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_U11 *)
    | id_U11 : symb
     (* id_n__0 *)
    | id_n__0 : symb
     (* id_s *)
    | id_s : symb
     (* id_0 *)
    | id_0 : symb
     (* id_activate *)
    | id_activate : symb
     (* id_n__isNat *)
    | id_n__isNat : symb
     (* id_and *)
    | id_and : symb
     (* id_tt *)
    | id_tt : symb
     (* id_n__plus *)
    | id_n__plus : symb
     (* id_plus *)
    | id_plus : symb
     (* id_U21 *)
    | id_U21 : symb
     (* id_n__s *)
    | id_n__s : symb
     (* id_isNat *)
    | id_isNat : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_U11,id_U11 => true
      | id_n__0,id_n__0 => true
      | id_s,id_s => true
      | id_0,id_0 => true
      | id_activate,id_activate => true
      | id_n__isNat,id_n__isNat => true
      | id_and,id_and => true
      | id_tt,id_tt => true
      | id_n__plus,id_n__plus => true
      | id_plus,id_plus => true
      | id_U21,id_U21 => true
      | id_n__s,id_n__s => true
      | id_isNat,id_isNat => true
      | _,_ => false
      end.
  
  
   (* Proof of decidability of equality over symb *)
  Definition symb_eq_bool_ok(f1 f2:symb) :
   match symb_eq_bool f1 f2 with
     | true => f1 = f2
     | false => f1 <> f2
     end.
  Proof.
    intros f1 f2.
    
