Hypothesis P_id_mark_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
   
   Hypothesis P_id_isNeList_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_isNeList x13 <= P_id_isNeList x12.
   
   Hypothesis P_id_a__isList_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_a__isList x13 <= P_id_a__isList x12.
   
   Hypothesis P_id____monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
   
   Hypothesis P_id_a__isQid_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_a__isQid x13 <= P_id_a__isQid x12.
   
   Hypothesis P_id_isNePal_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
   
   Hypothesis P_id_a__isPal_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_a__isPal x13 <= P_id_a__isPal x12.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id_and x13 x15 <= P_id_and x12 x14.
   
   Hypothesis P_id_a__isNePal_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_a__isNePal x13 <= P_id_a__isNePal x12.
   
   Hypothesis P_id_a__isNeList_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->
      P_id_a__isNeList x13 <= P_id_a__isNeList x12.
   
   Hypothesis P_id_a_____bounded :
    forall x12 x13, (A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a____ x13 x12.
   
   Hypothesis P_id_i_bounded : A0 <= P_id_i .
   
   Hypothesis P_id_isList_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_isList x12.
   
   Hypothesis P_id_a__and_bounded :
    forall x12 x13, (A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_a__and x13 x12.
   
   Hypothesis P_id_isQid_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_isQid x12.
   
   Hypothesis P_id_isPal_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_isPal x12.
   
   Hypothesis P_id_mark_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_mark x12.
   
   Hypothesis P_id_u_bounded : A0 <= P_id_u .
   
   Hypothesis P_id_isNeList_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_isNeList x12.
   
   Hypothesis P_id_a__isList_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_a__isList x12.
   
   Hypothesis P_id_a_bounded : A0 <= P_id_a .
   
   Hypothesis P_id____bounded :
    forall x12 x13, (A0 <= x12) ->(A0 <= x13) ->A0 <= P_id___ x13 x12.
   
   Hypothesis P_id_o_bounded : A0 <= P_id_o .
   
   Hypothesis P_id_a__isQid_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_a__isQid x12.
   
   Hypothesis P_id_tt_bounded : A0 <= P_id_tt .
   
   Hypothesis P_id_isNePal_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_isNePal x12.
   
   Hypothesis P_id_a__isPal_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_a__isPal x12.
   
   Hypothesis P_id_nil_bounded : A0 <= P_id_nil .
   
   Hypothesis P_id_and_bounded :
    forall x12 x13, (A0 <= x12) ->(A0 <= x13) ->A0 <= P_id_and x13 x12.
   
   Hypothesis P_id_a__isNePal_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_a__isNePal x12.
   
   Hypothesis P_id_a__isNeList_bounded :
    forall x12, (A0 <= x12) ->A0 <= P_id_a__isNeList x12.
   
   Hypothesis P_id_e_bounded : A0 <= P_id_e .
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
        P_id_a____ (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
       | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
        P_id_isList (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
        P_id_a__and (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
        P_id_isQid (measure x12)
       | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
        P_id_isPal (measure x12)
       | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
        P_id_mark (measure x12)
       | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
       | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
        P_id_isNeList (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
        P_id_a__isList (measure x12)
       | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
       | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
        P_id___ (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
       | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
        P_id_a__isQid (measure x12)
       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
       | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
        P_id_isNePal (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
        P_id_a__isPal (measure x12)
       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
       | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
        P_id_and (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
        P_id_a__isNePal (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
        P_id_a__isNeList (measure x12)
       | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
                    P_id_a____ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                   | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                    P_id_isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
                    P_id_a__and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                    P_id_isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                    P_id_isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                    P_id_mark (measure x12)
                   | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                   | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                    P_id_isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                    P_id_a__isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                   | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                    P_id___ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                   | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                    P_id_a__isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                    P_id_isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                    P_id_a__isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                    P_id_and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
                    P_id_a__isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
                    P_id_a__isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                   | _ => A0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Definition Pols f : InterpGen.Pol_type A (InterpGen.get_arity f) := 
     match f with
       | algebra.F.id_a____ => P_id_a____
       | algebra.F.id_i => P_id_i
       | algebra.F.id_isList => P_id_isList
       | algebra.F.id_a__and => P_id_a__and
       | algebra.F.id_isQid => P_id_isQid
       | algebra.F.id_isPal => P_id_isPal
       | algebra.F.id_mark => P_id_mark
       | algebra.F.id_u => P_id_u
       | algebra.F.id_isNeList => P_id_isNeList
       | algebra.F.id_a__isList => P_id_a__isList
       | algebra.F.id_a => P_id_a
       | algebra.F.id___ => P_id___
       | algebra.F.id_o => P_id_o
       | algebra.F.id_a__isQid => P_id_a__isQid
       | algebra.F.id_tt => P_id_tt
       | algebra.F.id_isNePal => P_id_isNePal
       | algebra.F.id_a__isPal => P_id_a__isPal
       | algebra.F.id_nil => P_id_nil
       | algebra.F.id_and => P_id_and
       | algebra.F.id_a__isNePal => P_id_a__isNePal
       | algebra.F.id_a__isNeList => P_id_a__isNeList
       | algebra.F.id_e => P_id_e
       end.
   
   Lemma same_measure : forall t, measure t = InterpGen.measure A0 Pols t.
   Proof.
     fix 1 .
     intros [a| f l].
     simpl in |-*.
     unfold eq_rect_r, eq_rect, sym_eq in |-*.
     reflexivity .
     
     refine match f with
              | algebra.F.id_a____ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_i => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_isList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isQid =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isPal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_isNeList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id___ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_o => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_a__isQid =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_isNePal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isPal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNePal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNeList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_e => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_i_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_o_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_e_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_a_____monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isQid_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isPal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNeList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isList_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id____monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_a__isQid_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isPal_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNeList_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_i_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_o_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_e_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_A__AND : A ->A ->A.
   
   Hypothesis P_id_A__ISNEPAL : A ->A.
   
   Hypothesis P_id_A__ISNELIST : A ->A.
   
   Hypothesis P_id_A____ : A ->A ->A.
   
   Hypothesis P_id_A__ISLIST : A ->A.
   
   Hypothesis P_id_A__ISPAL : A ->A.
   
   Hypothesis P_id_A__ISQID : A ->A.
   
   Hypothesis P_id_MARK : A ->A.
   
   Hypothesis P_id_A__AND_monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
   
   Hypothesis P_id_A__ISNEPAL_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
   
   Hypothesis P_id_A__ISNELIST_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->
      P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
   
   Hypothesis P_id_A_____monotonic :
    forall x12 x14 x13 x15, 
     (A0 <= x15)/\ (x15 <= x14) ->
      (A0 <= x13)/\ (x13 <= x12) ->P_id_A____ x13 x15 <= P_id_A____ x12 x14.
   
   Hypothesis P_id_A__ISLIST_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
   
   Hypothesis P_id_A__ISPAL_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
   
   Hypothesis P_id_A__ISQID_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_A__ISQID x13 <= P_id_A__ISQID x12.
   
   Hypothesis P_id_MARK_monotonic :
    forall x12 x13, 
     (A0 <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
        P_id_A__AND (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
        P_id_A__ISNEPAL (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
        P_id_A__ISNELIST (measure x12)
       | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
        P_id_A____ (measure x13) (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
        P_id_A__ISLIST (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
        P_id_A__ISPAL (measure x12)
       | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
        P_id_A__ISQID (measure x12)
       | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
        P_id_MARK (measure x12)
       | _ => 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 x12;apply (P_id_A__ISNELIST x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12;apply (P_id_A__ISNEPAL x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12;apply (P_id_A__ISPAL x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12;apply (P_id_A__ISQID x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12;apply (P_id_A__ISLIST x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x12;apply (P_id_MARK x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12;apply (P_id_A__AND x13 x12).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x13 x12;apply (P_id_A____ x13 x12).
     discriminate H.
   Defined.
   
   Lemma same_marked_measure :
    forall t, 
     marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols 
                         (ddp.defined_dec _ _ 
                           R_xml_0_deep_rew.R_xml_0_rules_included) t.
   Proof.
     intros [a| f l].
     simpl in |-*.
     unfold eq_rect_r, eq_rect, sym_eq in |-*.
     reflexivity .
     
     refine match f with
              | algebra.F.id_a____ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_i => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_isList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_isQid =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isPal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_u => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_isNeList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id___ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_o => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_a__isQid =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_isNePal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isPal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_a__isNePal =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_a__isNeList =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_e => match l with
                                    | 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 .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     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 .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     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 .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__and (x13::
                             x12::nil)) =>
                           P_id_A__AND (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNePal 
                             (x12::nil)) =>
                           P_id_A__ISNEPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNeList 
                             (x12::nil)) =>
                           P_id_A__ISNELIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a____ (x13::
                             x12::nil)) =>
                           P_id_A____ (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isList 
                             (x12::nil)) =>
                           P_id_A__ISLIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isPal 
                             (x12::nil)) =>
                           P_id_A__ISPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x12::nil)) =>
                           P_id_A__ISQID (measure x12)
                          | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                           P_id_MARK (measure x12)
                          | _ => 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_a_____monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isQid_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isPal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNeList_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isList_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id____monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_a__isQid_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isPal_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNeList_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_i_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_u_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id____bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_o_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_e_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     clear f.
     intros f.
     clear H.
     intros H.
     generalize H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNELIST_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISPAL_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISQID_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__ISLIST_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_MARK_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A__AND_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_A_____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_a____ : Z ->Z ->Z.
   
   Hypothesis P_id_i : Z.
   
   Hypothesis P_id_isList : Z ->Z.
   
   Hypothesis P_id_a__and : Z ->Z ->Z.
   
   Hypothesis P_id_isQid : Z ->Z.
   
   Hypothesis P_id_isPal : Z ->Z.
   
   Hypothesis P_id_mark : Z ->Z.
   
   Hypothesis P_id_u : Z.
   
   Hypothesis P_id_isNeList : Z ->Z.
   
   Hypothesis P_id_a__isList : Z ->Z.
   
   Hypothesis P_id_a : Z.
   
   Hypothesis P_id___ : Z ->Z ->Z.
   
   Hypothesis P_id_o : Z.
   
   Hypothesis P_id_a__isQid : Z ->Z.
   
   Hypothesis P_id_tt : Z.
   
   Hypothesis P_id_isNePal : Z ->Z.
   
   Hypothesis P_id_a__isPal : Z ->Z.
   
   Hypothesis P_id_nil : Z.
   
   Hypothesis P_id_and : Z ->Z ->Z.
   
   Hypothesis P_id_a__isNePal : Z ->Z.
   
   Hypothesis P_id_a__isNeList : Z ->Z.
   
   Hypothesis P_id_e : Z.
   
   Hypothesis P_id_a_____monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->
       P_id_a____ x13 x15 <= P_id_a____ x12 x14.
   
   Hypothesis P_id_isList_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
   
   Hypothesis P_id_a__and_monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->
       P_id_a__and x13 x15 <= P_id_a__and x12 x14.
   
   Hypothesis P_id_isQid_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
   
   Hypothesis P_id_isPal_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
   
   Hypothesis P_id_mark_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
   
   Hypothesis P_id_isNeList_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_isNeList x13 <= P_id_isNeList x12.
   
   Hypothesis P_id_a__isList_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_a__isList x13 <= P_id_a__isList x12.
   
   Hypothesis P_id____monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
   
   Hypothesis P_id_a__isQid_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_a__isQid x13 <= P_id_a__isQid x12.
   
   Hypothesis P_id_isNePal_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
   
   Hypothesis P_id_a__isPal_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_a__isPal x13 <= P_id_a__isPal x12.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->
       P_id_and x13 x15 <= P_id_and x12 x14.
   
   Hypothesis P_id_a__isNePal_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_a__isNePal x13 <= P_id_a__isNePal x12.
   
   Hypothesis P_id_a__isNeList_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_a__isNeList x13 <= P_id_a__isNeList x12.
   
