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_test *)
    | id_test : symb
     (* id_match_1 *)
    | id_match_1 : symb
     (* id_match_0 *)
    | id_match_0 : symb
     (* id_quick *)
    | id_quick : symb
     (* id_False *)
    | id_False : symb
     (* id_match_2 *)
    | id_match_2 : symb
     (* id_Cons *)
    | id_Cons : symb
     (* id_match_5 *)
    | id_match_5 : symb
     (* id_True *)
    | id_True : symb
     (* id_Pair *)
    | id_Pair : symb
     (* id_Nil *)
    | id_Nil : symb
     (* id_match_4 *)
    | id_match_4 : symb
     (* id_append *)
    | id_append : symb
     (* id_match_3 *)
    | id_match_3 : symb
     (* id_part *)
    | id_part : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_test,id_test => true
      | id_match_1,id_match_1 => true
      | id_match_0,id_match_0 => true
      | id_quick,id_quick => true
      | id_False,id_False => true
      | id_match_2,id_match_2 => true
      | id_Cons,id_Cons => true
      | id_match_5,id_match_5 => true
      | id_True,id_True => true
      | id_Pair,id_Pair => true
      | id_Nil,id_Nil => true
      | id_match_4,id_match_4 => true
      | id_append,id_append => true
      | id_match_3,id_match_3 => true
      | id_part,id_part => 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_test,id_test => refl_equal _
             | id_match_1,id_match_1 => refl_equal _
             | id_match_0,id_match_0 => refl_equal _
             | id_quick,id_quick => refl_equal _
             | id_False,id_False => refl_equal _
             | id_match_2,id_match_2 => refl_equal _
             | id_Cons,id_Cons => refl_equal _
             | id_match_5,id_match_5 => refl_equal _
             | id_True,id_True => refl_equal _
             | id_Pair,id_Pair => refl_equal _
             | id_Nil,id_Nil => refl_equal _
             | id_match_4,id_match_4 => refl_equal _
             | id_append,id_append => refl_equal _
             | id_match_3,id_match_3 => refl_equal _
             | id_part,id_part => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_test => term.Free 2
      | id_match_1 => term.Free 3
      | id_match_0 => term.Free 3
      | id_quick => term.Free 1
      | id_False => term.Free 0
      | id_match_2 => term.Free 5
      | id_Cons => term.Free 2
      | id_match_5 => term.Free 4
      | id_True => term.Free 0
      | id_Pair => term.Free 2
      | id_Nil => term.Free 0
      | id_match_4 => term.Free 2
      | id_append => term.Free 2
      | id_match_3 => term.Free 7
      | id_part => term.Free 2
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_test,id_test => true
      | id_test,id_match_1 => false
      | id_test,id_match_0 => false
      | id_test,id_quick => false
      | id_test,id_False => false
      | id_test,id_match_2 => false
      | id_test,id_Cons => false
      | id_test,id_match_5 => false
      | id_test,id_True => false
      | id_test,id_Pair => false
      | id_test,id_Nil => false
      | id_test,id_match_4 => false
      | id_test,id_append => false
      | id_test,id_match_3 => false
      | id_test,id_part => false
      | id_match_1,id_test => true
      | id_match_1,id_match_1 => true
      | id_match_1,id_match_0 => false
      | id_match_1,id_quick => false
      | id_match_1,id_False => false
      | id_match_1,id_match_2 => false
      | id_match_1,id_Cons => false
      | id_match_1,id_match_5 => false
      | id_match_1,id_True => false
      | id_match_1,id_Pair => false
      | id_match_1,id_Nil => false
      | id_match_1,id_match_4 => false
      | id_match_1,id_append => false
      | id_match_1,id_match_3 => false
      | id_match_1,id_part => false
      | id_match_0,id_test => true
      | id_match_0,id_match_1 => true
      | id_match_0,id_match_0 => true
      | id_match_0,id_quick => false
      | id_match_0,id_False => false
      | id_match_0,id_match_2 => false
      | id_match_0,id_Cons => false
      | id_match_0,id_match_5 => false
      | id_match_0,id_True => false
      | id_match_0,id_Pair => false
      | id_match_0,id_Nil => false
      | id_match_0,id_match_4 => false
      | id_match_0,id_append => false
      | id_match_0,id_match_3 => false
      | id_match_0,id_part => false
      | id_quick,id_test => true
      | id_quick,id_match_1 => true
      | id_quick,id_match_0 => true
      | id_quick,id_quick => true
      | id_quick,id_False => false
      | id_quick,id_match_2 => false
      | id_quick,id_Cons => false
      | id_quick,id_match_5 => false
      | id_quick,id_True => false
      | id_quick,id_Pair => false
      | id_quick,id_Nil => false
      | id_quick,id_match_4 => false
      | id_quick,id_append => false
      | id_quick,id_match_3 => false
      | id_quick,id_part => false
      | id_False,id_test => true
      | id_False,id_match_1 => true
      | id_False,id_match_0 => true
      | id_False,id_quick => true
      | id_False,id_False => true
      | id_False,id_match_2 => false
      | id_False,id_Cons => false
      | id_False,id_match_5 => false
      | id_False,id_True => false
      | id_False,id_Pair => false
      | id_False,id_Nil => false
      | id_False,id_match_4 => false
      | id_False,id_append => false
      | id_False,id_match_3 => false
      | id_False,id_part => false
      | id_match_2,id_test => true
      | id_match_2,id_match_1 => true
      | id_match_2,id_match_0 => true
      | id_match_2,id_quick => true
      | id_match_2,id_False => true
      | id_match_2,id_match_2 => true
      | id_match_2,id_Cons => false
      | id_match_2,id_match_5 => false
      | id_match_2,id_True => false
      | id_match_2,id_Pair => false
      | id_match_2,id_Nil => false
      | id_match_2,id_match_4 => false
      | id_match_2,id_append => false
      | id_match_2,id_match_3 => false
      | id_match_2,id_part => false
      | id_Cons,id_test => true
      | id_Cons,id_match_1 => true
      | id_Cons,id_match_0 => true
      | id_Cons,id_quick => true
      | id_Cons,id_False => true
      | id_Cons,id_match_2 => true
      | id_Cons,id_Cons => true
      | id_Cons,id_match_5 => false
      | id_Cons,id_True => false
      | id_Cons,id_Pair => false
      | id_Cons,id_Nil => false
      | id_Cons,id_match_4 => false
      | id_Cons,id_append => false
      | id_Cons,id_match_3 => false
      | id_Cons,id_part => false
      | id_match_5,id_test => true
      | id_match_5,id_match_1 => true
      | id_match_5,id_match_0 => true
      | id_match_5,id_quick => true
      | id_match_5,id_False => true
      | id_match_5,id_match_2 => true
      | id_match_5,id_Cons => true
      | id_match_5,id_match_5 => true
      | id_match_5,id_True => false
      | id_match_5,id_Pair => false
      | id_match_5,id_Nil => false
      | id_match_5,id_match_4 => false
      | id_match_5,id_append => false
      | id_match_5,id_match_3 => false
      | id_match_5,id_part => false
      | id_True,id_test => true
      | id_True,id_match_1 => true
      | id_True,id_match_0 => true
      | id_True,id_quick => true
      | id_True,id_False => true
      | id_True,id_match_2 => true
      | id_True,id_Cons => true
      | id_True,id_match_5 => true
      | id_True,id_True => true
      | id_True,id_Pair => false
      | id_True,id_Nil => false
      | id_True,id_match_4 => false
      | id_True,id_append => false
      | id_True,id_match_3 => false
      | id_True,id_part => false
      | id_Pair,id_test => true
      | id_Pair,id_match_1 => true
      | id_Pair,id_match_0 => true
      | id_Pair,id_quick => true
      | id_Pair,id_False => true
      | id_Pair,id_match_2 => true
      | id_Pair,id_Cons => true
      | id_Pair,id_match_5 => true
      | id_Pair,id_True => true
      | id_Pair,id_Pair => true
      | id_Pair,id_Nil => false
      | id_Pair,id_match_4 => false
      | id_Pair,id_append => false
      | id_Pair,id_match_3 => false
      | id_Pair,id_part => false
      | id_Nil,id_test => true
      | id_Nil,id_match_1 => true
      | id_Nil,id_match_0 => true
      | id_Nil,id_quick => true
      | id_Nil,id_False => true
      | id_Nil,id_match_2 => true
      | id_Nil,id_Cons => true
      | id_Nil,id_match_5 => true
      | id_Nil,id_True => true
      | id_Nil,id_Pair => true
      | id_Nil,id_Nil => true
      | id_Nil,id_match_4 => false
      | id_Nil,id_append => false
      | id_Nil,id_match_3 => false
      | id_Nil,id_part => false
      | id_match_4,id_test => true
      | id_match_4,id_match_1 => true
      | id_match_4,id_match_0 => true
      | id_match_4,id_quick => true
      | id_match_4,id_False => true
      | id_match_4,id_match_2 => true
      | id_match_4,id_Cons => true
      | id_match_4,id_match_5 => true
      | id_match_4,id_True => true
      | id_match_4,id_Pair => true
      | id_match_4,id_Nil => true
      | id_match_4,id_match_4 => true
      | id_match_4,id_append => false
      | id_match_4,id_match_3 => false
      | id_match_4,id_part => false
      | id_append,id_test => true
      | id_append,id_match_1 => true
      | id_append,id_match_0 => true
      | id_append,id_quick => true
      | id_append,id_False => true
      | id_append,id_match_2 => true
      | id_append,id_Cons => true
      | id_append,id_match_5 => true
      | id_append,id_True => true
      | id_append,id_Pair => true
      | id_append,id_Nil => true
      | id_append,id_match_4 => true
      | id_append,id_append => true
      | id_append,id_match_3 => false
      | id_append,id_part => false
      | id_match_3,id_test => true
      | id_match_3,id_match_1 => true
      | id_match_3,id_match_0 => true
      | id_match_3,id_quick => true
      | id_match_3,id_False => true
      | id_match_3,id_match_2 => true
      | id_match_3,id_Cons => true
      | id_match_3,id_match_5 => true
      | id_match_3,id_True => true
      | id_match_3,id_Pair => true
      | id_match_3,id_Nil => true
      | id_match_3,id_match_4 => true
      | id_match_3,id_append => true
      | id_match_3,id_match_3 => true
      | id_match_3,id_part => false
      | id_part,id_test => true
      | id_part,id_match_1 => true
      | id_part,id_match_0 => true
      | id_part,id_quick => true
      | id_part,id_False => true
      | id_part,id_match_2 => true
      | id_part,id_Cons => true
      | id_part,id_match_5 => true
      | id_part,id_True => true
      | id_part,id_Pair => true
      | id_part,id_Nil => true
      | id_part,id_match_4 => true
      | id_part,id_append => true
      | id_part,id_match_3 => true
      | id_part,id_part => 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 := 
    (* test(x_0_,y_) -> True *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_True nil) 
     (algebra.Alg.Term algebra.F.id_test ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 2)::nil))
    (* test(x_0_,y_) -> False *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_False nil) 
     (algebra.Alg.Term algebra.F.id_test ((algebra.Alg.Var 1)::
      (algebra.Alg.Var 2)::nil))
    (* append(l1_2_,l2_1_) -> match_0(l1_2_,l2_1_,l1_2_) *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_0 
                   ((algebra.Alg.Var 3)::(algebra.Alg.Var 4)::
                   (algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Var 3)::
      (algebra.Alg.Var 4)::nil))
    (* match_0(l1_2_,l2_1_,Nil) -> l2_1_ *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Var 4) 
     (algebra.Alg.Term algebra.F.id_match_0 ((algebra.Alg.Var 3)::
      (algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id_Nil nil)::nil))
   