    refine match f1 as u1,f2 as u2 return 
             match symb_eq_bool u1 u2 return 
               Prop with
               | true => u1 = u2
               | false => u1 <> u2
               end with
             | id_U11,id_U11 => refl_equal _
             | id_n__0,id_n__0 => refl_equal _
             | id_s,id_s => refl_equal _
             | id_0,id_0 => refl_equal _
             | id_activate,id_activate => refl_equal _
             | id_n__isNat,id_n__isNat => refl_equal _
             | id_and,id_and => refl_equal _
             | id_tt,id_tt => refl_equal _
             | id_n__plus,id_n__plus => refl_equal _
             | id_plus,id_plus => refl_equal _
             | id_U21,id_U21 => refl_equal _
             | id_n__s,id_n__s => refl_equal _
             | id_isNat,id_isNat => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_U11 => term.Free 2
      | id_n__0 => term.Free 0
      | id_s => term.Free 1
      | id_0 => term.Free 0
      | id_activate => term.Free 1
      | id_n__isNat => term.Free 1
      | id_and => term.Free 2
      | id_tt => term.Free 0
      | id_n__plus => term.Free 2
      | id_plus => term.Free 2
      | id_U21 => term.Free 3
      | id_n__s => term.Free 1
      | id_isNat => term.Free 1
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_U11,id_U11 => true
      | id_U11,id_n__0 => false
      | id_U11,id_s => false
      | id_U11,id_0 => false
      | id_U11,id_activate => false
      | id_U11,id_n__isNat => false
      | id_U11,id_and => false
      | id_U11,id_tt => false
      | id_U11,id_n__plus => false
      | id_U11,id_plus => false
      | id_U11,id_U21 => false
      | id_U11,id_n__s => false
      | id_U11,id_isNat => false
      | id_n__0,id_U11 => true
      | id_n__0,id_n__0 => true
      | id_n__0,id_s => false
      | id_n__0,id_0 => false
      | id_n__0,id_activate => false
      | id_n__0,id_n__isNat => false
      | id_n__0,id_and => false
      | id_n__0,id_tt => false
      | id_n__0,id_n__plus => false
      | id_n__0,id_plus => false
      | id_n__0,id_U21 => false
      | id_n__0,id_n__s => false
      | id_n__0,id_isNat => false
      | id_s,id_U11 => true
      | id_s,id_n__0 => true
      | id_s,id_s => true
      | id_s,id_0 => false
      | id_s,id_activate => false
      | id_s,id_n__isNat => false
      | id_s,id_and => false
      | id_s,id_tt => false
      | id_s,id_n__plus => false
      | id_s,id_plus => false
      | id_s,id_U21 => false
      | id_s,id_n__s => false
      | id_s,id_isNat => false
      | id_0,id_U11 => true
      | id_0,id_n__0 => true
      | id_0,id_s => true
      | id_0,id_0 => true
      | id_0,id_activate => false
      | id_0,id_n__isNat => false
      | id_0,id_and => false
      | id_0,id_tt => false
      | id_0,id_n__plus => false
      | id_0,id_plus => false
      | id_0,id_U21 => false
      | id_0,id_n__s => false
      | id_0,id_isNat => false
      | id_activate,id_U11 => true
      | id_activate,id_n__0 => true
      | id_activate,id_s => true
      | id_activate,id_0 => true
      | id_activate,id_activate => true
      | id_activate,id_n__isNat => false
      | id_activate,id_and => false
      | id_activate,id_tt => false
      | id_activate,id_n__plus => false
      | id_activate,id_plus => false
      | id_activate,id_U21 => false
      | id_activate,id_n__s => false
      | id_activate,id_isNat => false
      | id_n__isNat,id_U11 => true
      | id_n__isNat,id_n__0 => true
      | id_n__isNat,id_s => true
      | id_n__isNat,id_0 => true
      | id_n__isNat,id_activate => true
      | id_n__isNat,id_n__isNat => true
      | id_n__isNat,id_and => false
      | id_n__isNat,id_tt => false
      | id_n__isNat,id_n__plus => false
      | id_n__isNat,id_plus => false
      | id_n__isNat,id_U21 => false
      | id_n__isNat,id_n__s => false
      | id_n__isNat,id_isNat => false
      | id_and,id_U11 => true
      | id_and,id_n__0 => true
      | id_and,id_s => true
      | id_and,id_0 => true
      | id_and,id_activate => true
      | id_and,id_n__isNat => true
      | id_and,id_and => true
      | id_and,id_tt => false
      | id_and,id_n__plus => false
      | id_and,id_plus => false
      | id_and,id_U21 => false
      | id_and,id_n__s => false
      | id_and,id_isNat => false
      | id_tt,id_U11 => true
      | id_tt,id_n__0 => true
      | id_tt,id_s => true
      | id_tt,id_0 => true
      | id_tt,id_activate => true
      | id_tt,id_n__isNat => true
      | id_tt,id_and => true
      | id_tt,id_tt => true
      | id_tt,id_n__plus => false
      | id_tt,id_plus => false
      | id_tt,id_U21 => false
      | id_tt,id_n__s => false
      | id_tt,id_isNat => false
      | id_n__plus,id_U11 => true
      | id_n__plus,id_n__0 => true
      | id_n__plus,id_s => true
      | id_n__plus,id_0 => true
      | id_n__plus,id_activate => true
      | id_n__plus,id_n__isNat => true
      | id_n__plus,id_and => true
      | id_n__plus,id_tt => true
      | id_n__plus,id_n__plus => true
      | id_n__plus,id_plus => false
      | id_n__plus,id_U21 => false
      | id_n__plus,id_n__s => false
      | id_n__plus,id_isNat => false
      | id_plus,id_U11 => true
      | id_plus,id_n__0 => true
      | id_plus,id_s => true
      | id_plus,id_0 => true
      | id_plus,id_activate => true
      | id_plus,id_n__isNat => true
      | id_plus,id_and => true
      | id_plus,id_tt => true
      | id_plus,id_n__plus => true
      | id_plus,id_plus => true
      | id_plus,id_U21 => false
      | id_plus,id_n__s => false
      | id_plus,id_isNat => false
      | id_U21,id_U11 => true
      | id_U21,id_n__0 => true
      | id_U21,id_s => true
      | id_U21,id_0 => true
      | id_U21,id_activate => true
      | id_U21,id_n__isNat => true
      | id_U21,id_and => true
      | id_U21,id_tt => true
      | id_U21,id_n__plus => true
      | id_U21,id_plus => true
      | id_U21,id_U21 => true
      | id_U21,id_n__s => false
      | id_U21,id_isNat => false
      | id_n__s,id_U11 => true
      | id_n__s,id_n__0 => true
      | id_n__s,id_s => true
      | id_n__s,id_0 => true
      | id_n__s,id_activate => true
      | id_n__s,id_n__isNat => true
      | id_n__s,id_and => true
      | id_n__s,id_tt => true
      | id_n__s,id_n__plus => true
      | id_n__s,id_plus => true
      | id_n__s,id_U21 => true
      | id_n__s,id_n__s => true
      | id_n__s,id_isNat => false
      | id_isNat,id_U11 => true
      | id_isNat,id_n__0 => true
      | id_isNat,id_s => true
      | id_isNat,id_0 => true
      | id_isNat,id_activate => true
      | id_isNat,id_n__isNat => true
      | id_isNat,id_and => true
      | id_isNat,id_tt => true
      | id_isNat,id_n__plus => true
      | id_isNat,id_plus => true
      | id_isNat,id_U21 => true
      | id_isNat,id_n__s => true
      | id_isNat,id_isNat => true
      end.
  