   Hypothesis P_id_a_____bounded :
    forall x12 x13, 
     (min_value <= x12) ->
      (min_value <= x13) ->min_value <= P_id_a____ x13 x12.
   
   Hypothesis P_id_i_bounded : min_value <= P_id_i .
   
   Hypothesis P_id_isList_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_isList x12.
   
   Hypothesis P_id_a__and_bounded :
    forall x12 x13, 
     (min_value <= x12) ->
      (min_value <= x13) ->min_value <= P_id_a__and x13 x12.
   
   Hypothesis P_id_isQid_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_isQid x12.
   
   Hypothesis P_id_isPal_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_isPal x12.
   
   Hypothesis P_id_mark_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_mark x12.
   
   Hypothesis P_id_u_bounded : min_value <= P_id_u .
   
   Hypothesis P_id_isNeList_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_isNeList x12.
   
   Hypothesis P_id_a__isList_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_a__isList x12.
   
   Hypothesis P_id_a_bounded : min_value <= P_id_a .
   
   Hypothesis P_id____bounded :
    forall x12 x13, 
     (min_value <= x12) ->(min_value <= x13) ->min_value <= P_id___ x13 x12.
   
   Hypothesis P_id_o_bounded : min_value <= P_id_o .
   
   Hypothesis P_id_a__isQid_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_a__isQid x12.
   
   Hypothesis P_id_tt_bounded : min_value <= P_id_tt .
   
   Hypothesis P_id_isNePal_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_isNePal x12.
   
   Hypothesis P_id_a__isPal_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_a__isPal x12.
   
   Hypothesis P_id_nil_bounded : min_value <= P_id_nil .
   
   Hypothesis P_id_and_bounded :
    forall x12 x13, 
     (min_value <= x12) ->(min_value <= x13) ->min_value <= P_id_and x13 x12.
   
   Hypothesis P_id_a__isNePal_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_a__isNePal x12.
   
   Hypothesis P_id_a__isNeList_bounded :
    forall x12, (min_value <= x12) ->min_value <= P_id_a__isNeList x12.
   
   Hypothesis P_id_e_bounded : min_value <= P_id_e .
   
   Definition measure  := 
     Interp.measure min_value P_id_a____ P_id_i P_id_isList P_id_a__and 
      P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
      P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal P_id_a__isPal 
      P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList P_id_e.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
                    P_id_a____ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                   | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                    P_id_isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
                    P_id_a__and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                    P_id_isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                    P_id_isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                    P_id_mark (measure x12)
                   | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                   | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                    P_id_isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                    P_id_a__isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                   | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                    P_id___ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                   | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                    P_id_a__isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                    P_id_isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                    P_id_a__isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                    P_id_and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
                    P_id_a__isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
                    P_id_a__isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                   | _ => 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_a_____monotonic;assumption.
     intros ;apply P_id_isList_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_isQid_monotonic;assumption.
     intros ;apply P_id_isPal_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_isNeList_monotonic;assumption.
     intros ;apply P_id_a__isList_monotonic;assumption.
     intros ;apply P_id____monotonic;assumption.
     intros ;apply P_id_a__isQid_monotonic;assumption.
     intros ;apply P_id_isNePal_monotonic;assumption.
     intros ;apply P_id_a__isPal_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_a__isNePal_monotonic;assumption.
     intros ;apply P_id_a__isNeList_monotonic;assumption.
     intros ;apply P_id_a_____bounded;assumption.
     intros ;apply P_id_i_bounded;assumption.
     intros ;apply P_id_isList_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_isQid_bounded;assumption.
     intros ;apply P_id_isPal_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_u_bounded;assumption.
     intros ;apply P_id_isNeList_bounded;assumption.
     intros ;apply P_id_a__isList_bounded;assumption.
     intros ;apply P_id_a_bounded;assumption.
     intros ;apply P_id____bounded;assumption.
     intros ;apply P_id_o_bounded;assumption.
     intros ;apply P_id_a__isQid_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_isNePal_bounded;assumption.
     intros ;apply P_id_a__isPal_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_a__isNePal_bounded;assumption.
     intros ;apply P_id_a__isNeList_bounded;assumption.
     intros ;apply P_id_e_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_A__AND : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISNEPAL : Z ->Z.
   
   Hypothesis P_id_A__ISNELIST : Z ->Z.
   
   Hypothesis P_id_A____ : Z ->Z ->Z.
   
   Hypothesis P_id_A__ISLIST : Z ->Z.
   
   Hypothesis P_id_A__ISPAL : Z ->Z.
   
   Hypothesis P_id_A__ISQID : Z ->Z.
   
   Hypothesis P_id_MARK : Z ->Z.
   
   Hypothesis P_id_A__AND_monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->
       P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
   
   Hypothesis P_id_A__ISNEPAL_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
   
   Hypothesis P_id_A__ISNELIST_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
   
   Hypothesis P_id_A_____monotonic :
    forall x12 x14 x13 x15, 
     (min_value <= x15)/\ (x15 <= x14) ->
      (min_value <= x13)/\ (x13 <= x12) ->
       P_id_A____ x13 x15 <= P_id_A____ x12 x14.
   
   Hypothesis P_id_A__ISLIST_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
   
   Hypothesis P_id_A__ISPAL_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
   
   Hypothesis P_id_A__ISQID_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->
      P_id_A__ISQID x13 <= P_id_A__ISQID x12.
   
   Hypothesis P_id_MARK_monotonic :
    forall x12 x13, 
     (min_value <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_a____ P_id_i P_id_isList 
      P_id_a__and P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList 
      P_id_a__isList P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt 
      P_id_isNePal P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal 
      P_id_a__isNeList P_id_e P_id_A__AND P_id_A__ISNEPAL P_id_A__ISNELIST 
      P_id_A____ P_id_A__ISLIST P_id_A__ISPAL P_id_A__ISQID P_id_MARK.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__and (x13::
                             x12::nil)) =>
                           P_id_A__AND (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNePal 
                             (x12::nil)) =>
                           P_id_A__ISNEPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNeList 
                             (x12::nil)) =>
                           P_id_A__ISNELIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a____ (x13::
                             x12::nil)) =>
                           P_id_A____ (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isList 
                             (x12::nil)) =>
                           P_id_A__ISLIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isPal 
                             (x12::nil)) =>
                           P_id_A__ISPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x12::nil)) =>
                           P_id_A__ISQID (measure x12)
                          | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                           P_id_MARK (measure x12)
                          | _ => 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_a_____monotonic;assumption.
     intros ;apply P_id_isList_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_isQid_monotonic;assumption.
     intros ;apply P_id_isPal_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_isNeList_monotonic;assumption.
     intros ;apply P_id_a__isList_monotonic;assumption.
     intros ;apply P_id____monotonic;assumption.
     intros ;apply P_id_a__isQid_monotonic;assumption.
     intros ;apply P_id_isNePal_monotonic;assumption.
     intros ;apply P_id_a__isPal_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_a__isNePal_monotonic;assumption.
     intros ;apply P_id_a__isNeList_monotonic;assumption.
     intros ;apply P_id_a_____bounded;assumption.
     intros ;apply P_id_i_bounded;assumption.
     intros ;apply P_id_isList_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_isQid_bounded;assumption.
     intros ;apply P_id_isPal_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_u_bounded;assumption.
     intros ;apply P_id_isNeList_bounded;assumption.
     intros ;apply P_id_a__isList_bounded;assumption.
     intros ;apply P_id_a_bounded;assumption.
     intros ;apply P_id____bounded;assumption.
     intros ;apply P_id_o_bounded;assumption.
     intros ;apply P_id_a__isQid_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_isNePal_bounded;assumption.
     intros ;apply P_id_a__isPal_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_a__isNePal_bounded;assumption.
     intros ;apply P_id_a__isNeList_bounded;assumption.
     intros ;apply P_id_e_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__AND_monotonic;assumption.
     intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
     intros ;apply P_id_A__ISNELIST_monotonic;assumption.
     intros ;apply P_id_A_____monotonic;assumption.
     intros ;apply P_id_A__ISLIST_monotonic;assumption.
     intros ;apply P_id_A__ISPAL_monotonic;assumption.
     intros ;apply P_id_A__ISQID_monotonic;assumption.
     intros ;apply P_id_MARK_monotonic;assumption.
   Qed.
   
   End S.
End InterpZ.

Module WF_R_xml_0_deep_rew.
 Inductive DP_R_xml_0  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* <a____(__(X_,Y_),Z_),a____(mark(X_),a____(mark(Y_),mark(Z_)))> *)
   | DP_R_xml_0_0 :
    forall x12 x2 x1 x13 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                   algebra.F.id_mark (x1::nil))::(algebra.Alg.Term 
                   algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark 
                   (x2::nil))::(algebra.Alg.Term algebra.F.id_mark 
                   (x3::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(__(X_,Y_),Z_),mark(X_)> *)
   | DP_R_xml_0_1 :
    forall x12 x2 x1 x13 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(__(X_,Y_),Z_),a____(mark(Y_),mark(Z_))> *)
   | DP_R_xml_0_2 :
    forall x12 x2 x1 x13 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                   algebra.F.id_mark (x2::nil))::(algebra.Alg.Term 
                   algebra.F.id_mark (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(__(X_,Y_),Z_),mark(Y_)> *)
   | DP_R_xml_0_3 :
    forall x12 x2 x1 x13 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(__(X_,Y_),Z_),mark(Z_)> *)
   | DP_R_xml_0_4 :
    forall x12 x2 x1 x13 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(X_,nil),mark(X_)> *)
   | DP_R_xml_0_5 :
    forall x12 x1 x13, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_nil nil) 
        x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a____(nil,X_),mark(X_)> *)
   | DP_R_xml_0_6 :
    forall x12 x1 x13, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_nil nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    (* <a__and(tt,X_),mark(X_)> *)
   | DP_R_xml_0_7 :
    forall x12 x1 x13, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x13) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x12) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil))
    (* <a__isList(V_),a__isNeList(V_)> *)
   | DP_R_xml_0_8 :
    forall x4 x12, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
   
    (* <a__isList(__(V1_,V2_)),a__and(a__isList(V1_),isList(V2_))> *)
   | DP_R_xml_0_9 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isList (x5::nil))::(algebra.Alg.Term 
                  algebra.F.id_isList (x6::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
    (* <a__isList(__(V1_,V2_)),a__isList(V1_)> *)
   | DP_R_xml_0_10 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
    (* <a__isNeList(V_),a__isQid(V_)> *)
   | DP_R_xml_0_11 :
    forall x4 x12, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
   
    (* <a__isNeList(__(V1_,V2_)),a__and(a__isList(V1_),isNeList(V2_))> *)
   | DP_R_xml_0_12 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isList (x5::nil))::(algebra.Alg.Term 
                  algebra.F.id_isNeList (x6::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
    (* <a__isNeList(__(V1_,V2_)),a__isList(V1_)> *)
   | DP_R_xml_0_13 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
   
    (* <a__isNeList(__(V1_,V2_)),a__and(a__isNeList(V1_),isList(V2_))> *)
   | DP_R_xml_0_14 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isNeList (x5::nil))::(algebra.Alg.Term 
                  algebra.F.id_isList (x6::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
    (* <a__isNeList(__(V1_,V2_)),a__isNeList(V1_)> *)
   | DP_R_xml_0_15 :
    forall x12 x6 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
    (* <a__isNePal(V_),a__isQid(V_)> *)
   | DP_R_xml_0_16 :
    forall x4 x12, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
   
    (* <a__isNePal(__(I_,__(P_,I_))),a__and(a__isQid(I_),isPal(P_))> *)
   | DP_R_xml_0_17 :
    forall x8 x12 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
        algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_a__isQid (x7::nil))::(algebra.Alg.Term 
                  algebra.F.id_isPal (x8::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
    (* <a__isNePal(__(I_,__(P_,I_))),a__isQid(I_)> *)
   | DP_R_xml_0_18 :
    forall x8 x12 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
        algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
    (* <a__isPal(V_),a__isNePal(V_)> *)
   | DP_R_xml_0_19 :
    forall x4 x12, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x4 x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
    (* <mark(__(X1_,X2_)),a____(mark(X1_),mark(X2_))> *)
   | DP_R_xml_0_20 :
    forall x12 x10 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term 
                  algebra.F.id_mark (x9::nil))::(algebra.Alg.Term 
                  algebra.F.id_mark (x10::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(__(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_21 :
    forall x12 x10 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(__(X1_,X2_)),mark(X2_)> *)
   | DP_R_xml_0_22 :
    forall x12 x10 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x10::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
   | DP_R_xml_0_23 :
    forall x12 x10 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__and ((algebra.Alg.Term 
                  algebra.F.id_mark (x9::nil))::x10::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(and(X1_,X2_)),mark(X1_)> *)
   | DP_R_xml_0_24 :
    forall x12 x10 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(isList(X_)),a__isList(X_)> *)
   | DP_R_xml_0_25 :
    forall x12 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(isNeList(X_)),a__isNeList(X_)> *)
   | DP_R_xml_0_26 :
    forall x12 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(isQid(X_)),a__isQid(X_)> *)
   | DP_R_xml_0_27 :
    forall x12 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isQid (x1::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(isNePal(X_)),a__isNePal(X_)> *)
   | DP_R_xml_0_28 :
    forall x12 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNePal (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    (* <mark(isPal(X_)),a__isPal(X_)> *)
   | DP_R_xml_0_29 :
    forall x12 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPal (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_mark (x12::nil))
 .
 