    (* match_0(l1_2_,l2_1_,Cons(x_,l_)) -> Cons(x_,append(l_,l2_1_)) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 5)::
                   (algebra.Alg.Term algebra.F.id_append 
                   ((algebra.Alg.Var 6)::(algebra.Alg.Var 4)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_0 ((algebra.Alg.Var 3)::
      (algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id_Cons 
      ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil))
    (* part(a_4_,l_3_) -> match_1(a_4_,l_3_,l_3_) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_1 
                   ((algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
                   (algebra.Alg.Var 8)::nil)) 
     (algebra.Alg.Term algebra.F.id_part ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::nil))
    (* match_1(a_4_,l_3_,Nil) -> Pair(Nil,Nil) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Term 
                   algebra.F.id_Nil nil)::(algebra.Alg.Term algebra.F.id_Nil 
                   nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_match_1 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_Nil nil)::nil))
   
    (* match_1(a_4_,l_3_,Cons(x_,l'_)) -> match_2(x_,l'_,a_4_,l_3_,part(a_4_,l'_)) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_2 
                   ((algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_part ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 9)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_1 ((algebra.Alg.Var 7)::
      (algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_Cons 
      ((algebra.Alg.Var 5)::(algebra.Alg.Var 9)::nil))::nil))
   
    (* match_2(x_,l'_,a_4_,l_3_,Pair(l1_,l2_)) -> match_3(l1_,l2_,x_,l'_,a_4_,l_3_,test(a_4_,x_)) *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_3 
                   ((algebra.Alg.Var 10)::(algebra.Alg.Var 11)::
                   (algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
                   (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
                   (algebra.Alg.Term algebra.F.id_test ((algebra.Alg.Var 7)::
                   (algebra.Alg.Var 5)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_2 ((algebra.Alg.Var 5)::
      (algebra.Alg.Var 9)::(algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
      (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Var 10)::
      (algebra.Alg.Var 11)::nil))::nil))
   
    (* match_3(l1_,l2_,x_,l'_,a_4_,l_3_,False) -> Pair(Cons(x_,l1_),l2_) *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Term 
                   algebra.F.id_Cons ((algebra.Alg.Var 5)::
                   (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 11)::nil)) 
     (algebra.Alg.Term algebra.F.id_match_3 ((algebra.Alg.Var 10)::
      (algebra.Alg.Var 11)::(algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_False nil)::nil))
   
    (* match_3(l1_,l2_,x_,l'_,a_4_,l_3_,True) -> Pair(l1_,Cons(x_,l2_)) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Var 10)::
                   (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 5)::
                   (algebra.Alg.Var 11)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_3 ((algebra.Alg.Var 10)::
      (algebra.Alg.Var 11)::(algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
      (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::(algebra.Alg.Term 
      algebra.F.id_True nil)::nil))
    (* quick(l_5_) -> match_4(l_5_,l_5_) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_4 
                   ((algebra.Alg.Var 12)::(algebra.Alg.Var 12)::nil)) 
     (algebra.Alg.Term algebra.F.id_quick ((algebra.Alg.Var 12)::nil))
    (* match_4(l_5_,Nil) -> Nil *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_Nil nil) 
     (algebra.Alg.Term algebra.F.id_match_4 ((algebra.Alg.Var 12)::
      (algebra.Alg.Term algebra.F.id_Nil nil)::nil))
   
    (* match_4(l_5_,Cons(a_,l'_)) -> match_5(a_,l'_,l_5_,part(a_,l'_)) *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_match_5 
                   ((algebra.Alg.Var 13)::(algebra.Alg.Var 9)::
                   (algebra.Alg.Var 12)::(algebra.Alg.Term algebra.F.id_part 
                   ((algebra.Alg.Var 13)::(algebra.Alg.Var 9)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_4 ((algebra.Alg.Var 12)::
      (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 13)::
      (algebra.Alg.Var 9)::nil))::nil))
   
    (* match_5(a_,l'_,l_5_,Pair(l1_,l2_)) -> append(quick(l1_),Cons(a_,quick(l2_))) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Term 
                   algebra.F.id_quick ((algebra.Alg.Var 10)::nil))::
                   (algebra.Alg.Term algebra.F.id_Cons 
                   ((algebra.Alg.Var 13)::(algebra.Alg.Term 
                   algebra.F.id_quick 
                   ((algebra.Alg.Var 11)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match_5 ((algebra.Alg.Var 13)::
      (algebra.Alg.Var 9)::(algebra.Alg.Var 12)::(algebra.Alg.Term 
      algebra.F.id_Pair ((algebra.Alg.Var 10)::
      (algebra.Alg.Var 11)::nil))::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_test ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil)),(algebra.Alg.Term algebra.F.id_True nil))::
    nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_test ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil)),(algebra.Alg.Term algebra.F.id_False nil))::
    R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Var 3)::
     (algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_match_0 ((algebra.Alg.Var 3)::
     (algebra.Alg.Var 4)::(algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_1
   .
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_match_0 ((algebra.Alg.Var 3)::
     (algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id_Nil nil)::nil)),
    (algebra.Alg.Var 4))::R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_match_0 ((algebra.Alg.Var 3)::
     (algebra.Alg.Var 4)::(algebra.Alg.Term algebra.F.id_Cons 
     ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 5)::
     (algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 4)::nil))::nil)))::R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_part ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::nil)),
    (algebra.Alg.Term algebra.F.id_match_1 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_4
   .
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_match_1 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_Nil nil)::nil)),
    (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Term algebra.F.id_Nil 
     nil)::(algebra.Alg.Term algebra.F.id_Nil nil)::nil)))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_match_1 ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_Cons 
     ((algebra.Alg.Var 5)::(algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_match_2 ((algebra.Alg.Var 5)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
     (algebra.Alg.Term algebra.F.id_part ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 9)::nil))::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_match_2 ((algebra.Alg.Var 5)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 7)::(algebra.Alg.Var 8)::
     (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Var 10)::
     (algebra.Alg.Var 11)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_match_3 ((algebra.Alg.Var 10)::
     (algebra.Alg.Var 11)::(algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_test ((algebra.Alg.Var 7)::
     (algebra.Alg.Var 5)::nil))::nil)))::R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_match_3 ((algebra.Alg.Var 10)::
     (algebra.Alg.Var 11)::(algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_False nil)::nil)),
    (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Term algebra.F.id_Cons 
     ((algebra.Alg.Var 5)::(algebra.Alg.Var 10)::nil))::
     (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_match_3 ((algebra.Alg.Var 10)::
     (algebra.Alg.Var 11)::(algebra.Alg.Var 5)::(algebra.Alg.Var 9)::
     (algebra.Alg.Var 7)::(algebra.Alg.Var 8)::(algebra.Alg.Term 
     algebra.F.id_True nil)::nil)),
    (algebra.Alg.Term algebra.F.id_Pair ((algebra.Alg.Var 10)::
     (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 5)::
     (algebra.Alg.Var 11)::nil))::nil)))::R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_quick ((algebra.Alg.Var 12)::nil)),
    (algebra.Alg.Term algebra.F.id_match_4 ((algebra.Alg.Var 12)::
     (algebra.Alg.Var 12)::nil)))::R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_match_4 ((algebra.Alg.Var 12)::
     (algebra.Alg.Term algebra.F.id_Nil nil)::nil)),
    (algebra.Alg.Term algebra.F.id_Nil nil))::R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_match_4 ((algebra.Alg.Var 12)::
     (algebra.Alg.Term algebra.F.id_Cons ((algebra.Alg.Var 13)::
     (algebra.Alg.Var 9)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_match_5 ((algebra.Alg.Var 13)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 12)::(algebra.Alg.Term 
     algebra.F.id_part ((algebra.Alg.Var 13)::
     (algebra.Alg.Var 9)::nil))::nil)))::R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_match_5 ((algebra.Alg.Var 13)::
     (algebra.Alg.Var 9)::(algebra.Alg.Var 12)::(algebra.Alg.Term 
     algebra.F.id_Pair ((algebra.Alg.Var 10)::
     (algebra.Alg.Var 11)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Term 
     algebra.F.id_quick ((algebra.Alg.Var 10)::nil))::(algebra.Alg.Term 
     algebra.F.id_Cons ((algebra.Alg.Var 13)::(algebra.Alg.Term 
     algebra.F.id_quick ((algebra.Alg.Var 11)::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_13.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_14.
 
 
 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.or16_equiv in H|-.
   case H;clear H;intros H.
   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_5 (x15 x16 x17 x18 x19:Prop) :
  Prop := 
   | conj_5 : x15->x16->x17->x18->x19->and_5 x15 x16 x17 x18 x19
 .
 
 
 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_5 (t = (algebra.Alg.Term algebra.F.id_False nil) ->
           t' = (algebra.Alg.Term algebra.F.id_False nil)) 
     (forall x16 x18, 
      t = (algebra.Alg.Term algebra.F.id_Cons (x16::x18::nil)) ->
       exists x15,
         exists x17,
           t' = (algebra.Alg.Term algebra.F.id_Cons (x15::x17::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x15 x16)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x17 x18)) 
     (t = (algebra.Alg.Term algebra.F.id_True nil) ->
      t' = (algebra.Alg.Term algebra.F.id_True nil)) 
     (forall x16 x18, 
      t = (algebra.Alg.Term algebra.F.id_Pair (x16::x18::nil)) ->
       exists x15,
         exists x17,
           t' = (algebra.Alg.Term algebra.F.id_Pair (x15::x17::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x15 x16)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x17 x18)) 
     (t = (algebra.Alg.Term algebra.F.id_Nil nil) ->
      t' = (algebra.Alg.Term algebra.F.id_Nil 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 x16 x18 H;exists x16;exists x18;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x16 x18 H;exists x16;exists x18;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_False H_id_Cons H_id_True H_id_Pair H_id_Nil].
   constructor.
   
   clear H_id_Cons H_id_True H_id_Pair H_id_Nil;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_False H_id_True H_id_Pair H_id_Nil;intros x16 x18 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 x16 |- _ =>
      destruct (H_id_Cons y x18 (refl_equal _)) as [x15 [x17]];intros ;
       intuition;exists x15;exists x17;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x18 |- _ =>
      destruct (H_id_Cons x16 y (refl_equal _)) as [x15 [x17]];intros ;
       intuition;exists x15;exists x17;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_False H_id_Cons H_id_Pair H_id_Nil;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_False H_id_Cons H_id_True H_id_Nil;intros x16 x18 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 x16 |- _ =>
      destruct (H_id_Pair y x18 (refl_equal _)) as [x15 [x17]];intros ;
       intuition;exists x15;exists x17;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x18 |- _ =>
      destruct (H_id_Pair x16 y (refl_equal _)) as [x15 [x17]];intros ;
       intuition;exists x15;exists x17;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_False H_id_Cons H_id_True H_id_Pair;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_False_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_False nil) ->
     t' = (algebra.Alg.Term algebra.F.id_False nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_Cons_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x16 x18, 
     t = (algebra.Alg.Term algebra.F.id_Cons (x16::x18::nil)) ->
      exists x15,
        exists x17,
          t' = (algebra.Alg.Term algebra.F.id_Cons (x15::x17::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x15 x16)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x17 x18).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_True_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_True nil) ->
     t' = (algebra.Alg.Term algebra.F.id_True nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_Pair_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x16 x18, 
     t = (algebra.Alg.Term algebra.F.id_Pair (x16::x18::nil)) ->
      exists x15,
        exists x17,
          t' = (algebra.Alg.Term algebra.F.id_Pair (x15::x17::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x15 x16)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x17 x18).
 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.
 