  
  Module Symb.
   Definition A  := symb.
   
   Definition eq_A  := @eq A.
   
   
   Definition eq_proof : equivalence A eq_A.
   Proof.
     constructor.
     red ;reflexivity .
     red ;intros ;transitivity y ;assumption.
     red ;intros ;symmetry ;assumption.
   Defined.
   
   
   Add Relation A eq_A 
  reflexivity proved by (@equiv_refl _ _ eq_proof)
    symmetry proved by (@equiv_sym _ _ eq_proof)
      transitivity proved by (@equiv_trans _ _ eq_proof) as EQA
.
   
   Definition eq_bool  := symb_eq_bool.
   
   Definition eq_bool_ok  := symb_eq_bool_ok.
  End Symb.
  
  Export Symb.
 End F.
 
 Module Alg := term.Make'(F)(term_extension.IntVars).
 
 Module Alg_ext := term_extension.Make(Alg).
 
 Module EQT := equational_theory.Make(Alg).
 
 Module EQT_ext := equational_extension.Make(EQT).
End algebra.

Module R_xml_0_deep_rew.
 Inductive R_xml_0_rules  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* U11(tt,N_) -> activate(N_) *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_activate 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Term algebra.F.id_tt 
      nil)::(algebra.Alg.Var 1)::nil))
   
    (* U21(tt,M_,N_) -> s(plus(activate(N_),activate(M_))) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                   algebra.F.id_plus ((algebra.Alg.Term 
                   algebra.F.id_activate ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Term algebra.F.id_activate 
                   ((algebra.Alg.Var 2)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Term algebra.F.id_tt 
      nil)::(algebra.Alg.Var 2)::(algebra.Alg.Var 1)::nil))
    (* and(tt,X_) -> activate(X_) *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_activate 
                   ((algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_tt 
      nil)::(algebra.Alg.Var 3)::nil))
    (* isNat(n__0) -> tt *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) 
     (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
      algebra.F.id_n__0 nil)::nil))
   
    (* isNat(n__plus(V1_,V2_)) -> and(isNat(activate(V1_)),n__isNat(activate(V2_))) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                   algebra.F.id_isNat ((algebra.Alg.Term 
                   algebra.F.id_activate ((algebra.Alg.Var 4)::nil))::nil))::
                   (algebra.Alg.Term algebra.F.id_n__isNat 
                   ((algebra.Alg.Term algebra.F.id_activate 
                   ((algebra.Alg.Var 5)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
      algebra.F.id_n__plus ((algebra.Alg.Var 4)::
      (algebra.Alg.Var 5)::nil))::nil))
    (* isNat(n__s(V1_)) -> isNat(activate(V1_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
                   algebra.F.id_activate ((algebra.Alg.Var 4)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
      algebra.F.id_n__s ((algebra.Alg.Var 4)::nil))::nil))
    (* plus(N_,0) -> U11(isNat(N_),N_) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Term 
                   algebra.F.id_isNat ((algebra.Alg.Var 1)::nil))::
                   (algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id_0 nil)::nil))
   