 Module ddp := dp.MakeDP(algebra.EQT).
 
 
 Lemma R_xml_0_dp_step_spec :
  forall x y, 
   (ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) ->
    exists f,
      exists l1,
        exists l2,
          y = algebra.Alg.Term f l2/\ 
          (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                            R_xml_0_deep_rew.R_xml_0_rules)
                                                           ) l1 l2)/\ 
          (ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)).
 Proof.
   intros x y H.
   induction H.
   inversion H.
   subst.
   destruct t0.
   refine ((False_ind) _ _).
   refine (R_xml_0_deep_rew.R_xml_0_non_var H0).
   simpl in H|-*.
   exists a.
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   repeat (constructor).
   assumption.
   exists f.
   exists l2.
   exists l1.
   constructor.
   constructor.
   constructor.
   constructor.
   rewrite  <- closure.rwr_list_trans_clos_one_step_list.
   assumption.
   assumption.
 Qed.
 
 
 Ltac included_dp_tac H :=
  injection H;clear H;intros;subst;
  repeat (match goal with 
  | H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=>           
  let x := fresh "x" in 
  let l := fresh "l" in 
  let h1 := fresh "h" in 
  let h2 := fresh "h" in 
  let h3 := fresh "h" in 
  destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _  H) as [x [l[h1[h2 h3]]]];clear H;subst
  | H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ => 
  rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H
  end
  );simpl;
  econstructor eassumption
 .
 
 
 Ltac dp_concl_tac h2 h cont_tac 
  t :=
  match t with
    | False => let h' := fresh "a" in 
                (set (h':=t) in *;cont_tac h';
                  repeat (
                  let e := type of h in 
                   (match e with
                      | ?t => unfold t in h|-;
                               (case h;
                                [abstract (clear h;intros h;injection h;
                                            clear h;intros ;subst;
                                            included_dp_tac h2)|
                                clear h;intros h;clear t])
                      | ?t => unfold t in h|-;elim h
                      end
                    )
                  ))
    | or ?a ?b => let cont_tac 
                   h' := let h'' := fresh "a" in 
                          (set (h'':=or a h') in *;cont_tac h'') in 
                   (dp_concl_tac h2 h cont_tac b)
    end
  .
 
 
 Module WF_DP_R_xml_0.
  Inductive DP_R_xml_0_non_scc_1  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <mark(isQid(X_)),a__isQid(X_)> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x12 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isQid (x1::nil)) x12) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::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 := 
     (* <a__isNePal(__(I_,__(P_,I_))),a__isQid(I_)> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x8 x12 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
         algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNePal (x12::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 := 
     (* <a__isNePal(V_),a__isQid(V_)> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x4 x12, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNePal (x12::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 := 
     (* <a__isNeList(V_),a__isQid(V_)> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall x4 x12, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNeList (x12::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_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <a____(__(X_,Y_),Z_),mark(X_)> *)
    | DP_R_xml_0_scc_5_0 :
     forall x12 x2 x1 x13 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    
     (* <mark(__(X1_,X2_)),a____(mark(X1_),mark(X2_))> *)
    | DP_R_xml_0_scc_5_1 :
     forall x12 x10 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a____ 
                         ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::
                         (algebra.Alg.Term algebra.F.id_mark 
                         (x10::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    
     (* <a____(__(X_,Y_),Z_),a____(mark(X_),a____(mark(Y_),mark(Z_)))> *)
    | DP_R_xml_0_scc_5_2 :
     forall x12 x2 x1 x13 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a____ 
                          ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::
                          (algebra.Alg.Term algebra.F.id_a____ 
                          ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::
                          (algebra.Alg.Term algebra.F.id_mark 
                          (x3::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    
     (* <a____(__(X_,Y_),Z_),a____(mark(Y_),mark(Z_))> *)
    | DP_R_xml_0_scc_5_3 :
     forall x12 x2 x1 x13 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a____ 
                          ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::
                          (algebra.Alg.Term algebra.F.id_mark 
                          (x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     (* <a____(__(X_,Y_),Z_),mark(Y_)> *)
    | DP_R_xml_0_scc_5_4 :
     forall x12 x2 x1 x13 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     (* <mark(__(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_5_5 :
     forall x12 x10 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <mark(__(X1_,X2_)),mark(X2_)> *)
    | DP_R_xml_0_scc_5_6 :
     forall x12 x10 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x10::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
    | DP_R_xml_0_scc_5_7 :
     forall x12 x10 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::
                         x10::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <a__and(tt,X_),mark(X_)> *)
    | DP_R_xml_0_scc_5_8 :
     forall x12 x1 x13, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil))
     (* <mark(and(X1_,X2_)),mark(X1_)> *)
    | DP_R_xml_0_scc_5_9 :
     forall x12 x10 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <mark(isList(X_)),a__isList(X_)> *)
    | DP_R_xml_0_scc_5_10 :
     forall x12 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isList (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <a__isList(V_),a__isNeList(V_)> *)
    | DP_R_xml_0_scc_5_11 :
     forall x4 x12, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
    
     (* <a__isNeList(__(V1_,V2_)),a__and(a__isList(V1_),isNeList(V2_))> *)
    | DP_R_xml_0_scc_5_12 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_a__isList 
                         (x5::nil))::(algebra.Alg.Term algebra.F.id_isNeList 
                         (x6::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     (* <a__isNeList(__(V1_,V2_)),a__isList(V1_)> *)
    | DP_R_xml_0_scc_5_13 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
    
     (* <a__isList(__(V1_,V2_)),a__and(a__isList(V1_),isList(V2_))> *)
    | DP_R_xml_0_scc_5_14 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_a__isList 
                         (x5::nil))::(algebra.Alg.Term algebra.F.id_isList 
                         (x6::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
     (* <a__isList(__(V1_,V2_)),a__isList(V1_)> *)
    | DP_R_xml_0_scc_5_15 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
    
     (* <a__isNeList(__(V1_,V2_)),a__and(a__isNeList(V1_),isList(V2_))> *)
    | DP_R_xml_0_scc_5_16 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_a__isNeList 
                         (x5::nil))::(algebra.Alg.Term algebra.F.id_isList 
                         (x6::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     (* <a__isNeList(__(V1_,V2_)),a__isNeList(V1_)> *)
    | DP_R_xml_0_scc_5_17 :
     forall x12 x6 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     (* <mark(isNeList(X_)),a__isNeList(X_)> *)
    | DP_R_xml_0_scc_5_18 :
     forall x12 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isNeList (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <mark(isNePal(X_)),a__isNePal(X_)> *)
    | DP_R_xml_0_scc_5_19 :
     forall x12 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isNePal (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    
     (* <a__isNePal(__(I_,__(P_,I_))),a__and(a__isQid(I_),isPal(P_))> *)
    | DP_R_xml_0_scc_5_20 :
     forall x8 x12 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
         algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__and 
                         ((algebra.Alg.Term algebra.F.id_a__isQid 
                         (x7::nil))::(algebra.Alg.Term algebra.F.id_isPal 
                         (x8::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
     (* <mark(isPal(X_)),a__isPal(X_)> *)
    | DP_R_xml_0_scc_5_21 :
     forall x12 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isPal (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     (* <a__isPal(V_),a__isNePal(V_)> *)
    | DP_R_xml_0_scc_5_22 :
     forall x4 x12, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x12) ->
       DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
     (* <a____(__(X_,Y_),Z_),mark(Z_)> *)
    | DP_R_xml_0_scc_5_23 :
     forall x12 x2 x1 x13 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     (* <a____(X_,nil),mark(X_)> *)
    | DP_R_xml_0_scc_5_24 :
     forall x12 x1 x13, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_nil nil) 
         x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     (* <a____(nil,X_),mark(X_)> *)
    | DP_R_xml_0_scc_5_25 :
     forall x12 x1 x13, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_nil nil) 
        x13) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x12) ->
        DP_R_xml_0_scc_5 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_5.
   Inductive DP_R_xml_0_scc_5_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a____(__(X_,Y_),Z_),a____(mark(X_),a____(mark(Y_),mark(Z_)))> *)
     | DP_R_xml_0_scc_5_large_0 :
      forall x12 x2 x1 x13 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x12) ->
         DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a____ 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x1::nil))::(algebra.Alg.Term 
                                 algebra.F.id_a____ ((algebra.Alg.Term 
                                 algebra.F.id_mark (x2::nil))::
                                 (algebra.Alg.Term algebra.F.id_mark 
                                 (x3::nil))::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     
      (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
     | DP_R_xml_0_scc_5_large_1 :
      forall x12 x10 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__and 
                                ((algebra.Alg.Term algebra.F.id_mark 
                                (x9::nil))::x10::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <a__and(tt,X_),mark(X_)> *)
     | DP_R_xml_0_scc_5_large_2 :
      forall x12 x1 x13, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil))
      (* <mark(and(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_5_large_3 :
      forall x12 x10 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <mark(isList(X_)),a__isList(X_)> *)
     | DP_R_xml_0_scc_5_large_4 :
      forall x12 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isList 
                                (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <a__isList(V_),a__isNeList(V_)> *)
     | DP_R_xml_0_scc_5_large_5 :
      forall x4 x12, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isNeList 
                                (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
      (* <mark(isNeList(X_)),a__isNeList(X_)> *)
     | DP_R_xml_0_scc_5_large_6 :
      forall x12 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isNeList 
                                (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <mark(isNePal(X_)),a__isNePal(X_)> *)
     | DP_R_xml_0_scc_5_large_7 :
      forall x12 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isNePal 
                                (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     
      (* <a__isNePal(__(I_,__(P_,I_))),a__and(a__isQid(I_),isPal(P_))> *)
     | DP_R_xml_0_scc_5_large_8 :
      forall x8 x12 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
          algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__and 
                                ((algebra.Alg.Term algebra.F.id_a__isQid 
                                (x7::nil))::(algebra.Alg.Term 
                                algebra.F.id_isPal (x8::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
      (* <mark(isPal(X_)),a__isPal(X_)> *)
     | DP_R_xml_0_scc_5_large_9 :
      forall x12 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isPal 
                                (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <a__isPal(V_),a__isNePal(V_)> *)
     | DP_R_xml_0_scc_5_large_10 :
      forall x4 x12, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x12) ->
        DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_a__isNePal 
                                (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
      (* <a____(X_,nil),mark(X_)> *)
     | DP_R_xml_0_scc_5_large_11 :
      forall x12 x1 x13, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_nil nil) 
          x12) ->
         DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
      (* <a____(nil,X_),mark(X_)> *)
     | DP_R_xml_0_scc_5_large_12 :
      forall x12 x1 x13, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_nil nil) 
         x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x12) ->
         DP_R_xml_0_scc_5_large (algebra.Alg.Term algebra.F.id_mark 
                                 (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_5_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <a____(__(X_,Y_),Z_),mark(X_)> *)
     | DP_R_xml_0_scc_5_strict_0 :
      forall x12 x2 x1 x13 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x12) ->
         DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
     