 
 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_False nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_False_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_Cons (?x16::?x15::nil)) |- 
    _ =>
     let x16 := fresh "x" in 
      (let x15 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_Cons_is_R_xml_0_constructor H (refl_equal _)) as 
               [x16 [x15 [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_True nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_True_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_Pair (?x16::?x15::nil)) |- 
    _ =>
     let x16 := fresh "x" in 
      (let x15 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_Pair_is_R_xml_0_constructor H (refl_equal _)) as 
               [x16 [x15 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  impossible_star_reduction_R_xml_0 ))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_Nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_Nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    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_False nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_False_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_Cons (?x16::?x15::nil)) |- 
    _ =>
     let x16 := fresh "x" in 
      (let x15 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_Cons_is_R_xml_0_constructor H (refl_equal _)) as 
               [x16 [x15 [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_True nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_True_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_Pair (?x16::?x15::nil)) |- 
    _ =>
     let x16 := fresh "x" in 
      (let x15 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_Pair_is_R_xml_0_constructor H (refl_equal _)) as 
               [x16 [x15 [Heq [Hred2 Hred1]]]];
                (discriminate Heq)||
                (injection Heq;intros ;subst;clear Heq;clear H;
                  try (simplify_star_reduction_R_xml_0 )))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_Nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_Nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    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_test : A ->A ->A.
   
   Hypothesis P_id_match_1 : A ->A ->A ->A.
   
   Hypothesis P_id_match_0 : A ->A ->A ->A.
   
   Hypothesis P_id_quick : A ->A.
   
   Hypothesis P_id_False : A.
   
   Hypothesis P_id_match_2 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_Cons : A ->A ->A.
   
   Hypothesis P_id_match_5 : A ->A ->A ->A ->A.
   
   Hypothesis P_id_True : A.
   
   Hypothesis P_id_Pair : A ->A ->A.
   
   Hypothesis P_id_Nil : A.
   
   Hypothesis P_id_match_4 : A ->A ->A.
   
   Hypothesis P_id_append : A ->A ->A.
   
   Hypothesis P_id_match_3 : A ->A ->A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_part : A ->A ->A.
   
   Hypothesis P_id_test_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_test x16 x18 <= P_id_test x15 x17.
   
   Hypothesis P_id_match_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (A0 <= x20)/\ (x20 <= x19) ->
      (A0 <= x18)/\ (x18 <= x17) ->
       (A0 <= x16)/\ (x16 <= x15) ->
        P_id_match_1 x16 x18 x20 <= P_id_match_1 x15 x17 x19.
   
   Hypothesis P_id_match_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (A0 <= x20)/\ (x20 <= x19) ->
      (A0 <= x18)/\ (x18 <= x17) ->
       (A0 <= x16)/\ (x16 <= x15) ->
        P_id_match_0 x16 x18 x20 <= P_id_match_0 x15 x17 x19.
   
   Hypothesis P_id_quick_monotonic :
    forall x16 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->P_id_quick x16 <= P_id_quick x15.
   
   Hypothesis P_id_match_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (A0 <= x24)/\ (x24 <= x23) ->
      (A0 <= x22)/\ (x22 <= x21) ->
       (A0 <= x20)/\ (x20 <= x19) ->
        (A0 <= x18)/\ (x18 <= x17) ->
         (A0 <= x16)/\ (x16 <= x15) ->
          P_id_match_2 x16 x18 x20 x22 x24 <= P_id_match_2 x15 x17 x19 
                                               x21 x23.
   
   Hypothesis P_id_Cons_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_Cons x16 x18 <= P_id_Cons x15 x17.
   
   Hypothesis P_id_match_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (A0 <= x22)/\ (x22 <= x21) ->
      (A0 <= x20)/\ (x20 <= x19) ->
       (A0 <= x18)/\ (x18 <= x17) ->
        (A0 <= x16)/\ (x16 <= x15) ->
         P_id_match_5 x16 x18 x20 x22 <= P_id_match_5 x15 x17 x19 x21.
   
   Hypothesis P_id_Pair_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_Pair x16 x18 <= P_id_Pair x15 x17.
   
   Hypothesis P_id_match_4_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->
       P_id_match_4 x16 x18 <= P_id_match_4 x15 x17.
   
   Hypothesis P_id_append_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_append x16 x18 <= P_id_append x15 x17.
   
   Hypothesis P_id_match_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (A0 <= x28)/\ (x28 <= x27) ->
      (A0 <= x26)/\ (x26 <= x25) ->
       (A0 <= x24)/\ (x24 <= x23) ->
        (A0 <= x22)/\ (x22 <= x21) ->
         (A0 <= x20)/\ (x20 <= x19) ->
          (A0 <= x18)/\ (x18 <= x17) ->
           (A0 <= x16)/\ (x16 <= x15) ->
            P_id_match_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_match_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   
   Hypothesis P_id_part_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_part x16 x18 <= P_id_part x15 x17.
   
   Hypothesis P_id_test_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_test x16 x15.
   
   Hypothesis P_id_match_1_bounded :
    forall x16 x17 x15, 
     (A0 <= x15) ->(A0 <= x16) ->(A0 <= x17) ->A0 <= P_id_match_1 x17 x16 x15.
   
   Hypothesis P_id_match_0_bounded :
    forall x16 x17 x15, 
     (A0 <= x15) ->(A0 <= x16) ->(A0 <= x17) ->A0 <= P_id_match_0 x17 x16 x15.
   
   Hypothesis P_id_quick_bounded :
    forall x15, (A0 <= x15) ->A0 <= P_id_quick x15.
   
   Hypothesis P_id_False_bounded : A0 <= P_id_False .
   
   Hypothesis P_id_match_2_bounded :
    forall x16 x18 x17 x19 x15, 
     (A0 <= x15) ->
      (A0 <= x16) ->
       (A0 <= x17) ->
        (A0 <= x18) ->(A0 <= x19) ->A0 <= P_id_match_2 x19 x18 x17 x16 x15.
   
   Hypothesis P_id_Cons_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_Cons x16 x15.
   
   Hypothesis P_id_match_5_bounded :
    forall x16 x18 x17 x15, 
     (A0 <= x15) ->
      (A0 <= x16) ->
       (A0 <= x17) ->(A0 <= x18) ->A0 <= P_id_match_5 x18 x17 x16 x15.
   
   Hypothesis P_id_True_bounded : A0 <= P_id_True .
   
   Hypothesis P_id_Pair_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_Pair x16 x15.
   
   Hypothesis P_id_Nil_bounded : A0 <= P_id_Nil .
   
   Hypothesis P_id_match_4_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_match_4 x16 x15.
   
   Hypothesis P_id_append_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_append x16 x15.
   
   Hypothesis P_id_match_3_bounded :
    forall x16 x20 x18 x17 x21 x19 x15, 
     (A0 <= x15) ->
      (A0 <= x16) ->
       (A0 <= x17) ->
        (A0 <= x18) ->
         (A0 <= x19) ->
          (A0 <= x20) ->
           (A0 <= x21) ->A0 <= P_id_match_3 x21 x20 x19 x18 x17 x16 x15.
   
   Hypothesis P_id_part_bounded :
    forall x16 x15, (A0 <= x15) ->(A0 <= x16) ->A0 <= P_id_part x16 x15.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
        P_id_test (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::nil)) =>
        P_id_match_1 (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::x15::nil)) =>
        P_id_match_0 (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
        P_id_quick (measure x15)
       | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
       | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
          x15::nil)) =>
        P_id_match_2 (measure x19) (measure x18) (measure x17) (measure x16) 
         (measure x15)
       | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
        P_id_Cons (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil)) =>
        P_id_match_5 (measure x18) (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
       | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
        P_id_Pair (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
       | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
        P_id_match_4 (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
        P_id_append (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::x18::x17::
          x16::x15::nil)) =>
        P_id_match_3 (measure x21) (measure x20) (measure x19) (measure x18) 
         (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
        P_id_part (measure x16) (measure x15)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
                    P_id_test (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::
                      x15::nil)) =>
                    P_id_match_1 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::
                      x15::nil)) =>
                    P_id_match_0 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                    P_id_quick (measure x15)
                   | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
                   | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::
                      x16::x15::nil)) =>
                    P_id_match_2 (measure x19) (measure x18) (measure x17) 
                     (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
                    P_id_Cons (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
                      x15::nil)) =>
                    P_id_match_5 (measure x18) (measure x17) (measure x16) 
                     (measure x15)
                   | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
                   | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
                    P_id_Pair (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
                   | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
                    P_id_match_4 (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
                    P_id_append (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::
                      x18::x17::x16::x15::nil)) =>
                    P_id_match_3 (measure x21) (measure x20) (measure x19) 
                     (measure x18) (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
                    P_id_part (measure x16) (measure x15)
                   | _ => 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_test => P_id_test
       | algebra.F.id_match_1 => P_id_match_1
       | algebra.F.id_match_0 => P_id_match_0
       | algebra.F.id_quick => P_id_quick
       | algebra.F.id_False => P_id_False
       | algebra.F.id_match_2 => P_id_match_2
       | algebra.F.id_Cons => P_id_Cons
       | algebra.F.id_match_5 => P_id_match_5
       | algebra.F.id_True => P_id_True
       | algebra.F.id_Pair => P_id_Pair
       | algebra.F.id_Nil => P_id_Nil
       | algebra.F.id_match_4 => P_id_match_4
       | algebra.F.id_append => P_id_append
       | algebra.F.id_match_3 => P_id_match_3
       | algebra.F.id_part => P_id_part
       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_test =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_match_0 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_quick =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_False => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_match_2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_Cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_5 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::_ => _
                 end
              | algebra.F.id_True => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_Pair =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_Nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_match_4 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_append =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_3 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_part =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_test_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_quick_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_False_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_5_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_True_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Pair_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_4_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_append_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_3_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_part_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_test_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_0_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_quick_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_match_2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_Cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_5_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_Pair_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_match_4_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_append_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_3_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_part_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_test_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_quick_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_False_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_5_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_True_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Pair_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_4_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_append_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_3_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_part_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_MATCH_0 : A ->A ->A ->A.
   
   Hypothesis P_id_MATCH_5 : A ->A ->A ->A ->A.
   
   Hypothesis P_id_MATCH_3 : A ->A ->A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_MATCH_1 : A ->A ->A ->A.
   
   Hypothesis P_id_QUICK : A ->A.
   
   Hypothesis P_id_PART : A ->A ->A.
   
   Hypothesis P_id_TEST : A ->A ->A.
   
   Hypothesis P_id_MATCH_2 : A ->A ->A ->A ->A ->A.
   
   Hypothesis P_id_MATCH_4 : A ->A ->A.
   
   Hypothesis P_id_APPEND : A ->A ->A.
   
   Hypothesis P_id_MATCH_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (A0 <= x20)/\ (x20 <= x19) ->
      (A0 <= x18)/\ (x18 <= x17) ->
       (A0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_0 x16 x18 x20 <= P_id_MATCH_0 x15 x17 x19.
   
   Hypothesis P_id_MATCH_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (A0 <= x22)/\ (x22 <= x21) ->
      (A0 <= x20)/\ (x20 <= x19) ->
       (A0 <= x18)/\ (x18 <= x17) ->
        (A0 <= x16)/\ (x16 <= x15) ->
         P_id_MATCH_5 x16 x18 x20 x22 <= P_id_MATCH_5 x15 x17 x19 x21.
   