    (* plus(N_,s(M_)) -> U21(and(isNat(M_),n__isNat(N_)),M_,N_) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Term 
                   algebra.F.id_and ((algebra.Alg.Term algebra.F.id_isNat 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_n__isNat ((algebra.Alg.Var 1)::nil))::nil))::
                   (algebra.Alg.Var 2)::(algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 2)::nil))::nil))
    (* 0 -> n__0 *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__0 nil) 
     (algebra.Alg.Term algebra.F.id_0 nil)
    (* plus(X1_,X2_) -> n__plus(X1_,X2_) *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__plus 
                   ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)) 
     (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))
    (* isNat(X_) -> n__isNat(X_) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__isNat 
                   ((algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 3)::nil))
    (* s(X_) -> n__s(X_) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__s 
                   ((algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 3)::nil))
    (* activate(n__0) -> 0 *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 nil) 
     (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
      algebra.F.id_n__0 nil)::nil))
   
    (* activate(n__plus(X1_,X2_)) -> plus(activate(X1_),activate(X2_)) *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Term 
                   algebra.F.id_activate ((algebra.Alg.Var 6)::nil))::
                   (algebra.Alg.Term algebra.F.id_activate 
                   ((algebra.Alg.Var 7)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
      algebra.F.id_n__plus ((algebra.Alg.Var 6)::
      (algebra.Alg.Var 7)::nil))::nil))
    (* activate(n__isNat(X_)) -> isNat(X_) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNat 
                   ((algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
      algebra.F.id_n__isNat ((algebra.Alg.Var 3)::nil))::nil))
    (* activate(n__s(X_)) -> s(activate(X_)) *)
   | R_xml_0_rule_15 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                   algebra.F.id_activate ((algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
      algebra.F.id_n__s ((algebra.Alg.Var 3)::nil))::nil))
    (* activate(X_) -> X_ *)
   | R_xml_0_rule_16 :
    R_xml_0_rules (algebra.Alg.Var 3) 
     (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 3)::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 1)::nil)))::
    nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 2)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term algebra.F.id_plus 
     ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Term algebra.F.id_activate 
     ((algebra.Alg.Var 2)::nil))::nil))::nil)))::R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_tt 
     nil)::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_activate ((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_isNat ((algebra.Alg.Term 
     algebra.F.id_n__0 nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil))::
    R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
     algebra.F.id_n__plus ((algebra.Alg.Var 4)::
     (algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_isNat 
     ((algebra.Alg.Term algebra.F.id_activate 
     ((algebra.Alg.Var 4)::nil))::nil))::(algebra.Alg.Term 
     algebra.F.id_n__isNat ((algebra.Alg.Term algebra.F.id_activate 
     ((algebra.Alg.Var 5)::nil))::nil))::nil)))::R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
     algebra.F.id_n__s ((algebra.Alg.Var 4)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
     algebra.F.id_activate ((algebra.Alg.Var 4)::nil))::nil)))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_0 nil)::nil)),
    (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Term algebra.F.id_isNat 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Term algebra.F.id_and 
     ((algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Term algebra.F.id_n__isNat 
     ((algebra.Alg.Var 1)::nil))::nil))::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_0 nil),
    (algebra.Alg.Term algebra.F.id_n__0 nil))::R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)),
    (algebra.Alg.Term algebra.F.id_n__plus ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_n__isNat ((algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_n__s ((algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
     algebra.F.id_n__0 nil)::nil)),(algebra.Alg.Term algebra.F.id_0 nil))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
     algebra.F.id_n__plus ((algebra.Alg.Var 6)::
     (algebra.Alg.Var 7)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Term 
     algebra.F.id_activate ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Term 
     algebra.F.id_activate ((algebra.Alg.Var 7)::nil))::nil)))::
    R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
     algebra.F.id_n__isNat ((algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term 
     algebra.F.id_n__s ((algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
     algebra.F.id_activate ((algebra.Alg.Var 3)::nil))::nil)))::
    R_xml_0_rule_as_list_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Var 3))::R_xml_0_rule_as_list_15.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_16.
 