      (* <mark(__(X1_,X2_)),a____(mark(X1_),mark(X2_))> *)
     | DP_R_xml_0_scc_5_strict_1 :
      forall x12 x10 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a____ 
                                 ((algebra.Alg.Term algebra.F.id_mark 
                                 (x9::nil))::(algebra.Alg.Term 
                                 algebra.F.id_mark (x10::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     
      (* <a____(__(X_,Y_),Z_),a____(mark(Y_),mark(Z_))> *)
     | DP_R_xml_0_scc_5_strict_2 :
      forall x12 x2 x1 x13 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x12) ->
         DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a____ 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  (x2::nil))::(algebra.Alg.Term 
                                  algebra.F.id_mark (x3::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
      (* <a____(__(X_,Y_),Z_),mark(Y_)> *)
     | DP_R_xml_0_scc_5_strict_3 :
      forall x12 x2 x1 x13 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x12) ->
         DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x2::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
      (* <mark(__(X1_,X2_)),mark(X1_)> *)
     | DP_R_xml_0_scc_5_strict_4 :
      forall x12 x10 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_mark 
                                 (x9::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      (* <mark(__(X1_,X2_)),mark(X2_)> *)
     | DP_R_xml_0_scc_5_strict_5 :
      forall x12 x10 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_mark 
                                 (x10::nil)) 
         (algebra.Alg.Term algebra.F.id_mark (x12::nil))
     
      (* <a__isNeList(__(V1_,V2_)),a__and(a__isList(V1_),isNeList(V2_))> *)
     | DP_R_xml_0_scc_5_strict_6 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term algebra.F.id_a__isList 
                                 (x5::nil))::(algebra.Alg.Term 
                                 algebra.F.id_isNeList (x6::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     
      (* <a__isNeList(__(V1_,V2_)),a__isList(V1_)> *)
     | DP_R_xml_0_scc_5_strict_7 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__isList 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     
      (* <a__isList(__(V1_,V2_)),a__and(a__isList(V1_),isList(V2_))> *)
     | DP_R_xml_0_scc_5_strict_8 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term algebra.F.id_a__isList 
                                 (x5::nil))::(algebra.Alg.Term 
                                 algebra.F.id_isList (x6::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
     
      (* <a__isList(__(V1_,V2_)),a__isList(V1_)> *)
     | DP_R_xml_0_scc_5_strict_9 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__isList 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
     
      (* <a__isNeList(__(V1_,V2_)),a__and(a__isNeList(V1_),isList(V2_))> *)
     | DP_R_xml_0_scc_5_strict_10 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__and 
                                 ((algebra.Alg.Term algebra.F.id_a__isNeList 
                                 (x5::nil))::(algebra.Alg.Term 
                                 algebra.F.id_isList (x6::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
     
      (* <a__isNeList(__(V1_,V2_)),a__isNeList(V1_)> *)
     | DP_R_xml_0_scc_5_strict_11 :
      forall x12 x6 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x12) ->
        DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_a__isNeList 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil))
      (* <a____(__(X_,Y_),Z_),mark(Z_)> *)
     | DP_R_xml_0_scc_5_strict_12 :
      forall x12 x2 x1 x13 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x12) ->
         DP_R_xml_0_scc_5_strict (algebra.Alg.Term algebra.F.id_mark 
                                  (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_5_large.
    Inductive DP_R_xml_0_scc_5_large_non_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <mark(isNeList(X_)),a__isNeList(X_)> *)
      | DP_R_xml_0_scc_5_large_non_scc_1_0 :
       forall x12 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x12) ->
         DP_R_xml_0_scc_5_large_non_scc_1 (algebra.Alg.Term 
                                           algebra.F.id_a__isNeList 
                                           (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_non_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_non_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__isList(V_),a__isNeList(V_)> *)
      | DP_R_xml_0_scc_5_large_non_scc_2_0 :
       forall x4 x12, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x12) ->
         DP_R_xml_0_scc_5_large_non_scc_2 (algebra.Alg.Term 
                                           algebra.F.id_a__isNeList 
                                           (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isList (x12::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_non_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <mark(isList(X_)),a__isList(X_)> *)
      | DP_R_xml_0_scc_5_large_non_scc_3_0 :
       forall x12 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x12) ->
         DP_R_xml_0_scc_5_large_non_scc_3 (algebra.Alg.Term 
                                           algebra.F.id_a__isList (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_non_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_non_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_5_large_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_scc_4  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a__and(tt,X_),mark(X_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_0 :
       forall x12 x1 x13, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term algebra.F.id_mark 
                                        (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil))
      
       (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_1 :
       forall x12 x10 x9, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term algebra.F.id_a__and 
                                       ((algebra.Alg.Term algebra.F.id_mark 
                                       (x9::nil))::x10::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       (* <mark(and(X1_,X2_)),mark(X1_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_2 :
       forall x12 x10 x9, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term algebra.F.id_mark 
                                       (x9::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      
       (* <mark(isNePal(X_)),a__isNePal(X_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_3 :
       forall x12 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term 
                                       algebra.F.id_a__isNePal (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      
       (* <a__isNePal(__(I_,__(P_,I_))),a__and(a__isQid(I_),isPal(P_))> *)
      | DP_R_xml_0_scc_5_large_scc_4_4 :
       forall x8 x12 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
           algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term algebra.F.id_a__and 
                                       ((algebra.Alg.Term 
                                       algebra.F.id_a__isQid (x7::nil))::
                                       (algebra.Alg.Term algebra.F.id_isPal 
                                       (x8::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
       (* <mark(isPal(X_)),a__isPal(X_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_5 :
       forall x12 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term 
                                       algebra.F.id_a__isPal (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       (* <a__isPal(V_),a__isNePal(V_)> *)
      | DP_R_xml_0_scc_5_large_scc_4_6 :
       forall x4 x12, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x12) ->
         DP_R_xml_0_scc_5_large_scc_4 (algebra.Alg.Term 
                                       algebra.F.id_a__isNePal (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_5_large_scc_4.
     Inductive DP_R_xml_0_scc_5_large_scc_4_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <mark(and(X1_,X2_)),mark(X1_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_large_0 :
        forall x12 x10 x9, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
          DP_R_xml_0_scc_5_large_scc_4_large (algebra.Alg.Term 
                                              algebra.F.id_mark (x9::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       
        (* <mark(isNePal(X_)),a__isNePal(X_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_large_1 :
        forall x12 x1, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
          DP_R_xml_0_scc_5_large_scc_4_large (algebra.Alg.Term 
                                              algebra.F.id_a__isNePal 
                                              (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       
        (* <mark(isPal(X_)),a__isPal(X_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_large_2 :
        forall x12 x1, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
          DP_R_xml_0_scc_5_large_scc_4_large (algebra.Alg.Term 
                                              algebra.F.id_a__isPal 
                                              (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       
        (* <a__isPal(V_),a__isNePal(V_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_large_3 :
        forall x4 x12, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x12) ->
          DP_R_xml_0_scc_5_large_scc_4_large (algebra.Alg.Term 
                                              algebra.F.id_a__isNePal 
                                              (x4::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_5_large_scc_4_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <a__and(tt,X_),mark(X_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_strict_0 :
        forall x12 x1 x13, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_tt nil) 
           x13) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x1 x12) ->
           DP_R_xml_0_scc_5_large_scc_4_strict (algebra.Alg.Term 
                                                algebra.F.id_mark (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil))
       
        (* <mark(and(X1_,X2_)),a__and(mark(X1_),X2_)> *)
       | DP_R_xml_0_scc_5_large_scc_4_strict_1 :
        forall x12 x10 x9, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
          DP_R_xml_0_scc_5_large_scc_4_strict (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_mark (x9::nil))::
                                               x10::nil)) 
           (algebra.Alg.Term algebra.F.id_mark (x12::nil))
       
        (* <a__isNePal(__(I_,__(P_,I_))),a__and(a__isQid(I_),isPal(P_))> *)
       | DP_R_xml_0_scc_5_large_scc_4_strict_2 :
        forall x8 x12 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term 
            algebra.F.id___ (x8::x7::nil))::nil)) x12) ->
          DP_R_xml_0_scc_5_large_scc_4_strict (algebra.Alg.Term 
                                               algebra.F.id_a__and 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_a__isQid 
                                               (x7::nil))::(algebra.Alg.Term 
                                               algebra.F.id_isPal 
                                               (x8::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_5_large_scc_4_large.
      Inductive DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <a__isPal(V_),a__isNePal(V_)> *)
        | DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1_0 :
         forall x4 x12, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x12) ->
           DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1 (algebra.Alg.Term 
                                                         algebra.F.id_a__isNePal 
                                                         (x4::nil)) 
            (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1 :
       forall x y, 
        (DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1 x y) ->
         Acc WF_DP_R_xml_0_scc_5_large_scc_4.DP_R_xml_0_scc_5_large_scc_4_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(isPal(X_)),a__isPal(X_)> *)
        | DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2_0 :
         forall x12 x1, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x12) ->
           DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2 (algebra.Alg.Term 
                                                         algebra.F.id_a__isPal 
                                                         (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2 :
       forall x y, 
        (DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2 x y) ->
         Acc WF_DP_R_xml_0_scc_5_large_scc_4.DP_R_xml_0_scc_5_large_scc_4_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_5_large_scc_4_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(isNePal(X_)),a__isNePal(X_)> *)
        | DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3_0 :
         forall x12 x1, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x12) ->
           DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3 (algebra.Alg.Term 
                                                         algebra.F.id_a__isNePal 
                                                         (x1::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      .
      
      
      Lemma acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3 :
       forall x y, 
        (DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3 x y) ->
         Acc WF_DP_R_xml_0_scc_5_large_scc_4.DP_R_xml_0_scc_5_large_scc_4_large 
          x.
      Proof.
        intros x y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
      Qed.
      
      
      Inductive DP_R_xml_0_scc_5_large_scc_4_large_scc_4  :
       algebra.Alg.term ->algebra.Alg.term ->Prop := 
         (* <mark(and(X1_,X2_)),mark(X1_)> *)
        | DP_R_xml_0_scc_5_large_scc_4_large_scc_4_0 :
         forall x12 x10 x9, 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_and (x9::x10::nil)) x12) ->
           DP_R_xml_0_scc_5_large_scc_4_large_scc_4 (algebra.Alg.Term 
                                                     algebra.F.id_mark 
                                                     (x9::nil)) 
            (algebra.Alg.Term algebra.F.id_mark (x12::nil))
      .
      
      
      Module WF_DP_R_xml_0_scc_5_large_scc_4_large_scc_4.
       Open Scope Z_scope.
       
       Import ring_extention.
       
       Notation Local "a <= b" := (Zle a b).
       
       Notation Local "a < b" := (Zlt a b).
       
       Definition P_id_a____ (x12:Z) (x13:Z) := 2 + 2* x12 + 1* x13.
       
       Definition P_id_i  := 3.
       
       Definition P_id_isList (x12:Z) := 2* x12.
       
       Definition P_id_a__and (x12:Z) (x13:Z) := 1 + 2* x12 + 1* x13.
       
       Definition P_id_isQid (x12:Z) := 1* x12.
       
       Definition P_id_isPal (x12:Z) := 2* x12.
       
       Definition P_id_mark (x12:Z) := 1* x12.
       
       Definition P_id_u  := 3.
       
       Definition P_id_isNeList (x12:Z) := 2* x12.
       