   Hypothesis P_id_MATCH_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (A0 <= x28)/\ (x28 <= x27) ->
      (A0 <= x26)/\ (x26 <= x25) ->
       (A0 <= x24)/\ (x24 <= x23) ->
        (A0 <= x22)/\ (x22 <= x21) ->
         (A0 <= x20)/\ (x20 <= x19) ->
          (A0 <= x18)/\ (x18 <= x17) ->
           (A0 <= x16)/\ (x16 <= x15) ->
            P_id_MATCH_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_MATCH_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   
   Hypothesis P_id_MATCH_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (A0 <= x20)/\ (x20 <= x19) ->
      (A0 <= x18)/\ (x18 <= x17) ->
       (A0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_1 x16 x18 x20 <= P_id_MATCH_1 x15 x17 x19.
   
   Hypothesis P_id_QUICK_monotonic :
    forall x16 x15, 
     (A0 <= x16)/\ (x16 <= x15) ->P_id_QUICK x16 <= P_id_QUICK x15.
   
   Hypothesis P_id_PART_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_PART x16 x18 <= P_id_PART x15 x17.
   
   Hypothesis P_id_TEST_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_TEST x16 x18 <= P_id_TEST x15 x17.
   
   Hypothesis P_id_MATCH_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (A0 <= x24)/\ (x24 <= x23) ->
      (A0 <= x22)/\ (x22 <= x21) ->
       (A0 <= x20)/\ (x20 <= x19) ->
        (A0 <= x18)/\ (x18 <= x17) ->
         (A0 <= x16)/\ (x16 <= x15) ->
          P_id_MATCH_2 x16 x18 x20 x22 x24 <= P_id_MATCH_2 x15 x17 x19 
                                               x21 x23.
   
   Hypothesis P_id_MATCH_4_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->
       P_id_MATCH_4 x16 x18 <= P_id_MATCH_4 x15 x17.
   
   Hypothesis P_id_APPEND_monotonic :
    forall x16 x18 x17 x15, 
     (A0 <= x18)/\ (x18 <= x17) ->
      (A0 <= x16)/\ (x16 <= x15) ->P_id_APPEND x16 x18 <= P_id_APPEND x15 x17.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::x15::nil)) =>
        P_id_MATCH_0 (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil)) =>
        P_id_MATCH_5 (measure x18) (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::x18::x17::
          x16::x15::nil)) =>
        P_id_MATCH_3 (measure x21) (measure x20) (measure x19) (measure x18) 
         (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::nil)) =>
        P_id_MATCH_1 (measure x17) (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
        P_id_QUICK (measure x15)
       | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
        P_id_PART (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
        P_id_TEST (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
          x15::nil)) =>
        P_id_MATCH_2 (measure x19) (measure x18) (measure x17) (measure x16) 
         (measure x15)
       | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
        P_id_MATCH_4 (measure x16) (measure x15)
       | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
        P_id_APPEND (measure x16) (measure x15)
       | _ => 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 x16 x15;apply (P_id_PART x16 x15).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     
     vm_compute in |-*;intros x21 x20 x19 x18 x17 x16 x15;
      apply (P_id_MATCH_3 x21 x20 x19 x18 x17 x16 x15).
     
     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 x16 x15;apply (P_id_APPEND x16 x15).
     
     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 x16 x15;apply (P_id_MATCH_4 x16 x15).
     
     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 x18 x17 x16 x15;
      apply (P_id_MATCH_5 x18 x17 x16 x15).
     
     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 x19 x18 x17 x16 x15;
      apply (P_id_MATCH_2 x19 x18 x17 x16 x15).
     
     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 x15;apply (P_id_QUICK x15).
     
     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 x17 x16 x15;apply (P_id_MATCH_0 x17 x16 x15).
     
     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 x17 x16 x15;apply (P_id_MATCH_1 x17 x16 x15).
     
     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 x16 x15;apply (P_id_TEST x16 x15).
     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_test =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_match_0 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_quick =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_False => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_match_2 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_Cons =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_5 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::_ => _
                 end
              | algebra.F.id_True => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_Pair =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_Nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_match_4 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_append =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_match_3 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::nil => _
                 | _::_::_::_::_::nil => _
                 | _::_::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_::nil => _
                 | _::_::_::_::_::_::_::_::_ => _
                 end
              | algebra.F.id_part =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              end.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     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 .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     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 .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     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 .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match_0 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_0 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_5 (x18::
                             x17::x16::x15::nil)) =>
                           P_id_MATCH_5 (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_3 (x21::
                             x20::x19::x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_3 (measure x21) (measure x20) 
                            (measure x19) (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_1 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_1 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                           P_id_QUICK (measure x15)
                          | (algebra.Alg.Term algebra.F.id_part (x16::
                             x15::nil)) =>
                           P_id_PART (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_test (x16::
                             x15::nil)) =>
                           P_id_TEST (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_2 (x19::
                             x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_2 (measure x19) (measure x18) 
                            (measure x17) (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_4 (x16::
                             x15::nil)) =>
                           P_id_MATCH_4 (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_append (x16::
                             x15::nil)) =>
                           P_id_APPEND (measure x16) (measure x15)
                          | _ => 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_test_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_0_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_quick_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_match_2_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_Cons_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_5_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_Pair_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_match_4_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_append_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_3_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_part_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_test_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_quick_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_False_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_5_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_True_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Pair_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_Nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_4_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_append_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_3_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_part_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_PART_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_MATCH_3_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_APPEND_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_MATCH_4_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_MATCH_5_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_MATCH_2_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_QUICK_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_MATCH_0_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_MATCH_1_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_TEST_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_test : Z ->Z ->Z.
   
   Hypothesis P_id_match_1 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_match_0 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_quick : Z ->Z.
   
   Hypothesis P_id_False : Z.
   
   Hypothesis P_id_match_2 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_Cons : Z ->Z ->Z.
   
   Hypothesis P_id_match_5 : Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_True : Z.
   
   Hypothesis P_id_Pair : Z ->Z ->Z.
   
   Hypothesis P_id_Nil : Z.
   
   Hypothesis P_id_match_4 : Z ->Z ->Z.
   
   Hypothesis P_id_append : Z ->Z ->Z.
   
   Hypothesis P_id_match_3 : Z ->Z ->Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_part : Z ->Z ->Z.
   
   Hypothesis P_id_test_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_test x16 x18 <= P_id_test x15 x17.
   
   Hypothesis P_id_match_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (min_value <= x20)/\ (x20 <= x19) ->
      (min_value <= x18)/\ (x18 <= x17) ->
       (min_value <= x16)/\ (x16 <= x15) ->
        P_id_match_1 x16 x18 x20 <= P_id_match_1 x15 x17 x19.
   
   Hypothesis P_id_match_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (min_value <= x20)/\ (x20 <= x19) ->
      (min_value <= x18)/\ (x18 <= x17) ->
       (min_value <= x16)/\ (x16 <= x15) ->
        P_id_match_0 x16 x18 x20 <= P_id_match_0 x15 x17 x19.
   
   Hypothesis P_id_quick_monotonic :
    forall x16 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->P_id_quick x16 <= P_id_quick x15.
   
   Hypothesis P_id_match_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (min_value <= x24)/\ (x24 <= x23) ->
      (min_value <= x22)/\ (x22 <= x21) ->
       (min_value <= x20)/\ (x20 <= x19) ->
        (min_value <= x18)/\ (x18 <= x17) ->
         (min_value <= x16)/\ (x16 <= x15) ->
          P_id_match_2 x16 x18 x20 x22 x24 <= P_id_match_2 x15 x17 x19 
                                               x21 x23.
   
   Hypothesis P_id_Cons_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_Cons x16 x18 <= P_id_Cons x15 x17.
   
   Hypothesis P_id_match_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (min_value <= x22)/\ (x22 <= x21) ->
      (min_value <= x20)/\ (x20 <= x19) ->
       (min_value <= x18)/\ (x18 <= x17) ->
        (min_value <= x16)/\ (x16 <= x15) ->
         P_id_match_5 x16 x18 x20 x22 <= P_id_match_5 x15 x17 x19 x21.
   
   Hypothesis P_id_Pair_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_Pair x16 x18 <= P_id_Pair x15 x17.
   
   Hypothesis P_id_match_4_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_match_4 x16 x18 <= P_id_match_4 x15 x17.
   
   Hypothesis P_id_append_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_append x16 x18 <= P_id_append x15 x17.
   
   Hypothesis P_id_match_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (min_value <= x28)/\ (x28 <= x27) ->
      (min_value <= x26)/\ (x26 <= x25) ->
       (min_value <= x24)/\ (x24 <= x23) ->
        (min_value <= x22)/\ (x22 <= x21) ->
         (min_value <= x20)/\ (x20 <= x19) ->
          (min_value <= x18)/\ (x18 <= x17) ->
           (min_value <= x16)/\ (x16 <= x15) ->
            P_id_match_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_match_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   
   Hypothesis P_id_part_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_part x16 x18 <= P_id_part x15 x17.
   
   Hypothesis P_id_test_bounded :
    forall x16 x15, 
     (min_value <= x15) ->(min_value <= x16) ->min_value <= P_id_test x16 x15.
   
   Hypothesis P_id_match_1_bounded :
    forall x16 x17 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->
       (min_value <= x17) ->min_value <= P_id_match_1 x17 x16 x15.
   
   Hypothesis P_id_match_0_bounded :
    forall x16 x17 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->
       (min_value <= x17) ->min_value <= P_id_match_0 x17 x16 x15.
   
   Hypothesis P_id_quick_bounded :
    forall x15, (min_value <= x15) ->min_value <= P_id_quick x15.
   
   Hypothesis P_id_False_bounded : min_value <= P_id_False .
   
   Hypothesis P_id_match_2_bounded :
    forall x16 x18 x17 x19 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->
       (min_value <= x17) ->
        (min_value <= x18) ->
         (min_value <= x19) ->min_value <= P_id_match_2 x19 x18 x17 x16 x15.
   
   Hypothesis P_id_Cons_bounded :
    forall x16 x15, 
     (min_value <= x15) ->(min_value <= x16) ->min_value <= P_id_Cons x16 x15.
   
   Hypothesis P_id_match_5_bounded :
    forall x16 x18 x17 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->
       (min_value <= x17) ->
        (min_value <= x18) ->min_value <= P_id_match_5 x18 x17 x16 x15.
   
   Hypothesis P_id_True_bounded : min_value <= P_id_True .
   
   Hypothesis P_id_Pair_bounded :
    forall x16 x15, 
     (min_value <= x15) ->(min_value <= x16) ->min_value <= P_id_Pair x16 x15.
   
   Hypothesis P_id_Nil_bounded : min_value <= P_id_Nil .
   
   Hypothesis P_id_match_4_bounded :
    forall x16 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->min_value <= P_id_match_4 x16 x15.
   
   Hypothesis P_id_append_bounded :
    forall x16 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->min_value <= P_id_append x16 x15.
   
   Hypothesis P_id_match_3_bounded :
    forall x16 x20 x18 x17 x21 x19 x15, 
     (min_value <= x15) ->
      (min_value <= x16) ->
       (min_value <= x17) ->
        (min_value <= x18) ->
         (min_value <= x19) ->
          (min_value <= x20) ->
           (min_value <= x21) ->
            min_value <= P_id_match_3 x21 x20 x19 x18 x17 x16 x15.
   