 
 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.or18_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 17.
   injection H;intros ;subst;constructor 16.
   injection H;intros ;subst;constructor 15.
   injection H;intros ;subst;constructor 14.
   injection H;intros ;subst;constructor 13.
   injection H;intros ;subst;constructor 12.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   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 (x9 x10 x11 x12 x13:Prop) :
  Prop := 
   | conj_5 : x9->x10->x11->x12->x13->and_5 x9 x10 x11 x12 x13
 .
 
 
 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_n__0 nil) ->
           t' = (algebra.Alg.Term algebra.F.id_n__0 nil)) 
     (forall x10, 
      t = (algebra.Alg.Term algebra.F.id_n__isNat (x10::nil)) ->
       exists x9,
         t' = (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10))
       
     (t = (algebra.Alg.Term algebra.F.id_tt nil) ->
      t' = (algebra.Alg.Term algebra.F.id_tt nil)) 
     (forall x10 x12, 
      t = (algebra.Alg.Term algebra.F.id_n__plus (x10::x12::nil)) ->
       exists x9,
         exists x11,
           t' = (algebra.Alg.Term algebra.F.id_n__plus (x9::x11::nil))/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x9 x10)/\ 
           (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
             x11 x12)) 
     (forall x10, 
      t = (algebra.Alg.Term algebra.F.id_n__s (x10::nil)) ->
       exists x9,
         t' = (algebra.Alg.Term algebra.F.id_n__s (x9::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10))
      .
 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 x10 H;exists x10;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x10 x12 H;exists x10;exists x12;intuition;constructor 1.
   intros x10 H;exists x10;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_n__0 H_id_n__isNat H_id_tt H_id_n__plus H_id_n__s].
   constructor.
   
   clear H_id_n__isNat H_id_tt H_id_n__plus H_id_n__s;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_n__0 H_id_tt H_id_n__plus H_id_n__s;intros x10 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 x10 |- _ =>
      destruct (H_id_n__isNat y (refl_equal _)) as [x9];intros ;intuition;
       exists x9;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_n__0 H_id_n__isNat H_id_n__plus H_id_n__s;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_n__0 H_id_n__isNat H_id_tt H_id_n__s;intros x10 x12 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x10 |- _ =>
      destruct (H_id_n__plus y x12 (refl_equal _)) as [x9 [x11]];intros ;
       intuition;exists x9;exists x11;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x12 |- _ =>
      destruct (H_id_n__plus x10 y (refl_equal _)) as [x9 [x11]];intros ;
       intuition;exists x9;exists x11;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_n__0 H_id_n__isNat H_id_tt H_id_n__plus;intros x10 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 x10 |- _ =>
      destruct (H_id_n__s y (refl_equal _)) as [x9];intros ;intuition;
       exists x9;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
 Qed.
 
 
 Lemma id_n__0_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_n__0 nil) ->
     t' = (algebra.Alg.Term algebra.F.id_n__0 nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_n__isNat_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x10, 
     t = (algebra.Alg.Term algebra.F.id_n__isNat (x10::nil)) ->
      exists x9,
        t' = (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_tt_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_tt nil) ->
     t' = (algebra.Alg.Term algebra.F.id_tt nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_n__plus_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x10 x12, 
     t = (algebra.Alg.Term algebra.F.id_n__plus (x10::x12::nil)) ->
      exists x9,
        exists x11,
          t' = (algebra.Alg.Term algebra.F.id_n__plus (x9::x11::nil))/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x9 x10)/\ 
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
            x11 x12).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_n__s_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x10, 
     t = (algebra.Alg.Term algebra.F.id_n__s (x10::nil)) ->
      exists x9,
        t' = (algebra.Alg.Term algebra.F.id_n__s (x9::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10).
 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_n__0 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_n__0_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n__isNat (?x9::nil)) |- _ =>
     let x9 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_n__isNat_is_R_xml_0_constructor H (refl_equal _)) as 
           [x9 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n__plus (?x10::?x9::nil)) |- 
    _ =>
     let x10 := fresh "x" in 
      (let x9 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_n__plus_is_R_xml_0_constructor H (refl_equal _))
                as [x10 [x9 [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_n__s (?x9::nil)) |- _ =>
     let x9 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_n__s_is_R_xml_0_constructor H (refl_equal _)) as 
           [x9 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    end
  .
 