       Definition P_id_a__isList (x12:Z) := 2* x12.
       
       Definition P_id_a  := 3.
       
       Definition P_id___ (x12:Z) (x13:Z) := 2 + 2* x12 + 1* x13.
       
       Definition P_id_o  := 1.
       
       Definition P_id_a__isQid (x12:Z) := 1* x12.
       
       Definition P_id_tt  := 0.
       
       Definition P_id_isNePal (x12:Z) := 2* x12.
       
       Definition P_id_a__isPal (x12:Z) := 2* x12.
       
       Definition P_id_nil  := 0.
       
       Definition P_id_and (x12:Z) (x13:Z) := 1 + 2* x12 + 1* x13.
       
       Definition P_id_a__isNePal (x12:Z) := 2* x12.
       
       Definition P_id_a__isNeList (x12:Z) := 2* x12.
       
       Definition P_id_e  := 3.
       
       Lemma P_id_a_____monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->
           P_id_a____ x13 x15 <= P_id_a____ x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isList_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__and_monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->
           P_id_a__and x13 x15 <= P_id_a__and x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isQid_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isPal_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_mark_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNeList_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_isNeList x13 <= P_id_isNeList x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isList_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_a__isList x13 <= P_id_a__isList x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id____monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isQid_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_a__isQid x13 <= P_id_a__isQid x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNePal_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isPal_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_a__isPal x13 <= P_id_a__isPal x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_and_monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->P_id_and x13 x15 <= P_id_and x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNePal_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->
          P_id_a__isNePal x13 <= P_id_a__isNePal x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNeList_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->
          P_id_a__isNeList x13 <= P_id_a__isNeList x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a_____bounded :
        forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a____ x13 x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_i_bounded : 0 <= P_id_i .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isList_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_isList x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__and_bounded :
        forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__and x13 x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isQid_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_isQid x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isPal_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_isPal x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_mark_bounded : forall x12, (0 <= x12) ->0 <= P_id_mark x12.
       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_bounded : 0 <= P_id_u .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNeList_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_isNeList x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isList_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_a__isList x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a_bounded : 0 <= P_id_a .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id____bounded :
        forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id___ x13 x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_o_bounded : 0 <= P_id_o .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isQid_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_a__isQid x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_tt_bounded : 0 <= P_id_tt .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_isNePal_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_isNePal x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isPal_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_a__isPal x12.
       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_and_bounded :
        forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_and x13 x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNePal_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_a__isNePal x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_a__isNeList_bounded :
        forall x12, (0 <= x12) ->0 <= P_id_a__isNeList x12.
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_e_bounded : 0 <= P_id_e .
       Proof.
         intros .
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Definition measure  := 
         InterpZ.measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and 
          P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList 
          P_id_a__isList P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt 
          P_id_isNePal P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal 
          P_id_a__isNeList P_id_e.
       
       Lemma measure_equation :
        forall t, 
         measure t = match t with
                       | (algebra.Alg.Term algebra.F.id_a____ (x13::
                          x12::nil)) =>
                        P_id_a____ (measure x13) (measure x12)
                       | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                       | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                        P_id_isList (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a__and (x13::
                          x12::nil)) =>
                        P_id_a__and (measure x13) (measure x12)
                       | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                        P_id_isQid (measure x12)
                       | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                        P_id_isPal (measure x12)
                       | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                        P_id_mark (measure x12)
                       | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                       | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                        P_id_isNeList (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                        P_id_a__isList (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                       | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                        P_id___ (measure x13) (measure x12)
                       | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                       | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                        P_id_a__isQid (measure x12)
                       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                       | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                        P_id_isNePal (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                        P_id_a__isPal (measure x12)
                       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                       | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                        P_id_and (measure x13) (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a__isNePal 
                          (x12::nil)) =>
                        P_id_a__isNePal (measure x12)
                       | (algebra.Alg.Term algebra.F.id_a__isNeList 
                          (x12::nil)) =>
                        P_id_a__isNeList (measure x12)
                       | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                       | _ => 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_a_____monotonic;assumption.
         intros ;apply P_id_isList_monotonic;assumption.
         intros ;apply P_id_a__and_monotonic;assumption.
         intros ;apply P_id_isQid_monotonic;assumption.
         intros ;apply P_id_isPal_monotonic;assumption.
         intros ;apply P_id_mark_monotonic;assumption.
         intros ;apply P_id_isNeList_monotonic;assumption.
         intros ;apply P_id_a__isList_monotonic;assumption.
         intros ;apply P_id____monotonic;assumption.
         intros ;apply P_id_a__isQid_monotonic;assumption.
         intros ;apply P_id_isNePal_monotonic;assumption.
         intros ;apply P_id_a__isPal_monotonic;assumption.
         intros ;apply P_id_and_monotonic;assumption.
         intros ;apply P_id_a__isNePal_monotonic;assumption.
         intros ;apply P_id_a__isNeList_monotonic;assumption.
         intros ;apply P_id_a_____bounded;assumption.
         intros ;apply P_id_i_bounded;assumption.
         intros ;apply P_id_isList_bounded;assumption.
         intros ;apply P_id_a__and_bounded;assumption.
         intros ;apply P_id_isQid_bounded;assumption.
         intros ;apply P_id_isPal_bounded;assumption.
         intros ;apply P_id_mark_bounded;assumption.
         intros ;apply P_id_u_bounded;assumption.
         intros ;apply P_id_isNeList_bounded;assumption.
         intros ;apply P_id_a__isList_bounded;assumption.
         intros ;apply P_id_a_bounded;assumption.
         intros ;apply P_id____bounded;assumption.
         intros ;apply P_id_o_bounded;assumption.
         intros ;apply P_id_a__isQid_bounded;assumption.
         intros ;apply P_id_tt_bounded;assumption.
         intros ;apply P_id_isNePal_bounded;assumption.
         intros ;apply P_id_a__isPal_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_and_bounded;assumption.
         intros ;apply P_id_a__isNePal_bounded;assumption.
         intros ;apply P_id_a__isNeList_bounded;assumption.
         intros ;apply P_id_e_bounded;assumption.
         apply rules_monotonic.
       Qed.
       
       Definition P_id_A__AND (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__ISNEPAL (x12:Z) := 0.
       
       Definition P_id_A__ISNELIST (x12:Z) := 0.
       
       Definition P_id_A____ (x12:Z) (x13:Z) := 0.
       
       Definition P_id_A__ISLIST (x12:Z) := 0.
       
       Definition P_id_A__ISPAL (x12:Z) := 0.
       
       Definition P_id_A__ISQID (x12:Z) := 0.
       
       Definition P_id_MARK (x12:Z) := 2* x12.
       
       Lemma P_id_A__AND_monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->
           P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNEPAL_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->
          P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISNELIST_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->
          P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A_____monotonic :
        forall x12 x14 x13 x15, 
         (0 <= x15)/\ (x15 <= x14) ->
          (0 <= x13)/\ (x13 <= x12) ->
           P_id_A____ x13 x15 <= P_id_A____ x12 x14.
       Proof.
         intros x15 x14 x13 x12.
         intros [H_1 H_0].
         intros [H_3 H_2].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISLIST_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISPAL_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_A__ISQID_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISQID x13 <= P_id_A__ISQID x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Lemma P_id_MARK_monotonic :
        forall x12 x13, 
         (0 <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
       Proof.
         intros x13 x12.
         intros [H_1 H_0].
         
         cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
          (auto with zarith)||(repeat (translate_vars );prove_ineq ).
       Qed.
       
       Definition marked_measure  := 
         InterpZ.marked_measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and 
          P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList 
          P_id_a__isList P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt 
          P_id_isNePal P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal 
          P_id_a__isNeList P_id_e P_id_A__AND P_id_A__ISNEPAL 
          P_id_A__ISNELIST P_id_A____ P_id_A__ISLIST P_id_A__ISPAL 
          P_id_A__ISQID P_id_MARK.
       
       Lemma marked_measure_equation :
        forall t, 
         marked_measure t = match t with
                              | (algebra.Alg.Term algebra.F.id_a__and (x13::
                                 x12::nil)) =>
                               P_id_A__AND (measure x13) (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a__isNePal 
                                 (x12::nil)) =>
                               P_id_A__ISNEPAL (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a__isNeList 
                                 (x12::nil)) =>
                               P_id_A__ISNELIST (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a____ (x13::
                                 x12::nil)) =>
                               P_id_A____ (measure x13) (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a__isList 
                                 (x12::nil)) =>
                               P_id_A__ISLIST (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a__isPal 
                                 (x12::nil)) =>
                               P_id_A__ISPAL (measure x12)
                              | (algebra.Alg.Term algebra.F.id_a__isQid 
                                 (x12::nil)) =>
                               P_id_A__ISQID (measure x12)
                              | (algebra.Alg.Term algebra.F.id_mark 
                                 (x12::nil)) =>
                               P_id_MARK (measure x12)
                              | _ => 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_a_____monotonic;assumption.
         intros ;apply P_id_isList_monotonic;assumption.
         intros ;apply P_id_a__and_monotonic;assumption.
         intros ;apply P_id_isQid_monotonic;assumption.
         intros ;apply P_id_isPal_monotonic;assumption.
         intros ;apply P_id_mark_monotonic;assumption.
         intros ;apply P_id_isNeList_monotonic;assumption.
         intros ;apply P_id_a__isList_monotonic;assumption.
         intros ;apply P_id____monotonic;assumption.
         intros ;apply P_id_a__isQid_monotonic;assumption.
         intros ;apply P_id_isNePal_monotonic;assumption.
         intros ;apply P_id_a__isPal_monotonic;assumption.
         intros ;apply P_id_and_monotonic;assumption.
         intros ;apply P_id_a__isNePal_monotonic;assumption.
         intros ;apply P_id_a__isNeList_monotonic;assumption.
         intros ;apply P_id_a_____bounded;assumption.
         intros ;apply P_id_i_bounded;assumption.
         intros ;apply P_id_isList_bounded;assumption.
         intros ;apply P_id_a__and_bounded;assumption.
         intros ;apply P_id_isQid_bounded;assumption.
         intros ;apply P_id_isPal_bounded;assumption.
         intros ;apply P_id_mark_bounded;assumption.
         intros ;apply P_id_u_bounded;assumption.
         intros ;apply P_id_isNeList_bounded;assumption.
         intros ;apply P_id_a__isList_bounded;assumption.
         intros ;apply P_id_a_bounded;assumption.
         intros ;apply P_id____bounded;assumption.
         intros ;apply P_id_o_bounded;assumption.
         intros ;apply P_id_a__isQid_bounded;assumption.
         intros ;apply P_id_tt_bounded;assumption.
         intros ;apply P_id_isNePal_bounded;assumption.
         intros ;apply P_id_a__isPal_bounded;assumption.
         intros ;apply P_id_nil_bounded;assumption.
         intros ;apply P_id_and_bounded;assumption.
         intros ;apply P_id_a__isNePal_bounded;assumption.
         intros ;apply P_id_a__isNeList_bounded;assumption.
         intros ;apply P_id_e_bounded;assumption.
         apply rules_monotonic.
         intros ;apply P_id_A__AND_monotonic;assumption.
         intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
         intros ;apply P_id_A__ISNELIST_monotonic;assumption.
         intros ;apply P_id_A_____monotonic;assumption.
         intros ;apply P_id_A__ISLIST_monotonic;assumption.
         intros ;apply P_id_A__ISPAL_monotonic;assumption.
         intros ;apply P_id_A__ISQID_monotonic;assumption.
         intros ;apply P_id_MARK_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_5_large_scc_4_large.DP_R_xml_0_scc_5_large_scc_4_large_scc_4
         .
       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_5_large_scc_4_large_scc_4.
      
      Definition wf_DP_R_xml_0_scc_5_large_scc_4_large_scc_4  := 
        WF_DP_R_xml_0_scc_5_large_scc_4_large_scc_4.wf.
      