   Hypothesis P_id_part_bounded :
    forall x16 x15, 
     (min_value <= x15) ->(min_value <= x16) ->min_value <= P_id_part x16 x15.
   
   Definition measure  := 
     Interp.measure min_value P_id_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
                    P_id_test (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::
                      x15::nil)) =>
                    P_id_match_1 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::
                      x15::nil)) =>
                    P_id_match_0 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                    P_id_quick (measure x15)
                   | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
                   | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::
                      x16::x15::nil)) =>
                    P_id_match_2 (measure x19) (measure x18) (measure x17) 
                     (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
                    P_id_Cons (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
                      x15::nil)) =>
                    P_id_match_5 (measure x18) (measure x17) (measure x16) 
                     (measure x15)
                   | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
                   | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
                    P_id_Pair (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
                   | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
                    P_id_match_4 (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
                    P_id_append (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::
                      x18::x17::x16::x15::nil)) =>
                    P_id_match_3 (measure x21) (measure x20) (measure x19) 
                     (measure x18) (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
                    P_id_part (measure x16) (measure x15)
                   | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_MATCH_0 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_5 : Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_3 : Z ->Z ->Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_1 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_QUICK : Z ->Z.
   
   Hypothesis P_id_PART : Z ->Z ->Z.
   
   Hypothesis P_id_TEST : Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_2 : Z ->Z ->Z ->Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_4 : Z ->Z ->Z.
   
   Hypothesis P_id_APPEND : Z ->Z ->Z.
   
   Hypothesis P_id_MATCH_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (min_value <= x20)/\ (x20 <= x19) ->
      (min_value <= x18)/\ (x18 <= x17) ->
       (min_value <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_0 x16 x18 x20 <= P_id_MATCH_0 x15 x17 x19.
   
   Hypothesis P_id_MATCH_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (min_value <= x22)/\ (x22 <= x21) ->
      (min_value <= x20)/\ (x20 <= x19) ->
       (min_value <= x18)/\ (x18 <= x17) ->
        (min_value <= x16)/\ (x16 <= x15) ->
         P_id_MATCH_5 x16 x18 x20 x22 <= P_id_MATCH_5 x15 x17 x19 x21.
   
   Hypothesis P_id_MATCH_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (min_value <= x28)/\ (x28 <= x27) ->
      (min_value <= x26)/\ (x26 <= x25) ->
       (min_value <= x24)/\ (x24 <= x23) ->
        (min_value <= x22)/\ (x22 <= x21) ->
         (min_value <= x20)/\ (x20 <= x19) ->
          (min_value <= x18)/\ (x18 <= x17) ->
           (min_value <= x16)/\ (x16 <= x15) ->
            P_id_MATCH_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_MATCH_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   
   Hypothesis P_id_MATCH_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (min_value <= x20)/\ (x20 <= x19) ->
      (min_value <= x18)/\ (x18 <= x17) ->
       (min_value <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_1 x16 x18 x20 <= P_id_MATCH_1 x15 x17 x19.
   
   Hypothesis P_id_QUICK_monotonic :
    forall x16 x15, 
     (min_value <= x16)/\ (x16 <= x15) ->P_id_QUICK x16 <= P_id_QUICK x15.
   
   Hypothesis P_id_PART_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_PART x16 x18 <= P_id_PART x15 x17.
   
   Hypothesis P_id_TEST_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_TEST x16 x18 <= P_id_TEST x15 x17.
   
   Hypothesis P_id_MATCH_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (min_value <= x24)/\ (x24 <= x23) ->
      (min_value <= x22)/\ (x22 <= x21) ->
       (min_value <= x20)/\ (x20 <= x19) ->
        (min_value <= x18)/\ (x18 <= x17) ->
         (min_value <= x16)/\ (x16 <= x15) ->
          P_id_MATCH_2 x16 x18 x20 x22 x24 <= P_id_MATCH_2 x15 x17 x19 
                                               x21 x23.
   
   Hypothesis P_id_MATCH_4_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_MATCH_4 x16 x18 <= P_id_MATCH_4 x15 x17.
   
   Hypothesis P_id_APPEND_monotonic :
    forall x16 x18 x17 x15, 
     (min_value <= x18)/\ (x18 <= x17) ->
      (min_value <= x16)/\ (x16 <= x15) ->
       P_id_APPEND x16 x18 <= P_id_APPEND x15 x17.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_test P_id_match_1 P_id_match_0 
      P_id_quick P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True 
      P_id_Pair P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part 
      P_id_MATCH_0 P_id_MATCH_5 P_id_MATCH_3 P_id_MATCH_1 P_id_QUICK 
      P_id_PART P_id_TEST P_id_MATCH_2 P_id_MATCH_4 P_id_APPEND.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match_0 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_0 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_5 (x18::
                             x17::x16::x15::nil)) =>
                           P_id_MATCH_5 (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_3 (x21::
                             x20::x19::x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_3 (measure x21) (measure x20) 
                            (measure x19) (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_1 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_1 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                           P_id_QUICK (measure x15)
                          | (algebra.Alg.Term algebra.F.id_part (x16::
                             x15::nil)) =>
                           P_id_PART (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_test (x16::
                             x15::nil)) =>
                           P_id_TEST (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_2 (x19::
                             x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_2 (measure x19) (measure x18) 
                            (measure x17) (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_4 (x16::
                             x15::nil)) =>
                           P_id_MATCH_4 (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_append (x16::
                             x15::nil)) =>
                           P_id_APPEND (measure x16) (measure x15)
                          | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_0_monotonic;assumption.
     intros ;apply P_id_MATCH_5_monotonic;assumption.
     intros ;apply P_id_MATCH_3_monotonic;assumption.
     intros ;apply P_id_MATCH_1_monotonic;assumption.
     intros ;apply P_id_QUICK_monotonic;assumption.
     intros ;apply P_id_PART_monotonic;assumption.
     intros ;apply P_id_TEST_monotonic;assumption.
     intros ;apply P_id_MATCH_2_monotonic;assumption.
     intros ;apply P_id_MATCH_4_monotonic;assumption.
     intros ;apply P_id_APPEND_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 := 
    (* <append(l1_2_,l2_1_),match_0(l1_2_,l2_1_,l1_2_)> *)
   | DP_R_xml_0_0 :
    forall x16 x4 x3 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x3 x16) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x15) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_0 (x3::x4::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_append (x16::x15::nil))
    (* <match_0(l1_2_,l2_1_,Cons(x_,l_)),append(l_,l2_1_)> *)
   | DP_R_xml_0_1 :
    forall x16 x4 x6 x17 x5 x3 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x3 x17) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_Cons (x5::x6::nil)) x15) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_append (x6::x4::nil)) 
         (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::x15::nil))
    (* <part(a_4_,l_3_),match_1(a_4_,l_3_,l_3_)> *)
   | DP_R_xml_0_2 :
    forall x16 x8 x7 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x7 x16) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x8 x15) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_1 (x7::x8::x8::nil)) 
        (algebra.Alg.Term algebra.F.id_part (x16::x15::nil))
   
    (* <match_1(a_4_,l_3_,Cons(x_,l'_)),match_2(x_,l'_,a_4_,l_3_,part(a_4_,l'_))> *)
   | DP_R_xml_0_3 :
    forall x16 x8 x17 x9 x5 x7 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x7 x17) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x8 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_Cons (x5::x9::nil)) x15) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_2 (x5::x9::x7::x8::
                    (algebra.Alg.Term algebra.F.id_part (x7::x9::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::nil))
    (* <match_1(a_4_,l_3_,Cons(x_,l'_)),part(a_4_,l'_)> *)
   | DP_R_xml_0_4 :
    forall x16 x8 x17 x9 x5 x7 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x7 x17) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x8 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_Cons (x5::x9::nil)) x15) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_part (x7::x9::nil)) 
         (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::nil))
   
    (* <match_2(x_,l'_,a_4_,l_3_,Pair(l1_,l2_)),match_3(l1_,l2_,x_,l'_,a_4_,l_3_,test(a_4_,x_))> *)
   | DP_R_xml_0_5 :
    forall x16 x8 x18 x10 x17 x9 x5 x19 x11 x7 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x5 x19) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x9 x18) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x7 x17) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x8 x16) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_3 (x10::x11::x5::
                      x9::x7::x8::(algebra.Alg.Term algebra.F.id_test (x7::
                      x5::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
            x15::nil))
   
    (* <match_2(x_,l'_,a_4_,l_3_,Pair(l1_,l2_)),test(a_4_,x_)> *)
   | DP_R_xml_0_6 :
    forall x16 x8 x18 x10 x17 x9 x5 x19 x11 x7 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x5 x19) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x9 x18) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x7 x17) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x8 x16) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
          DP_R_xml_0 (algebra.Alg.Term algebra.F.id_test (x7::x5::nil)) 
           (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
            x15::nil))
    (* <quick(l_5_),match_4(l_5_,l_5_)> *)
   | DP_R_xml_0_7 :
    forall x12 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x12 x15) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_4 (x12::x12::nil)) 
       (algebra.Alg.Term algebra.F.id_quick (x15::nil))
   
    (* <match_4(l_5_,Cons(a_,l'_)),match_5(a_,l'_,l_5_,part(a_,l'_))> *)
   | DP_R_xml_0_8 :
    forall x16 x12 x9 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x12 x16) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_Cons (x13::x9::nil)) x15) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match_5 (x13::x9::x12::
                   (algebra.Alg.Term algebra.F.id_part (x13::x9::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil))
    (* <match_4(l_5_,Cons(a_,l'_)),part(a_,l'_)> *)
   | DP_R_xml_0_9 :
    forall x16 x12 x9 x13 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x12 x16) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_Cons (x13::x9::nil)) x15) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_part (x13::x9::nil)) 
        (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil))
   
    (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),append(quick(l1_),Cons(a_,quick(l2_)))> *)
   | DP_R_xml_0_10 :
    forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x13 x18) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x9 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
         DP_R_xml_0 (algebra.Alg.Term algebra.F.id_append ((algebra.Alg.Term 
                     algebra.F.id_quick (x10::nil))::(algebra.Alg.Term 
                     algebra.F.id_Cons (x13::(algebra.Alg.Term 
                     algebra.F.id_quick (x11::nil))::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
    (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l1_)> *)
   | DP_R_xml_0_11 :
    forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x13 x18) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x9 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
         DP_R_xml_0 (algebra.Alg.Term algebra.F.id_quick (x10::nil)) 
          (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
    (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l2_)> *)
   | DP_R_xml_0_12 :
    forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x13 x18) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x9 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
         DP_R_xml_0 (algebra.Alg.Term algebra.F.id_quick (x11::nil)) 
          (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::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 := 
     (* <match_2(x_,l'_,a_4_,l_3_,Pair(l1_,l2_)),test(a_4_,x_)> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x16 x8 x18 x10 x17 x9 x5 x19 x11 x7 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x5 x19) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x9 x18) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x7 x17) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x8 x16) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
           DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_test (x7::
                                 x5::nil)) 
            (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
             x15::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 := 
     (* <match_2(x_,l'_,a_4_,l_3_,Pair(l1_,l2_)),match_3(l1_,l2_,x_,l'_,a_4_,l_3_,test(a_4_,x_))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x16 x8 x18 x10 x17 x9 x5 x19 x11 x7 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x5 x19) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x9 x18) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x7 x17) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x8 x16) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
           DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_match_3 (x10::
                                 x11::x5::x9::x7::x8::(algebra.Alg.Term 
                                 algebra.F.id_test (x7::x5::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::x16::
             x15::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 := 
     (* <match_1(a_4_,l_3_,Cons(x_,l'_)),match_2(x_,l'_,a_4_,l_3_,part(a_4_,l'_))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x16 x8 x17 x9 x5 x7 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x7 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Cons (x5::x9::nil)) x15) ->
         DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_match_2 (x5::
                               x9::x7::x8::(algebra.Alg.Term 
                               algebra.F.id_part (x7::x9::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::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;
      (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' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match_1(a_4_,l_3_,Cons(x_,l'_)),part(a_4_,l'_)> *)
    | DP_R_xml_0_scc_4_0 :
     forall x16 x8 x17 x9 x5 x7 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x7 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Cons (x5::x9::nil)) x15) ->
         DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_part (x7::x9::nil)) 
          (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::x15::nil))
     (* <part(a_4_,l_3_),match_1(a_4_,l_3_,l_3_)> *)
    | DP_R_xml_0_scc_4_1 :
     forall x16 x8 x7 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x7 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x8 x15) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_match_1 (x7::x8::
                          x8::nil)) 
         (algebra.Alg.Term algebra.F.id_part (x16::x15::nil))
  .
  