 
 Ltac simplify_star_reduction_R_xml_0  :=
  match goal with
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n__0 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_n__0_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n__isNat (?x9::nil)) |- _ =>
     let x9 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_n__isNat_is_R_xml_0_constructor H (refl_equal _)) as 
           [x9 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n__plus (?x10::?x9::nil)) |- 
    _ =>
     let x10 := fresh "x" in 
      (let x9 := fresh "x" in 
        (let Heq := fresh "Heq" in 
          (let Hred1 := fresh "Hred" in 
            (let Hred2 := fresh "Hred" in 
              (destruct (id_n__plus_is_R_xml_0_constructor H (refl_equal _))
                as [x10 [x9 [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_n__s (?x9::nil)) |- _ =>
     let x9 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_n__s_is_R_xml_0_constructor H (refl_equal _)) as 
           [x9 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    end
  .
End R_xml_0_deep_rew.

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

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

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

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

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

Module Interp.
 Section S.
   Require Import interp.
   
   Hypothesis A : Type.
   
   Hypothesis Ale Alt Aeq : A -> A -> Prop.
   
   Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
   
   Hypothesis A0 : A.
   
   Notation Local "a <= b" := (Ale a b).
   
   Hypothesis P_id_U11 : A ->A ->A.
   
   Hypothesis P_id_n__0 : A.
   
   Hypothesis P_id_s : A ->A.
   
   Hypothesis P_id_0 : A.
   
   Hypothesis P_id_activate : A ->A.
   
   Hypothesis P_id_n__isNat : A ->A.
   
   Hypothesis P_id_and : A ->A ->A.
   
   Hypothesis P_id_tt : A.
   
   Hypothesis P_id_n__plus : A ->A ->A.
   
   Hypothesis P_id_plus : A ->A ->A.
   
   Hypothesis P_id_U21 : A ->A ->A ->A.
   
   Hypothesis P_id_n__s : A ->A.
   
   Hypothesis P_id_isNat : A ->A.
   
   Hypothesis P_id_U11_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 x12 <= P_id_U11 x9 x11.
   
   Hypothesis P_id_s_monotonic :
    forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
   
   Hypothesis P_id_activate_monotonic :
    forall x10 x9, 
     (A0 <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
   
   Hypothesis P_id_n__isNat_monotonic :
    forall x10 x9, 
     (A0 <= x10)/\ (x10 <= x9) ->P_id_n__isNat x10 <= P_id_n__isNat x9.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_and x10 x12 <= P_id_and x9 x11.
   
   Hypothesis P_id_n__plus_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_n__plus x10 x12 <= P_id_n__plus x9 x11.
   
   Hypothesis P_id_plus_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_plus x10 x12 <= P_id_plus x9 x11.
   
   Hypothesis P_id_U21_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->
       (A0 <= x10)/\ (x10 <= x9) ->
        P_id_U21 x10 x12 x14 <= P_id_U21 x9 x11 x13.
   
   Hypothesis P_id_n__s_monotonic :
    forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_n__s x10 <= P_id_n__s x9.
   
   Hypothesis P_id_isNat_monotonic :
    forall x10 x9, 
     (A0 <= x10)/\ (x10 <= x9) ->P_id_isNat x10 <= P_id_isNat x9.
   
   Hypothesis P_id_U11_bounded :
    forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_U11 x10 x9.
   
   Hypothesis P_id_n__0_bounded : A0 <= P_id_n__0 .
   
   Hypothesis P_id_s_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_s x9.
   
   Hypothesis P_id_0_bounded : A0 <= P_id_0 .
   
   Hypothesis P_id_activate_bounded :
    forall x9, (A0 <= x9) ->A0 <= P_id_activate x9.
   
   Hypothesis P_id_n__isNat_bounded :
    forall x9, (A0 <= x9) ->A0 <= P_id_n__isNat x9.
   
   Hypothesis P_id_and_bounded :
    forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_and x10 x9.
   
   Hypothesis P_id_tt_bounded : A0 <= P_id_tt .
   
   Hypothesis P_id_n__plus_bounded :
    forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_n__plus x10 x9.
   
   Hypothesis P_id_plus_bounded :
    forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_plus x10 x9.
   