      
      Lemma acc_DP_R_xml_0_scc_5_large_scc_4_large_scc_4 :
       forall x y, 
        (DP_R_xml_0_scc_5_large_scc_4_large_scc_4 x y) ->
         Acc WF_DP_R_xml_0_scc_5_large_scc_4.DP_R_xml_0_scc_5_large_scc_4_large 
          x.
      Proof.
        intros x.
        pattern x.
        apply (@Acc_ind _ DP_R_xml_0_scc_5_large_scc_4_large_scc_4).
        intros x' _ Hrec y h.
        
        inversion h;clear h;subst;
         constructor;intros _y _h;inversion _h;clear _h;subst;
          (eapply Hrec;econstructor eassumption)||
          ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
             (eapply Hrec;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
        apply wf_DP_R_xml_0_scc_5_large_scc_4_large_scc_4.
      Qed.
      
      
      Lemma wf :
       well_founded WF_DP_R_xml_0_scc_5_large_scc_4.DP_R_xml_0_scc_5_large_scc_4_large
        .
      Proof.
        constructor;intros _y _h;inversion _h;clear _h;subst;
         (eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_1;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_non_scc_0;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_scc_4;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_scc_3;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_scc_2;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_large_scc_1;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_5_large_scc_4_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_5_large_scc_4_large.
     
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a____ (x12:Z) (x13:Z) := 2* x12 + 1* x13.
     
     Definition P_id_i  := 1.
     
     Definition P_id_isList (x12:Z) := 1* x12.
     
     Definition P_id_a__and (x12:Z) (x13:Z) := 2* x12 + 1* x13.
     
     Definition P_id_isQid (x12:Z) := 1* x12.
     
     Definition P_id_isPal (x12:Z) := 2* x12.
     
     Definition P_id_mark (x12:Z) := 1* x12.
     
     Definition P_id_u  := 1.
     
     Definition P_id_isNeList (x12:Z) := 1* x12.
     
     Definition P_id_a__isList (x12:Z) := 1* x12.
     
     Definition P_id_a  := 1.
     
     Definition P_id___ (x12:Z) (x13:Z) := 2* x12 + 1* x13.
     
     Definition P_id_o  := 1.
     
     Definition P_id_a__isQid (x12:Z) := 1* x12.
     
     Definition P_id_tt  := 1.
     
     Definition P_id_isNePal (x12:Z) := 2* x12.
     
     Definition P_id_a__isPal (x12:Z) := 2* x12.
     
     Definition P_id_nil  := 1.
     
     Definition P_id_and (x12:Z) (x13:Z) := 2* x12 + 1* x13.
     
     Definition P_id_a__isNePal (x12:Z) := 2* x12.
     
     Definition P_id_a__isNeList (x12:Z) := 1* x12.
     
     Definition P_id_e  := 1.
     
     Lemma P_id_a_____monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->P_id_a____ x13 x15 <= P_id_a____ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isList_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__and_monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->
         P_id_a__and x13 x15 <= P_id_a__and x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isQid_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isPal_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNeList_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_isNeList x13 <= P_id_isNeList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isList_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_a__isList x13 <= P_id_a__isList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id____monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isQid_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_a__isQid x13 <= P_id_a__isQid x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNePal_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isPal_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_a__isPal x13 <= P_id_a__isPal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->P_id_and x13 x15 <= P_id_and x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNePal_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_a__isNePal x13 <= P_id_a__isNePal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNeList_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->
        P_id_a__isNeList x13 <= P_id_a__isNeList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a_____bounded :
      forall x12 x13, (1 <= x12) ->(1 <= x13) ->1 <= P_id_a____ x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_i_bounded : 1 <= P_id_i .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isList_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_isList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__and_bounded :
      forall x12 x13, (1 <= x12) ->(1 <= x13) ->1 <= P_id_a__and x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isQid_bounded : forall x12, (1 <= x12) ->1 <= P_id_isQid x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isPal_bounded : forall x12, (1 <= x12) ->1 <= P_id_isPal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x12, (1 <= x12) ->1 <= P_id_mark x12.
     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_bounded : 1 <= P_id_u .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNeList_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_isNeList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isList_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_a__isList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a_bounded : 1 <= P_id_a .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id____bounded :
      forall x12 x13, (1 <= x12) ->(1 <= x13) ->1 <= P_id___ x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_o_bounded : 1 <= P_id_o .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isQid_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_a__isQid x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 1 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNePal_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_isNePal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isPal_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_a__isPal x12.
     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 : 1 <= 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_and_bounded :
      forall x12 x13, (1 <= x12) ->(1 <= x13) ->1 <= P_id_and x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNePal_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_a__isNePal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNeList_bounded :
      forall x12, (1 <= x12) ->1 <= P_id_a__isNeList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_e_bounded : 1 <= P_id_e .
     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 1 P_id_a____ P_id_i P_id_isList P_id_a__and 
        P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
        P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal 
        P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList 
        P_id_e.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
                      P_id_a____ (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                     | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                      P_id_isList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
                      P_id_a__and (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                      P_id_isQid (measure x12)
                     | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                      P_id_isPal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                      P_id_mark (measure x12)
                     | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                     | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                      P_id_isNeList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                      P_id_a__isList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                     | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                      P_id___ (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                     | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                      P_id_a__isQid (measure x12)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                      P_id_isNePal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                      P_id_a__isPal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                      P_id_and (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
                      P_id_a__isNePal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
                      P_id_a__isNeList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                     | _ => 1
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 1 <= 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_a_____monotonic;assumption.
       intros ;apply P_id_isList_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_isQid_monotonic;assumption.
       intros ;apply P_id_isPal_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_isNeList_monotonic;assumption.
       intros ;apply P_id_a__isList_monotonic;assumption.
       intros ;apply P_id____monotonic;assumption.
       intros ;apply P_id_a__isQid_monotonic;assumption.
       intros ;apply P_id_isNePal_monotonic;assumption.
       intros ;apply P_id_a__isPal_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_a__isNePal_monotonic;assumption.
       intros ;apply P_id_a__isNeList_monotonic;assumption.
       intros ;apply P_id_a_____bounded;assumption.
       intros ;apply P_id_i_bounded;assumption.
       intros ;apply P_id_isList_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_isQid_bounded;assumption.
       intros ;apply P_id_isPal_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_u_bounded;assumption.
       intros ;apply P_id_isNeList_bounded;assumption.
       intros ;apply P_id_a__isList_bounded;assumption.
       intros ;apply P_id_a_bounded;assumption.
       intros ;apply P_id____bounded;assumption.
       intros ;apply P_id_o_bounded;assumption.
       intros ;apply P_id_a__isQid_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_isNePal_bounded;assumption.
       intros ;apply P_id_a__isPal_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_a__isNePal_bounded;assumption.
       intros ;apply P_id_a__isNeList_bounded;assumption.
       intros ;apply P_id_e_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__AND (x12:Z) (x13:Z) := 2* x12 + 1* x13.
     
     Definition P_id_A__ISNEPAL (x12:Z) := 1 + 1* x12.
     
     Definition P_id_A__ISNELIST (x12:Z) := 0.
     
     Definition P_id_A____ (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISLIST (x12:Z) := 0.
     
     Definition P_id_A__ISPAL (x12:Z) := 1 + 2* x12.
     
     Definition P_id_A__ISQID (x12:Z) := 0.
     
     Definition P_id_MARK (x12:Z) := 1 + 1* x12.
     
     Lemma P_id_A__AND_monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->
         P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNEPAL_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNELIST_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->
        P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A_____monotonic :
      forall x12 x14 x13 x15, 
       (1 <= x15)/\ (x15 <= x14) ->
        (1 <= x13)/\ (x13 <= x12) ->P_id_A____ x13 x15 <= P_id_A____ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISLIST_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISPAL_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISQID_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_A__ISQID x13 <= P_id_A__ISQID x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x13, 
       (1 <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       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 1 P_id_a____ P_id_i P_id_isList P_id_a__and 
        P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
        P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal 
        P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList 
        P_id_e P_id_A__AND P_id_A__ISNEPAL P_id_A__ISNELIST P_id_A____ 
        P_id_A__ISLIST P_id_A__ISPAL P_id_A__ISQID P_id_MARK.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__and (x13::
                               x12::nil)) =>
                             P_id_A__AND (measure x13) (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isNePal 
                               (x12::nil)) =>
                             P_id_A__ISNEPAL (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isNeList 
                               (x12::nil)) =>
                             P_id_A__ISNELIST (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a____ (x13::
                               x12::nil)) =>
                             P_id_A____ (measure x13) (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isList 
                               (x12::nil)) =>
                             P_id_A__ISLIST (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isPal 
                               (x12::nil)) =>
                             P_id_A__ISPAL (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isQid 
                               (x12::nil)) =>
                             P_id_A__ISQID (measure x12)
                            | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                             P_id_MARK (measure x12)
                            | _ => 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_a_____monotonic;assumption.
       intros ;apply P_id_isList_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_isQid_monotonic;assumption.
       intros ;apply P_id_isPal_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_isNeList_monotonic;assumption.
       intros ;apply P_id_a__isList_monotonic;assumption.
       intros ;apply P_id____monotonic;assumption.
       intros ;apply P_id_a__isQid_monotonic;assumption.
       intros ;apply P_id_isNePal_monotonic;assumption.
       intros ;apply P_id_a__isPal_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_a__isNePal_monotonic;assumption.
       intros ;apply P_id_a__isNeList_monotonic;assumption.
       intros ;apply P_id_a_____bounded;assumption.
       intros ;apply P_id_i_bounded;assumption.
       intros ;apply P_id_isList_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_isQid_bounded;assumption.
       intros ;apply P_id_isPal_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_u_bounded;assumption.
       intros ;apply P_id_isNeList_bounded;assumption.
       intros ;apply P_id_a__isList_bounded;assumption.
       intros ;apply P_id_a_bounded;assumption.
       intros ;apply P_id____bounded;assumption.
       intros ;apply P_id_o_bounded;assumption.
       intros ;apply P_id_a__isQid_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_isNePal_bounded;assumption.
       intros ;apply P_id_a__isPal_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_a__isNePal_bounded;assumption.
       intros ;apply P_id_a__isNeList_bounded;assumption.
       intros ;apply P_id_e_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__AND_monotonic;assumption.
       intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
       intros ;apply P_id_A__ISNELIST_monotonic;assumption.
       intros ;apply P_id_A_____monotonic;assumption.
       intros ;apply P_id_A__ISLIST_monotonic;assumption.
       intros ;apply P_id_A__ISPAL_monotonic;assumption.
       intros ;apply P_id_A__ISQID_monotonic;assumption.
       intros ;apply P_id_MARK_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 1) (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 1)).
     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_5_large_scc_4_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_5_large_scc_4_strict lt.
     Proof.
       unfold Relation_Definitions.inclusion, lt in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- (Zwf.Zwf 1) _ (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 1)) 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_5_large_scc_4_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_5_large_scc_4_large le.
     Proof.
       unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
       
       intros a b H;destruct H;
        match goal with
          |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
           let l'' := algebra.Alg_ext.find_replacement l  in 
            ((apply (interp.le_trans (interp.o_Z 1)) 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_5_large_scc_4_large  := 
       WF_DP_R_xml_0_scc_5_large_scc_4_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_5_large.DP_R_xml_0_scc_5_large_scc_4.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_5_large_scc_4_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_5_large_scc_4_strict_in_lt;
          econstructor eassumption)||
        ((apply IHx;[econstructor eassumption|
          intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
           apply DP_R_xml_0_scc_5_large_scc_4_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_5_large_scc_4_large.
     Qed.
    End WF_DP_R_xml_0_scc_5_large_scc_4.
    