  
  Module WF_DP_R_xml_0_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_test (x15:Z) (x16:Z) := 1.
   
   Definition P_id_match_1 (x15:Z) (x16:Z) (x17:Z) := 1 + 1* x17.
   
   Definition P_id_match_0 (x15:Z) (x16:Z) (x17:Z) := 1* x16 + 1* x17.
   
   Definition P_id_quick (x15:Z) := 1* x15.
   
   Definition P_id_False  := 1.
   
   Definition P_id_match_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 
     2 + 1* x19.
   
   Definition P_id_Cons (x15:Z) (x16:Z) := 2 + 1* x16.
   
   Definition P_id_match_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 1 + 1* x18.
   
   Definition P_id_True  := 1.
   
   Definition P_id_Pair (x15:Z) (x16:Z) := 1 + 1* x15 + 1* x16.
   
   Definition P_id_Nil  := 0.
   
   Definition P_id_match_4 (x15:Z) (x16:Z) := 1* x16.
   
   Definition P_id_append (x15:Z) (x16:Z) := 1* x15 + 1* x16.
   
   Definition P_id_match_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 1 + 1* x15 + 1* x16 + 2* x21.
   
   Definition P_id_part (x15:Z) (x16:Z) := 1 + 1* x16.
   
   Lemma P_id_test_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_test x16 x18 <= P_id_test x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_1 x16 x18 x20 <= P_id_match_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_0 x16 x18 x20 <= P_id_match_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_quick x16 <= P_id_quick x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_match_2 x16 x18 x20 x22 x24 <= P_id_match_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Cons x16 x18 <= P_id_Cons x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_match_5 x16 x18 x20 x22 <= P_id_match_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Pair x16 x18 <= P_id_Pair x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_match_4 x16 x18 <= P_id_match_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_append x16 x18 <= P_id_append x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_match_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_match_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_part x16 x18 <= P_id_part x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_test_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_test x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_1 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_0 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_bounded : forall x15, (0 <= x15) ->0 <= P_id_quick x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_False_bounded : 0 <= P_id_False .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_bounded :
    forall x16 x18 x17 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->(0 <= x19) ->0 <= P_id_match_2 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Cons x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_bounded :
    forall x16 x18 x17 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->(0 <= x18) ->0 <= P_id_match_5 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_True_bounded : 0 <= P_id_True .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Pair x16 x15.
   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_match_4_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_match_4 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_append x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_bounded :
    forall x16 x20 x18 x17 x21 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->
         (0 <= x19) ->
          (0 <= x20) ->
           (0 <= x21) ->0 <= P_id_match_3 x21 x20 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_part x16 x15.
   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_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
                    P_id_test (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::
                      x15::nil)) =>
                    P_id_match_1 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::
                      x15::nil)) =>
                    P_id_match_0 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                    P_id_quick (measure x15)
                   | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
                   | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::
                      x16::x15::nil)) =>
                    P_id_match_2 (measure x19) (measure x18) (measure x17) 
                     (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
                    P_id_Cons (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
                      x15::nil)) =>
                    P_id_match_5 (measure x18) (measure x17) (measure x16) 
                     (measure x15)
                   | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
                   | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
                    P_id_Pair (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
                   | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
                    P_id_match_4 (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
                    P_id_append (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::
                      x18::x17::x16::x15::nil)) =>
                    P_id_match_3 (measure x21) (measure x20) (measure x19) 
                     (measure x18) (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
                    P_id_part (measure x16) (measure x15)
                   | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH_0 (x15:Z) (x16:Z) (x17:Z) := 0.
   
   Definition P_id_MATCH_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 0.
   
   Definition P_id_MATCH_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 0.
   
   Definition P_id_MATCH_1 (x15:Z) (x16:Z) (x17:Z) := 2 + 1* x17.
   
   Definition P_id_QUICK (x15:Z) := 0.
   
   Definition P_id_PART (x15:Z) (x16:Z) := 3 + 1* x16.
   
   Definition P_id_TEST (x15:Z) (x16:Z) := 0.
   
   Definition P_id_MATCH_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 0.
   
   Definition P_id_MATCH_4 (x15:Z) (x16:Z) := 0.
   
   Definition P_id_APPEND (x15:Z) (x16:Z) := 0.
   
   Lemma P_id_MATCH_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_0 x16 x18 x20 <= P_id_MATCH_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_MATCH_5 x16 x18 x20 x22 <= P_id_MATCH_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_MATCH_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_MATCH_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_1 x16 x18 x20 <= P_id_MATCH_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_QUICK_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_QUICK x16 <= P_id_QUICK x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PART_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_PART x16 x18 <= P_id_PART x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TEST_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_TEST x16 x18 <= P_id_TEST x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_MATCH_2 x16 x18 x20 x22 x24 <= P_id_MATCH_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_MATCH_4 x16 x18 <= P_id_MATCH_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_APPEND_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_APPEND x16 x18 <= P_id_APPEND x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part P_id_MATCH_0 
      P_id_MATCH_5 P_id_MATCH_3 P_id_MATCH_1 P_id_QUICK P_id_PART P_id_TEST 
      P_id_MATCH_2 P_id_MATCH_4 P_id_APPEND.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match_0 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_0 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_5 (x18::
                             x17::x16::x15::nil)) =>
                           P_id_MATCH_5 (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_3 (x21::
                             x20::x19::x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_3 (measure x21) (measure x20) 
                            (measure x19) (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_1 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_1 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                           P_id_QUICK (measure x15)
                          | (algebra.Alg.Term algebra.F.id_part (x16::
                             x15::nil)) =>
                           P_id_PART (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_test (x16::
                             x15::nil)) =>
                           P_id_TEST (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_2 (x19::
                             x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_2 (measure x19) (measure x18) 
                            (measure x17) (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_4 (x16::
                             x15::nil)) =>
                           P_id_MATCH_4 (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_append (x16::
                             x15::nil)) =>
                           P_id_APPEND (measure x16) (measure x15)
                          | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_0_monotonic;assumption.
     intros ;apply P_id_MATCH_5_monotonic;assumption.
     intros ;apply P_id_MATCH_3_monotonic;assumption.
     intros ;apply P_id_MATCH_1_monotonic;assumption.
     intros ;apply P_id_QUICK_monotonic;assumption.
     intros ;apply P_id_PART_monotonic;assumption.
     intros ;apply P_id_TEST_monotonic;assumption.
     intros ;apply P_id_MATCH_2_monotonic;assumption.
     intros ;apply P_id_MATCH_4_monotonic;assumption.
     intros ;apply P_id_APPEND_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_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_4.
  
  Definition wf_DP_R_xml_0_scc_4  := WF_DP_R_xml_0_scc_4.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_4 :
   forall x y, (DP_R_xml_0_scc_4 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_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_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
    apply wf_DP_R_xml_0_scc_4.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match_4(l_5_,Cons(a_,l'_)),part(a_,l'_)> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall x16 x12 x9 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x12 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_Cons (x13::x9::nil)) x15) ->
        DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_part (x13::
                              x9::nil)) 
         (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_5 :
   forall x y, 
    (DP_R_xml_0_non_scc_5 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_6  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match_0(l1_2_,l2_1_,Cons(x_,l_)),append(l_,l2_1_)> *)
    | DP_R_xml_0_scc_6_0 :
     forall x16 x4 x6 x17 x5 x3 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x17) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Cons (x5::x6::nil)) x15) ->
         DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_append (x6::
                           x4::nil)) 
          (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::x15::nil))
    
     (* <append(l1_2_,l2_1_),match_0(l1_2_,l2_1_,l1_2_)> *)
    | DP_R_xml_0_scc_6_1 :
     forall x16 x4 x3 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x15) ->
        DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_match_0 (x3::x4::
                          x3::nil)) 
         (algebra.Alg.Term algebra.F.id_append (x16::x15::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_6.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_test (x15:Z) (x16:Z) := 1.
   
   Definition P_id_match_1 (x15:Z) (x16:Z) (x17:Z) := 1 + 1* x17.
   
   Definition P_id_match_0 (x15:Z) (x16:Z) (x17:Z) := 1* x16 + 1* x17.
   
   Definition P_id_quick (x15:Z) := 1* x15.
   
   Definition P_id_False  := 1.
   
   Definition P_id_match_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 
     2 + 1* x19.
   
   Definition P_id_Cons (x15:Z) (x16:Z) := 2 + 1* x16.
   
   Definition P_id_match_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 1 + 1* x18.
   
   Definition P_id_True  := 1.
   
   Definition P_id_Pair (x15:Z) (x16:Z) := 1 + 1* x15 + 1* x16.
   
   Definition P_id_Nil  := 0.
   
   Definition P_id_match_4 (x15:Z) (x16:Z) := 1* x16.
   
   Definition P_id_append (x15:Z) (x16:Z) := 1* x15 + 1* x16.
   
   Definition P_id_match_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 1 + 1* x15 + 1* x16 + 2* x21.
   
   Definition P_id_part (x15:Z) (x16:Z) := 1 + 1* x16.
   
   Lemma P_id_test_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_test x16 x18 <= P_id_test x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_1 x16 x18 x20 <= P_id_match_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_0 x16 x18 x20 <= P_id_match_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_quick x16 <= P_id_quick x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_match_2 x16 x18 x20 x22 x24 <= P_id_match_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Cons x16 x18 <= P_id_Cons x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_match_5 x16 x18 x20 x22 <= P_id_match_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Pair x16 x18 <= P_id_Pair x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_match_4 x16 x18 <= P_id_match_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_append x16 x18 <= P_id_append x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_match_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_match_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_part x16 x18 <= P_id_part x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_test_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_test x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_1 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_0 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_bounded : forall x15, (0 <= x15) ->0 <= P_id_quick x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_False_bounded : 0 <= P_id_False .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_bounded :
    forall x16 x18 x17 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->(0 <= x19) ->0 <= P_id_match_2 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Cons x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_bounded :
    forall x16 x18 x17 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->(0 <= x18) ->0 <= P_id_match_5 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_True_bounded : 0 <= P_id_True .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Pair x16 x15.
   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_match_4_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_match_4 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_append x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_bounded :
    forall x16 x20 x18 x17 x21 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->
         (0 <= x19) ->
          (0 <= x20) ->
           (0 <= x21) ->0 <= P_id_match_3 x21 x20 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_part x16 x15.
   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_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
                    P_id_test (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::
                      x15::nil)) =>
                    P_id_match_1 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::
                      x15::nil)) =>
                    P_id_match_0 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                    P_id_quick (measure x15)
                   | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
                   | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::
                      x16::x15::nil)) =>
                    P_id_match_2 (measure x19) (measure x18) (measure x17) 
                     (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
                    P_id_Cons (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
                      x15::nil)) =>
                    P_id_match_5 (measure x18) (measure x17) (measure x16) 
                     (measure x15)
                   | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
                   | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
                    P_id_Pair (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
                   | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
                    P_id_match_4 (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
                    P_id_append (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::
                      x18::x17::x16::x15::nil)) =>
                    P_id_match_3 (measure x21) (measure x20) (measure x19) 
                     (measure x18) (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
                    P_id_part (measure x16) (measure x15)
                   | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH_0 (x15:Z) (x16:Z) (x17:Z) := 2 + 1* x17.
   
   Definition P_id_MATCH_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 0.
   
   Definition P_id_MATCH_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 0.
   
   Definition P_id_MATCH_1 (x15:Z) (x16:Z) (x17:Z) := 0.
   
   Definition P_id_QUICK (x15:Z) := 0.
   
   Definition P_id_PART (x15:Z) (x16:Z) := 0.
   
   Definition P_id_TEST (x15:Z) (x16:Z) := 0.
   
   Definition P_id_MATCH_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 0.
   
   Definition P_id_MATCH_4 (x15:Z) (x16:Z) := 0.
   
   Definition P_id_APPEND (x15:Z) (x16:Z) := 3 + 1* x15.
   
   Lemma P_id_MATCH_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_0 x16 x18 x20 <= P_id_MATCH_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_MATCH_5 x16 x18 x20 x22 <= P_id_MATCH_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_MATCH_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_MATCH_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_1 x16 x18 x20 <= P_id_MATCH_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_QUICK_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_QUICK x16 <= P_id_QUICK x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PART_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_PART x16 x18 <= P_id_PART x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TEST_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_TEST x16 x18 <= P_id_TEST x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_MATCH_2 x16 x18 x20 x22 x24 <= P_id_MATCH_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_MATCH_4 x16 x18 <= P_id_MATCH_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_APPEND_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_APPEND x16 x18 <= P_id_APPEND x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part P_id_MATCH_0 
      P_id_MATCH_5 P_id_MATCH_3 P_id_MATCH_1 P_id_QUICK P_id_PART P_id_TEST 
      P_id_MATCH_2 P_id_MATCH_4 P_id_APPEND.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match_0 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_0 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_5 (x18::
                             x17::x16::x15::nil)) =>
                           P_id_MATCH_5 (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_3 (x21::
                             x20::x19::x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_3 (measure x21) (measure x20) 
                            (measure x19) (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_1 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_1 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                           P_id_QUICK (measure x15)
                          | (algebra.Alg.Term algebra.F.id_part (x16::
                             x15::nil)) =>
                           P_id_PART (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_test (x16::
                             x15::nil)) =>
                           P_id_TEST (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_2 (x19::
                             x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_2 (measure x19) (measure x18) 
                            (measure x17) (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_4 (x16::
                             x15::nil)) =>
                           P_id_MATCH_4 (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_append (x16::
                             x15::nil)) =>
                           P_id_APPEND (measure x16) (measure x15)
                          | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_0_monotonic;assumption.
     intros ;apply P_id_MATCH_5_monotonic;assumption.
     intros ;apply P_id_MATCH_3_monotonic;assumption.
     intros ;apply P_id_MATCH_1_monotonic;assumption.
     intros ;apply P_id_QUICK_monotonic;assumption.
     intros ;apply P_id_PART_monotonic;assumption.
     intros ;apply P_id_TEST_monotonic;assumption.
     intros ;apply P_id_MATCH_2_monotonic;assumption.
     intros ;apply P_id_MATCH_4_monotonic;assumption.
     intros ;apply P_id_APPEND_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_6.
   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_6.
  
  Definition wf_DP_R_xml_0_scc_6  := WF_DP_R_xml_0_scc_6.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_6 :
   forall x y, (DP_R_xml_0_scc_6 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_6).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    apply wf_DP_R_xml_0_scc_6.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_7  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),append(quick(l1_),Cons(a_,quick(l2_)))> *)
    | DP_R_xml_0_non_scc_7_0 :
     forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x13 x18) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x9 x17) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x12 x16) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
          DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_append 
                                ((algebra.Alg.Term algebra.F.id_quick 
                                (x10::nil))::(algebra.Alg.Term 
                                algebra.F.id_Cons (x13::(algebra.Alg.Term 
                                algebra.F.id_quick (x11::nil))::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_7 :
   forall x y, 
    (DP_R_xml_0_non_scc_7 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_6;
        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_8  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l1_)> *)
    | DP_R_xml_0_scc_8_0 :
     forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x13 x18) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x9 x17) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x12 x16) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
          DP_R_xml_0_scc_8 (algebra.Alg.Term algebra.F.id_quick (x10::nil)) 
           (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
     (* <quick(l_5_),match_4(l_5_,l_5_)> *)
    | DP_R_xml_0_scc_8_1 :
     forall x12 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x12 x15) ->
       DP_R_xml_0_scc_8 (algebra.Alg.Term algebra.F.id_match_4 (x12::
                         x12::nil)) 
        (algebra.Alg.Term algebra.F.id_quick (x15::nil))
    
     (* <match_4(l_5_,Cons(a_,l'_)),match_5(a_,l'_,l_5_,part(a_,l'_))> *)
    | DP_R_xml_0_scc_8_2 :
     forall x16 x12 x9 x13 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x12 x16) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_Cons (x13::x9::nil)) x15) ->
        DP_R_xml_0_scc_8 (algebra.Alg.Term algebra.F.id_match_5 (x13::x9::
                          x12::(algebra.Alg.Term algebra.F.id_part (x13::
                          x9::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil))
    
     (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l2_)> *)
    | DP_R_xml_0_scc_8_3 :
     forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x13 x18) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x9 x17) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x12 x16) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
          DP_R_xml_0_scc_8 (algebra.Alg.Term algebra.F.id_quick (x11::nil)) 
           (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_8.
   Inductive DP_R_xml_0_scc_8_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l1_)> *)
     | DP_R_xml_0_scc_8_large_0 :
      forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x13 x18) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x17) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x12 x16) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
           DP_R_xml_0_scc_8_large (algebra.Alg.Term algebra.F.id_quick 
                                   (x10::nil)) 
            (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
      (* <quick(l_5_),match_4(l_5_,l_5_)> *)
     | DP_R_xml_0_scc_8_large_1 :
      forall x12 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x15) ->
        DP_R_xml_0_scc_8_large (algebra.Alg.Term algebra.F.id_match_4 (x12::
                                x12::nil)) 
         (algebra.Alg.Term algebra.F.id_quick (x15::nil))
     
      (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l2_)> *)
     | DP_R_xml_0_scc_8_large_2 :
      forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x13 x18) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x9 x17) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x12 x16) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     
            (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
           DP_R_xml_0_scc_8_large (algebra.Alg.Term algebra.F.id_quick 
                                   (x11::nil)) 
            (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::x15::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_8_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <match_4(l_5_,Cons(a_,l'_)),match_5(a_,l'_,l_5_,part(a_,l'_))> *)
     | DP_R_xml_0_scc_8_strict_0 :
      forall x16 x12 x9 x13 x15, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x12 x16) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_Cons (x13::x9::nil)) x15) ->
         DP_R_xml_0_scc_8_strict (algebra.Alg.Term algebra.F.id_match_5 
                                  (x13::x9::x12::(algebra.Alg.Term 
                                  algebra.F.id_part (x13::x9::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_8_large.
    Inductive DP_R_xml_0_scc_8_large_non_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <quick(l_5_),match_4(l_5_,l_5_)> *)
      | DP_R_xml_0_scc_8_large_non_scc_1_0 :
       forall x12 x15, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x12 x15) ->
         DP_R_xml_0_scc_8_large_non_scc_1 (algebra.Alg.Term 
                                           algebra.F.id_match_4 (x12::
                                           x12::nil)) 
          (algebra.Alg.Term algebra.F.id_quick (x15::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_8_large_non_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_8_large_non_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_8.DP_R_xml_0_scc_8_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_8_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l2_)> *)
      | DP_R_xml_0_scc_8_large_non_scc_2_0 :
       forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x13 x18) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x17) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x12 x16) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
            DP_R_xml_0_scc_8_large_non_scc_2 (algebra.Alg.Term 
                                              algebra.F.id_quick (x11::nil)) 
             (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
              x15::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_8_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_8_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_8.DP_R_xml_0_scc_8_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_8_large_non_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_8_large_non_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <match_5(a_,l'_,l_5_,Pair(l1_,l2_)),quick(l1_)> *)
      | DP_R_xml_0_scc_8_large_non_scc_3_0 :
       forall x16 x12 x18 x10 x17 x9 x13 x11 x15, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x13 x18) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x9 x17) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x12 x16) ->
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                      
             (algebra.Alg.Term algebra.F.id_Pair (x10::x11::nil)) x15) ->
            DP_R_xml_0_scc_8_large_non_scc_3 (algebra.Alg.Term 
                                              algebra.F.id_quick (x10::nil)) 
             (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
              x15::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_8_large_non_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_8_large_non_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_8.DP_R_xml_0_scc_8_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply acc_DP_R_xml_0_scc_8_large_non_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_8.DP_R_xml_0_scc_8_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_8_large_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_8_large_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_8_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_8_large_non_scc_0;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))).
    Qed.
   End WF_DP_R_xml_0_scc_8_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_test (x15:Z) (x16:Z) := 0.
   
   Definition P_id_match_1 (x15:Z) (x16:Z) (x17:Z) := 1* x17.
   
   Definition P_id_match_0 (x15:Z) (x16:Z) (x17:Z) := 1* x16 + 1* x17.
   
   Definition P_id_quick (x15:Z) := 1* x15.
   
   Definition P_id_False  := 0.
   
   Definition P_id_match_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 
     1 + 1* x19.
   
   Definition P_id_Cons (x15:Z) (x16:Z) := 1 + 1* x16.
   
   Definition P_id_match_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 1 + 1* x18.
   
   Definition P_id_True  := 0.
   
   Definition P_id_Pair (x15:Z) (x16:Z) := 1* x15 + 1* x16.
   
   Definition P_id_Nil  := 0.
   
   Definition P_id_match_4 (x15:Z) (x16:Z) := 1* x16.
   
   Definition P_id_append (x15:Z) (x16:Z) := 1* x15 + 1* x16.
   
   Definition P_id_match_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 1 + 1* x15 + 1* x16.
   
   Definition P_id_part (x15:Z) (x16:Z) := 1* x16.
   