    Definition wf_DP_R_xml_0_scc_5_large_scc_4  := 
      WF_DP_R_xml_0_scc_5_large_scc_4.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_scc_4 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_scc_4 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_5_large_scc_4).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
      apply wf_DP_R_xml_0_scc_5_large_scc_4.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_non_scc_5  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a____(nil,X_),mark(X_)> *)
      | DP_R_xml_0_scc_5_large_non_scc_5_0 :
       forall x12 x1 x13, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_nil nil) 
          x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x12) ->
          DP_R_xml_0_scc_5_large_non_scc_5 (algebra.Alg.Term 
                                            algebra.F.id_mark (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_non_scc_5 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_non_scc_5 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_5_large_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_non_scc_6  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a____(X_,nil),mark(X_)> *)
      | DP_R_xml_0_scc_5_large_non_scc_6_0 :
       forall x12 x1 x13, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    (algebra.Alg.Term algebra.F.id_nil nil) 
           x12) ->
          DP_R_xml_0_scc_5_large_non_scc_6 (algebra.Alg.Term 
                                            algebra.F.id_mark (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_non_scc_6 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_non_scc_6 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_5_large_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_5_large_scc_7  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <a____(__(X_,Y_),Z_),a____(mark(X_),a____(mark(Y_),mark(Z_)))> *)
      | DP_R_xml_0_scc_5_large_scc_7_0 :
       forall x12 x2 x1 x13 x3, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x13) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x12) ->
          DP_R_xml_0_scc_5_large_scc_7 (algebra.Alg.Term algebra.F.id_a____ 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x1::nil))::(algebra.Alg.Term 
                                        algebra.F.id_a____ 
                                        ((algebra.Alg.Term algebra.F.id_mark 
                                        (x2::nil))::(algebra.Alg.Term 
                                        algebra.F.id_mark 
                                        (x3::nil))::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_5_large_scc_7.
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_a____ (x12:Z) (x13:Z) := 1 + 2* x12 + 1* x13.
     
     Definition P_id_i  := 2.
     
     Definition P_id_isList (x12:Z) := 0.
     
     Definition P_id_a__and (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_isQid (x12:Z) := 0.
     
     Definition P_id_isPal (x12:Z) := 0.
     
     Definition P_id_mark (x12:Z) := 1* x12.
     
     Definition P_id_u  := 0.
     
     Definition P_id_isNeList (x12:Z) := 0.
     
     Definition P_id_a__isList (x12:Z) := 0.
     
     Definition P_id_a  := 2.
     
     Definition P_id___ (x12:Z) (x13:Z) := 1 + 2* x12 + 1* x13.
     
     Definition P_id_o  := 1.
     
     Definition P_id_a__isQid (x12:Z) := 0.
     
     Definition P_id_tt  := 0.
     
     Definition P_id_isNePal (x12:Z) := 0.
     
     Definition P_id_a__isPal (x12:Z) := 0.
     
     Definition P_id_nil  := 0.
     
     Definition P_id_and (x12:Z) (x13:Z) := 1* x13.
     
     Definition P_id_a__isNePal (x12:Z) := 0.
     
     Definition P_id_a__isNeList (x12:Z) := 0.
     
     Definition P_id_e  := 0.
     
     Lemma P_id_a_____monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->P_id_a____ x13 x15 <= P_id_a____ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isList_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__and_monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->
         P_id_a__and x13 x15 <= P_id_a__and x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isQid_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isPal_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNeList_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_isNeList x13 <= P_id_isNeList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isList_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_a__isList x13 <= P_id_a__isList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id____monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isQid_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_a__isQid x13 <= P_id_a__isQid x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNePal_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isPal_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_a__isPal x13 <= P_id_a__isPal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->P_id_and x13 x15 <= P_id_and x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNePal_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_a__isNePal x13 <= P_id_a__isNePal x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNeList_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->
        P_id_a__isNeList x13 <= P_id_a__isNeList x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a_____bounded :
      forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a____ x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_i_bounded : 0 <= P_id_i .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isList_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_isList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__and_bounded :
      forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__and x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isQid_bounded : forall x12, (0 <= x12) ->0 <= P_id_isQid x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isPal_bounded : forall x12, (0 <= x12) ->0 <= P_id_isPal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x12, (0 <= x12) ->0 <= P_id_mark x12.
     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_bounded : 0 <= P_id_u .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNeList_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_isNeList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isList_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_a__isList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a_bounded : 0 <= P_id_a .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id____bounded :
      forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id___ x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_o_bounded : 0 <= P_id_o .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isQid_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_a__isQid x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tt_bounded : 0 <= P_id_tt .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_isNePal_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_isNePal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isPal_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_a__isPal x12.
     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_and_bounded :
      forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_and x13 x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNePal_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_a__isNePal x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_a__isNeList_bounded :
      forall x12, (0 <= x12) ->0 <= P_id_a__isNeList x12.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_e_bounded : 0 <= P_id_e .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and 
        P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
        P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal 
        P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList 
        P_id_e.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
                      P_id_a____ (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                     | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                      P_id_isList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
                      P_id_a__and (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                      P_id_isQid (measure x12)
                     | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                      P_id_isPal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                      P_id_mark (measure x12)
                     | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                     | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                      P_id_isNeList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                      P_id_a__isList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                     | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                      P_id___ (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                     | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                      P_id_a__isQid (measure x12)
                     | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                     | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                      P_id_isNePal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                      P_id_a__isPal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                      P_id_and (measure x13) (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
                      P_id_a__isNePal (measure x12)
                     | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
                      P_id_a__isNeList (measure x12)
                     | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                     | _ => 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_a_____monotonic;assumption.
       intros ;apply P_id_isList_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_isQid_monotonic;assumption.
       intros ;apply P_id_isPal_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_isNeList_monotonic;assumption.
       intros ;apply P_id_a__isList_monotonic;assumption.
       intros ;apply P_id____monotonic;assumption.
       intros ;apply P_id_a__isQid_monotonic;assumption.
       intros ;apply P_id_isNePal_monotonic;assumption.
       intros ;apply P_id_a__isPal_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_a__isNePal_monotonic;assumption.
       intros ;apply P_id_a__isNeList_monotonic;assumption.
       intros ;apply P_id_a_____bounded;assumption.
       intros ;apply P_id_i_bounded;assumption.
       intros ;apply P_id_isList_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_isQid_bounded;assumption.
       intros ;apply P_id_isPal_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_u_bounded;assumption.
       intros ;apply P_id_isNeList_bounded;assumption.
       intros ;apply P_id_a__isList_bounded;assumption.
       intros ;apply P_id_a_bounded;assumption.
       intros ;apply P_id____bounded;assumption.
       intros ;apply P_id_o_bounded;assumption.
       intros ;apply P_id_a__isQid_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_isNePal_bounded;assumption.
       intros ;apply P_id_a__isPal_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_a__isNePal_bounded;assumption.
       intros ;apply P_id_a__isNeList_bounded;assumption.
       intros ;apply P_id_e_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_A__AND (x12:Z) (x13:Z) := 0.
     
     Definition P_id_A__ISNEPAL (x12:Z) := 0.
     
     Definition P_id_A__ISNELIST (x12:Z) := 0.
     
     Definition P_id_A____ (x12:Z) (x13:Z) := 1* x12.
     
     Definition P_id_A__ISLIST (x12:Z) := 0.
     
     Definition P_id_A__ISPAL (x12:Z) := 0.
     
     Definition P_id_A__ISQID (x12:Z) := 0.
     
     Definition P_id_MARK (x12:Z) := 0.
     
     Lemma P_id_A__AND_monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->
         P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNEPAL_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISNELIST_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->
        P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A_____monotonic :
      forall x12 x14 x13 x15, 
       (0 <= x15)/\ (x15 <= x14) ->
        (0 <= x13)/\ (x13 <= x12) ->P_id_A____ x13 x15 <= P_id_A____ x12 x14.
     Proof.
       intros x15 x14 x13 x12.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISLIST_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISPAL_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_A__ISQID_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISQID x13 <= P_id_A__ISQID x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MARK_monotonic :
      forall x12 x13, 
       (0 <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
     Proof.
       intros x13 x12.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and 
        P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
        P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal 
        P_id_a__isPal P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList 
        P_id_e P_id_A__AND P_id_A__ISNEPAL P_id_A__ISNELIST P_id_A____ 
        P_id_A__ISLIST P_id_A__ISPAL P_id_A__ISQID P_id_MARK.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_a__and (x13::
                               x12::nil)) =>
                             P_id_A__AND (measure x13) (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isNePal 
                               (x12::nil)) =>
                             P_id_A__ISNEPAL (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isNeList 
                               (x12::nil)) =>
                             P_id_A__ISNELIST (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a____ (x13::
                               x12::nil)) =>
                             P_id_A____ (measure x13) (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isList 
                               (x12::nil)) =>
                             P_id_A__ISLIST (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isPal 
                               (x12::nil)) =>
                             P_id_A__ISPAL (measure x12)
                            | (algebra.Alg.Term algebra.F.id_a__isQid 
                               (x12::nil)) =>
                             P_id_A__ISQID (measure x12)
                            | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                             P_id_MARK (measure x12)
                            | _ => 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_a_____monotonic;assumption.
       intros ;apply P_id_isList_monotonic;assumption.
       intros ;apply P_id_a__and_monotonic;assumption.
       intros ;apply P_id_isQid_monotonic;assumption.
       intros ;apply P_id_isPal_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_isNeList_monotonic;assumption.
       intros ;apply P_id_a__isList_monotonic;assumption.
       intros ;apply P_id____monotonic;assumption.
       intros ;apply P_id_a__isQid_monotonic;assumption.
       intros ;apply P_id_isNePal_monotonic;assumption.
       intros ;apply P_id_a__isPal_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id_a__isNePal_monotonic;assumption.
       intros ;apply P_id_a__isNeList_monotonic;assumption.
       intros ;apply P_id_a_____bounded;assumption.
       intros ;apply P_id_i_bounded;assumption.
       intros ;apply P_id_isList_bounded;assumption.
       intros ;apply P_id_a__and_bounded;assumption.
       intros ;apply P_id_isQid_bounded;assumption.
       intros ;apply P_id_isPal_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_u_bounded;assumption.
       intros ;apply P_id_isNeList_bounded;assumption.
       intros ;apply P_id_a__isList_bounded;assumption.
       intros ;apply P_id_a_bounded;assumption.
       intros ;apply P_id____bounded;assumption.
       intros ;apply P_id_o_bounded;assumption.
       intros ;apply P_id_a__isQid_bounded;assumption.
       intros ;apply P_id_tt_bounded;assumption.
       intros ;apply P_id_isNePal_bounded;assumption.
       intros ;apply P_id_a__isPal_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id_a__isNePal_bounded;assumption.
       intros ;apply P_id_a__isNeList_bounded;assumption.
       intros ;apply P_id_e_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_A__AND_monotonic;assumption.
       intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
       intros ;apply P_id_A__ISNELIST_monotonic;assumption.
       intros ;apply P_id_A_____monotonic;assumption.
       intros ;apply P_id_A__ISLIST_monotonic;assumption.
       intros ;apply P_id_A__ISPAL_monotonic;assumption.
       intros ;apply P_id_A__ISQID_monotonic;assumption.
       intros ;apply P_id_MARK_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_5_large.DP_R_xml_0_scc_5_large_scc_7.
     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_5_large_scc_7.
    
    Definition wf_DP_R_xml_0_scc_5_large_scc_7  := 
      WF_DP_R_xml_0_scc_5_large_scc_7.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_5_large_scc_7 :
     forall x y, 
      (DP_R_xml_0_scc_5_large_scc_7 x y) ->
       Acc WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_5_large_scc_7).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_6;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_5;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
      apply wf_DP_R_xml_0_scc_5_large_scc_7.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_5.DP_R_xml_0_scc_5_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_5_large_non_scc_6;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_5;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_4;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_3;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_2;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_1;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_5_large_non_scc_0;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_5_large_scc_7;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_5_large_scc_6;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_5_large_scc_5;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_5_large_scc_4;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_5_large_scc_3;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_5_large_scc_2;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_5_large_scc_1;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_5_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_5_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_a____ (x12:Z) (x13:Z) := 2 + 1* x12 + 1* x13.
   