   Lemma P_id_test_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_test x16 x18 <= P_id_test x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_1 x16 x18 x20 <= P_id_match_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_match_0 x16 x18 x20 <= P_id_match_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_quick x16 <= P_id_quick x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_match_2 x16 x18 x20 x22 x24 <= P_id_match_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Cons x16 x18 <= P_id_Cons x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_match_5 x16 x18 x20 x22 <= P_id_match_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_Pair x16 x18 <= P_id_Pair x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_match_4 x16 x18 <= P_id_match_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_append x16 x18 <= P_id_append x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_match_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_match_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_part x16 x18 <= P_id_part x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_test_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_test x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_1_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_1 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_0_bounded :
    forall x16 x17 x15, 
     (0 <= x15) ->(0 <= x16) ->(0 <= x17) ->0 <= P_id_match_0 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_quick_bounded : forall x15, (0 <= x15) ->0 <= P_id_quick x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_False_bounded : 0 <= P_id_False .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_2_bounded :
    forall x16 x18 x17 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->(0 <= x19) ->0 <= P_id_match_2 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Cons_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Cons x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_5_bounded :
    forall x16 x18 x17 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->(0 <= x18) ->0 <= P_id_match_5 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_True_bounded : 0 <= P_id_True .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Pair_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_Pair x16 x15.
   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_match_4_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_match_4 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_append_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_append x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_3_bounded :
    forall x16 x20 x18 x17 x21 x19 x15, 
     (0 <= x15) ->
      (0 <= x16) ->
       (0 <= x17) ->
        (0 <= x18) ->
         (0 <= x19) ->
          (0 <= x20) ->
           (0 <= x21) ->0 <= P_id_match_3 x21 x20 x19 x18 x17 x16 x15.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_part_bounded :
    forall x16 x15, (0 <= x15) ->(0 <= x16) ->0 <= P_id_part x16 x15.
   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_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_test (x16::x15::nil)) =>
                    P_id_test (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_1 (x17::x16::
                      x15::nil)) =>
                    P_id_match_1 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_0 (x17::x16::
                      x15::nil)) =>
                    P_id_match_0 (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                    P_id_quick (measure x15)
                   | (algebra.Alg.Term algebra.F.id_False nil) => P_id_False 
                   | (algebra.Alg.Term algebra.F.id_match_2 (x19::x18::x17::
                      x16::x15::nil)) =>
                    P_id_match_2 (measure x19) (measure x18) (measure x17) 
                     (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Cons (x16::x15::nil)) =>
                    P_id_Cons (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_5 (x18::x17::x16::
                      x15::nil)) =>
                    P_id_match_5 (measure x18) (measure x17) (measure x16) 
                     (measure x15)
                   | (algebra.Alg.Term algebra.F.id_True nil) => P_id_True 
                   | (algebra.Alg.Term algebra.F.id_Pair (x16::x15::nil)) =>
                    P_id_Pair (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_Nil nil) => P_id_Nil 
                   | (algebra.Alg.Term algebra.F.id_match_4 (x16::x15::nil)) =>
                    P_id_match_4 (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_append (x16::x15::nil)) =>
                    P_id_append (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_match_3 (x21::x20::x19::
                      x18::x17::x16::x15::nil)) =>
                    P_id_match_3 (measure x21) (measure x20) (measure x19) 
                     (measure x18) (measure x17) (measure x16) (measure x15)
                   | (algebra.Alg.Term algebra.F.id_part (x16::x15::nil)) =>
                    P_id_part (measure x16) (measure x15)
                   | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH_0 (x15:Z) (x16:Z) (x17:Z) := 0.
   
   Definition P_id_MATCH_5 (x15:Z) (x16:Z) (x17:Z) (x18:Z) := 2* x18.
   
   Definition P_id_MATCH_3 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) (x20:
     Z) (x21:Z) := 0.
   
   Definition P_id_MATCH_1 (x15:Z) (x16:Z) (x17:Z) := 0.
   
   Definition P_id_QUICK (x15:Z) := 2* x15.
   
   Definition P_id_PART (x15:Z) (x16:Z) := 0.
   
   Definition P_id_TEST (x15:Z) (x16:Z) := 0.
   
   Definition P_id_MATCH_2 (x15:Z) (x16:Z) (x17:Z) (x18:Z) (x19:Z) := 0.
   
   Definition P_id_MATCH_4 (x15:Z) (x16:Z) := 2* x16.
   
   Definition P_id_APPEND (x15:Z) (x16:Z) := 0.
   
   Lemma P_id_MATCH_0_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_0 x16 x18 x20 <= P_id_MATCH_0 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_5_monotonic :
    forall x16 x20 x18 x22 x17 x21 x19 x15, 
     (0 <= x22)/\ (x22 <= x21) ->
      (0 <= x20)/\ (x20 <= x19) ->
       (0 <= x18)/\ (x18 <= x17) ->
        (0 <= x16)/\ (x16 <= x15) ->
         P_id_MATCH_5 x16 x18 x20 x22 <= P_id_MATCH_5 x15 x17 x19 x21.
   Proof.
     intros x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_3_monotonic :
    forall x16 x24 x20 x28 x18 x26 x22 x17 x25 x21 x19 x27 x23 x15, 
     (0 <= x28)/\ (x28 <= x27) ->
      (0 <= x26)/\ (x26 <= x25) ->
       (0 <= x24)/\ (x24 <= x23) ->
        (0 <= x22)/\ (x22 <= x21) ->
         (0 <= x20)/\ (x20 <= x19) ->
          (0 <= x18)/\ (x18 <= x17) ->
           (0 <= x16)/\ (x16 <= x15) ->
            P_id_MATCH_3 x16 x18 x20 x22 x24 x26 x28 <= P_id_MATCH_3 
                                                         x15 x17 x19 
                                                         x21 x23 x25 
                                                         x27.
   Proof.
     intros x28 x27 x26 x25 x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     intros [H_11 H_10].
     intros [H_13 H_12].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_1_monotonic :
    forall x16 x20 x18 x17 x19 x15, 
     (0 <= x20)/\ (x20 <= x19) ->
      (0 <= x18)/\ (x18 <= x17) ->
       (0 <= x16)/\ (x16 <= x15) ->
        P_id_MATCH_1 x16 x18 x20 <= P_id_MATCH_1 x15 x17 x19.
   Proof.
     intros x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_QUICK_monotonic :
    forall x16 x15, 
     (0 <= x16)/\ (x16 <= x15) ->P_id_QUICK x16 <= P_id_QUICK x15.
   Proof.
     intros x16 x15.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PART_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_PART x16 x18 <= P_id_PART x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TEST_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_TEST x16 x18 <= P_id_TEST x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_2_monotonic :
    forall x16 x24 x20 x18 x22 x17 x21 x19 x23 x15, 
     (0 <= x24)/\ (x24 <= x23) ->
      (0 <= x22)/\ (x22 <= x21) ->
       (0 <= x20)/\ (x20 <= x19) ->
        (0 <= x18)/\ (x18 <= x17) ->
         (0 <= x16)/\ (x16 <= x15) ->
          P_id_MATCH_2 x16 x18 x20 x22 x24 <= P_id_MATCH_2 x15 x17 x19 
                                               x21 x23.
   Proof.
     intros x24 x23 x22 x21 x20 x19 x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     intros [H_7 H_6].
     intros [H_9 H_8].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MATCH_4_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->
       P_id_MATCH_4 x16 x18 <= P_id_MATCH_4 x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_APPEND_monotonic :
    forall x16 x18 x17 x15, 
     (0 <= x18)/\ (x18 <= x17) ->
      (0 <= x16)/\ (x16 <= x15) ->P_id_APPEND x16 x18 <= P_id_APPEND x15 x17.
   Proof.
     intros x18 x17 x16 x15.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_test P_id_match_1 P_id_match_0 P_id_quick 
      P_id_False P_id_match_2 P_id_Cons P_id_match_5 P_id_True P_id_Pair 
      P_id_Nil P_id_match_4 P_id_append P_id_match_3 P_id_part P_id_MATCH_0 
      P_id_MATCH_5 P_id_MATCH_3 P_id_MATCH_1 P_id_QUICK P_id_PART P_id_TEST 
      P_id_MATCH_2 P_id_MATCH_4 P_id_APPEND.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match_0 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_0 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_5 (x18::
                             x17::x16::x15::nil)) =>
                           P_id_MATCH_5 (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_3 (x21::
                             x20::x19::x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_3 (measure x21) (measure x20) 
                            (measure x19) (measure x18) (measure x17) 
                            (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_1 (x17::
                             x16::x15::nil)) =>
                           P_id_MATCH_1 (measure x17) (measure x16) 
                            (measure x15)
                          | (algebra.Alg.Term algebra.F.id_quick (x15::nil)) =>
                           P_id_QUICK (measure x15)
                          | (algebra.Alg.Term algebra.F.id_part (x16::
                             x15::nil)) =>
                           P_id_PART (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_test (x16::
                             x15::nil)) =>
                           P_id_TEST (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_2 (x19::
                             x18::x17::x16::x15::nil)) =>
                           P_id_MATCH_2 (measure x19) (measure x18) 
                            (measure x17) (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_match_4 (x16::
                             x15::nil)) =>
                           P_id_MATCH_4 (measure x16) (measure x15)
                          | (algebra.Alg.Term algebra.F.id_append (x16::
                             x15::nil)) =>
                           P_id_APPEND (measure x16) (measure x15)
                          | _ => 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_test_monotonic;assumption.
     intros ;apply P_id_match_1_monotonic;assumption.
     intros ;apply P_id_match_0_monotonic;assumption.
     intros ;apply P_id_quick_monotonic;assumption.
     intros ;apply P_id_match_2_monotonic;assumption.
     intros ;apply P_id_Cons_monotonic;assumption.
     intros ;apply P_id_match_5_monotonic;assumption.
     intros ;apply P_id_Pair_monotonic;assumption.
     intros ;apply P_id_match_4_monotonic;assumption.
     intros ;apply P_id_append_monotonic;assumption.
     intros ;apply P_id_match_3_monotonic;assumption.
     intros ;apply P_id_part_monotonic;assumption.
     intros ;apply P_id_test_bounded;assumption.
     intros ;apply P_id_match_1_bounded;assumption.
     intros ;apply P_id_match_0_bounded;assumption.
     intros ;apply P_id_quick_bounded;assumption.
     intros ;apply P_id_False_bounded;assumption.
     intros ;apply P_id_match_2_bounded;assumption.
     intros ;apply P_id_Cons_bounded;assumption.
     intros ;apply P_id_match_5_bounded;assumption.
     intros ;apply P_id_True_bounded;assumption.
     intros ;apply P_id_Pair_bounded;assumption.
     intros ;apply P_id_Nil_bounded;assumption.
     intros ;apply P_id_match_4_bounded;assumption.
     intros ;apply P_id_append_bounded;assumption.
     intros ;apply P_id_match_3_bounded;assumption.
     intros ;apply P_id_part_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_0_monotonic;assumption.
     intros ;apply P_id_MATCH_5_monotonic;assumption.
     intros ;apply P_id_MATCH_3_monotonic;assumption.
     intros ;apply P_id_MATCH_1_monotonic;assumption.
     intros ;apply P_id_QUICK_monotonic;assumption.
     intros ;apply P_id_PART_monotonic;assumption.
     intros ;apply P_id_TEST_monotonic;assumption.
     intros ;apply P_id_MATCH_2_monotonic;assumption.
     intros ;apply P_id_MATCH_4_monotonic;assumption.
     intros ;apply P_id_APPEND_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_8_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_8_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_8_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_8_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_8_large  := WF_DP_R_xml_0_scc_8_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_8.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_8_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_8_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_8_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_8_large.
   Qed.
  End WF_DP_R_xml_0_scc_8.
  
  Definition wf_DP_R_xml_0_scc_8  := WF_DP_R_xml_0_scc_8.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_8 :
   forall x y, (DP_R_xml_0_scc_8 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_8).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_7;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_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_8.
  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_7;
       econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
     ((eapply acc_DP_R_xml_0_non_scc_6;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_5;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_3;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_2;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_1;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_0;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_8;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_scc_7;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_scc_6;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_scc_5;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_scc_4;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_scc_3;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_scc_2;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_scc_1;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_0;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                      (fail)))))))))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


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