   Definition P_id_i  := 0.
   
   Definition P_id_isList (x12:Z) := 2* x12.
   
   Definition P_id_a__and (x12:Z) (x13:Z) := 1* x12 + 1* x13.
   
   Definition P_id_isQid (x12:Z) := 0.
   
   Definition P_id_isPal (x12:Z) := 0.
   
   Definition P_id_mark (x12:Z) := 1* x12.
   
   Definition P_id_u  := 1.
   
   Definition P_id_isNeList (x12:Z) := 2* x12.
   
   Definition P_id_a__isList (x12:Z) := 2* x12.
   
   Definition P_id_a  := 0.
   
   Definition P_id___ (x12:Z) (x13:Z) := 2 + 1* x12 + 1* x13.
   
   Definition P_id_o  := 0.
   
   Definition P_id_a__isQid (x12:Z) := 0.
   
   Definition P_id_tt  := 0.
   
   Definition P_id_isNePal (x12:Z) := 0.
   
   Definition P_id_a__isPal (x12:Z) := 0.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_and (x12:Z) (x13:Z) := 1* x12 + 1* x13.
   
   Definition P_id_a__isNePal (x12:Z) := 0.
   
   Definition P_id_a__isNeList (x12:Z) := 2* x12.
   
   Definition P_id_e  := 0.
   
   Lemma P_id_a_____monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id_a____ x13 x15 <= P_id_a____ x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isList_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_isList x13 <= P_id_isList x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__and_monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id_a__and x13 x15 <= P_id_a__and x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isQid_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_isQid x13 <= P_id_isQid x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isPal_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_isPal x13 <= P_id_isPal x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_mark x13 <= P_id_mark x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNeList_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_isNeList x13 <= P_id_isNeList x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isList_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_a__isList x13 <= P_id_a__isList x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id____monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id___ x13 x15 <= P_id___ x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isQid_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_a__isQid x13 <= P_id_a__isQid x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNePal_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_isNePal x13 <= P_id_isNePal x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isPal_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_a__isPal x13 <= P_id_a__isPal x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id_and x13 x15 <= P_id_and x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNePal_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_a__isNePal x13 <= P_id_a__isNePal x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNeList_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_a__isNeList x13 <= P_id_a__isNeList x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a_____bounded :
    forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a____ x13 x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_i_bounded : 0 <= P_id_i .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isList_bounded : forall x12, (0 <= x12) ->0 <= P_id_isList x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__and_bounded :
    forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_a__and x13 x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isQid_bounded : forall x12, (0 <= x12) ->0 <= P_id_isQid x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isPal_bounded : forall x12, (0 <= x12) ->0 <= P_id_isPal x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x12, (0 <= x12) ->0 <= P_id_mark x12.
   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_bounded : 0 <= P_id_u .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNeList_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_isNeList x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isList_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_a__isList x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a_bounded : 0 <= P_id_a .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id____bounded :
    forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id___ x13 x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_o_bounded : 0 <= P_id_o .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isQid_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_a__isQid x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 0 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNePal_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_isNePal x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isPal_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_a__isPal x12.
   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_and_bounded :
    forall x12 x13, (0 <= x12) ->(0 <= x13) ->0 <= P_id_and x13 x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNePal_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_a__isNePal x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_a__isNeList_bounded :
    forall x12, (0 <= x12) ->0 <= P_id_a__isNeList x12.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_e_bounded : 0 <= P_id_e .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and P_id_isQid 
      P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList P_id_a 
      P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal P_id_a__isPal 
      P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList P_id_e.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_a____ (x13::x12::nil)) =>
                    P_id_a____ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i 
                   | (algebra.Alg.Term algebra.F.id_isList (x12::nil)) =>
                    P_id_isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__and (x13::x12::nil)) =>
                    P_id_a__and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isQid (x12::nil)) =>
                    P_id_isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_isPal (x12::nil)) =>
                    P_id_isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                    P_id_mark (measure x12)
                   | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u 
                   | (algebra.Alg.Term algebra.F.id_isNeList (x12::nil)) =>
                    P_id_isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isList (x12::nil)) =>
                    P_id_a__isList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a 
                   | (algebra.Alg.Term algebra.F.id___ (x13::x12::nil)) =>
                    P_id___ (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o 
                   | (algebra.Alg.Term algebra.F.id_a__isQid (x12::nil)) =>
                    P_id_a__isQid (measure x12)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_isNePal (x12::nil)) =>
                    P_id_isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isPal (x12::nil)) =>
                    P_id_a__isPal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_and (x13::x12::nil)) =>
                    P_id_and (measure x13) (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNePal (x12::nil)) =>
                    P_id_a__isNePal (measure x12)
                   | (algebra.Alg.Term algebra.F.id_a__isNeList (x12::nil)) =>
                    P_id_a__isNeList (measure x12)
                   | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e 
                   | _ => 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_a_____monotonic;assumption.
     intros ;apply P_id_isList_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_isQid_monotonic;assumption.
     intros ;apply P_id_isPal_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_isNeList_monotonic;assumption.
     intros ;apply P_id_a__isList_monotonic;assumption.
     intros ;apply P_id____monotonic;assumption.
     intros ;apply P_id_a__isQid_monotonic;assumption.
     intros ;apply P_id_isNePal_monotonic;assumption.
     intros ;apply P_id_a__isPal_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_a__isNePal_monotonic;assumption.
     intros ;apply P_id_a__isNeList_monotonic;assumption.
     intros ;apply P_id_a_____bounded;assumption.
     intros ;apply P_id_i_bounded;assumption.
     intros ;apply P_id_isList_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_isQid_bounded;assumption.
     intros ;apply P_id_isPal_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_u_bounded;assumption.
     intros ;apply P_id_isNeList_bounded;assumption.
     intros ;apply P_id_a__isList_bounded;assumption.
     intros ;apply P_id_a_bounded;assumption.
     intros ;apply P_id____bounded;assumption.
     intros ;apply P_id_o_bounded;assumption.
     intros ;apply P_id_a__isQid_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_isNePal_bounded;assumption.
     intros ;apply P_id_a__isPal_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_a__isNePal_bounded;assumption.
     intros ;apply P_id_a__isNeList_bounded;assumption.
     intros ;apply P_id_e_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_A__AND (x12:Z) (x13:Z) := 1* x13.
   
   Definition P_id_A__ISNEPAL (x12:Z) := 0.
   
   Definition P_id_A__ISNELIST (x12:Z) := 2* x12.
   
   Definition P_id_A____ (x12:Z) (x13:Z) := 1* x12 + 1* x13.
   
   Definition P_id_A__ISLIST (x12:Z) := 2* x12.
   
   Definition P_id_A__ISPAL (x12:Z) := 0.
   
   Definition P_id_A__ISQID (x12:Z) := 0.
   
   Definition P_id_MARK (x12:Z) := 1* x12.
   
   Lemma P_id_A__AND_monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id_A__AND x13 x15 <= P_id_A__AND x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNEPAL_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISNEPAL x13 <= P_id_A__ISNEPAL x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISNELIST_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISNELIST x13 <= P_id_A__ISNELIST x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A_____monotonic :
    forall x12 x14 x13 x15, 
     (0 <= x15)/\ (x15 <= x14) ->
      (0 <= x13)/\ (x13 <= x12) ->P_id_A____ x13 x15 <= P_id_A____ x12 x14.
   Proof.
     intros x15 x14 x13 x12.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISLIST_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISLIST x13 <= P_id_A__ISLIST x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISPAL_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISPAL x13 <= P_id_A__ISPAL x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_A__ISQID_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_A__ISQID x13 <= P_id_A__ISQID x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MARK_monotonic :
    forall x12 x13, 
     (0 <= x13)/\ (x13 <= x12) ->P_id_MARK x13 <= P_id_MARK x12.
   Proof.
     intros x13 x12.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_a____ P_id_i P_id_isList P_id_a__and 
      P_id_isQid P_id_isPal P_id_mark P_id_u P_id_isNeList P_id_a__isList 
      P_id_a P_id___ P_id_o P_id_a__isQid P_id_tt P_id_isNePal P_id_a__isPal 
      P_id_nil P_id_and P_id_a__isNePal P_id_a__isNeList P_id_e P_id_A__AND 
      P_id_A__ISNEPAL P_id_A__ISNELIST P_id_A____ P_id_A__ISLIST 
      P_id_A__ISPAL P_id_A__ISQID P_id_MARK.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_a__and (x13::
                             x12::nil)) =>
                           P_id_A__AND (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNePal 
                             (x12::nil)) =>
                           P_id_A__ISNEPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isNeList 
                             (x12::nil)) =>
                           P_id_A__ISNELIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a____ (x13::
                             x12::nil)) =>
                           P_id_A____ (measure x13) (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isList 
                             (x12::nil)) =>
                           P_id_A__ISLIST (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isPal 
                             (x12::nil)) =>
                           P_id_A__ISPAL (measure x12)
                          | (algebra.Alg.Term algebra.F.id_a__isQid 
                             (x12::nil)) =>
                           P_id_A__ISQID (measure x12)
                          | (algebra.Alg.Term algebra.F.id_mark (x12::nil)) =>
                           P_id_MARK (measure x12)
                          | _ => 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_a_____monotonic;assumption.
     intros ;apply P_id_isList_monotonic;assumption.
     intros ;apply P_id_a__and_monotonic;assumption.
     intros ;apply P_id_isQid_monotonic;assumption.
     intros ;apply P_id_isPal_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_isNeList_monotonic;assumption.
     intros ;apply P_id_a__isList_monotonic;assumption.
     intros ;apply P_id____monotonic;assumption.
     intros ;apply P_id_a__isQid_monotonic;assumption.
     intros ;apply P_id_isNePal_monotonic;assumption.
     intros ;apply P_id_a__isPal_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_a__isNePal_monotonic;assumption.
     intros ;apply P_id_a__isNeList_monotonic;assumption.
     intros ;apply P_id_a_____bounded;assumption.
     intros ;apply P_id_i_bounded;assumption.
     intros ;apply P_id_isList_bounded;assumption.
     intros ;apply P_id_a__and_bounded;assumption.
     intros ;apply P_id_isQid_bounded;assumption.
     intros ;apply P_id_isPal_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_u_bounded;assumption.
     intros ;apply P_id_isNeList_bounded;assumption.
     intros ;apply P_id_a__isList_bounded;assumption.
     intros ;apply P_id_a_bounded;assumption.
     intros ;apply P_id____bounded;assumption.
     intros ;apply P_id_o_bounded;assumption.
     intros ;apply P_id_a__isQid_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_isNePal_bounded;assumption.
     intros ;apply P_id_a__isPal_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_a__isNePal_bounded;assumption.
     intros ;apply P_id_a__isNeList_bounded;assumption.
     intros ;apply P_id_e_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_A__AND_monotonic;assumption.
     intros ;apply P_id_A__ISNEPAL_monotonic;assumption.
     intros ;apply P_id_A__ISNELIST_monotonic;assumption.
     intros ;apply P_id_A_____monotonic;assumption.
     intros ;apply P_id_A__ISLIST_monotonic;assumption.
     intros ;apply P_id_A__ISPAL_monotonic;assumption.
     intros ;apply P_id_A__ISQID_monotonic;assumption.
     intros ;apply P_id_MARK_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_5_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_5_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_5_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_5_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_5_large  := WF_DP_R_xml_0_scc_5_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_5.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_5_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_5_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_5_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_5_large.
   Qed.
  End WF_DP_R_xml_0_scc_5.
  
  Definition wf_DP_R_xml_0_scc_5  := WF_DP_R_xml_0_scc_5.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_5 :
   forall x y, (DP_R_xml_0_scc_5 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_5).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_4;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_3;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_2;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_1;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
           (eapply Hrec;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))))).
    apply wf_DP_R_xml_0_scc_5.
  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_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_5;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_4;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_3;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_2;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_1;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_0;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


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