   Hypothesis P_id_U21_bounded :
    forall x10 x9 x11, 
     (A0 <= x9) ->(A0 <= x10) ->(A0 <= x11) ->A0 <= P_id_U21 x11 x10 x9.
   
   Hypothesis P_id_n__s_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_n__s x9.
   
   Hypothesis P_id_isNat_bounded :
    forall x9, (A0 <= x9) ->A0 <= P_id_isNat x9.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil)) =>
        P_id_U11 (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_n__0 nil) => P_id_n__0 
       | (algebra.Alg.Term algebra.F.id_s (x9::nil)) => P_id_s (measure x9)
       | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
       | (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
        P_id_activate (measure x9)
       | (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil)) =>
        P_id_n__isNat (measure x9)
       | (algebra.Alg.Term algebra.F.id_and (x10::x9::nil)) =>
        P_id_and (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
       | (algebra.Alg.Term algebra.F.id_n__plus (x10::x9::nil)) =>
        P_id_n__plus (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil)) =>
        P_id_plus (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil)) =>
        P_id_U21 (measure x11) (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_n__s (x9::nil)) =>
        P_id_n__s (measure x9)
       | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
        P_id_isNat (measure x9)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil)) =>
                    P_id_U11 (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__0 nil) => P_id_n__0 
                   | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                    P_id_s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
                    P_id_activate (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil)) =>
                    P_id_n__isNat (measure x9)
                   | (algebra.Alg.Term algebra.F.id_and (x10::x9::nil)) =>
                    P_id_and (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_n__plus (x10::x9::nil)) =>
                    P_id_n__plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil)) =>
                    P_id_plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil)) =>
                    P_id_U21 (measure x11) (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__s (x9::nil)) =>
                    P_id_n__s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                    P_id_isNat (measure x9)
                   | _ => 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_U11 => P_id_U11
       | algebra.F.id_n__0 => P_id_n__0
       | algebra.F.id_s => P_id_s
       | algebra.F.id_0 => P_id_0
       | algebra.F.id_activate => P_id_activate
       | algebra.F.id_n__isNat => P_id_n__isNat
       | algebra.F.id_and => P_id_and
       | algebra.F.id_tt => P_id_tt
       | algebra.F.id_n__plus => P_id_n__plus
       | algebra.F.id_plus => P_id_plus
       | algebra.F.id_U21 => P_id_U21
       | algebra.F.id_n__s => P_id_n__s
       | algebra.F.id_isNat => P_id_isNat
       end.
   
   Lemma same_measure : forall t, measure t = InterpGen.measure A0 Pols t.
   Proof.
     fix 1 .
     intros [a| f l].
     simpl in |-*.
     unfold eq_rect_r, eq_rect, sym_eq in |-*.
     reflexivity .
     
     refine match f with
              | algebra.F.id_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_n__0 => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_activate =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_n__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_n__plus =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_plus =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_n__s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Hypothesis rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     intros .
     do 2 (rewrite same_measure in |-*).
     
     apply InterpGen.measure_star_monotonic with (1:=Aop) (Pols:=Pols) 
     (rules:=R_xml_0_deep_rew.R_xml_0_rules).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_activate_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n__isNat_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_n__plus_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_plus_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n__s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_S : A ->A.
   
   Hypothesis P_id_0_hat_1 : A.
   
   Hypothesis P_id_AND : A ->A ->A.
   
   Hypothesis P_id_ACTIVATE : A ->A.
   
   Hypothesis P_id_PLUS : A ->A ->A.
   
   Hypothesis P_id_U11_hat_1 : A ->A ->A.
   
   Hypothesis P_id_ISNAT : A ->A.
   
   Hypothesis P_id_U21_hat_1 : A ->A ->A ->A.
   
   Hypothesis P_id_S_monotonic :
    forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_S x10 <= P_id_S x9.
   
   Hypothesis P_id_AND_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_AND x10 x12 <= P_id_AND x9 x11.
   
   Hypothesis P_id_ACTIVATE_monotonic :
    forall x10 x9, 
     (A0 <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
   
   Hypothesis P_id_PLUS_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->P_id_PLUS x10 x12 <= P_id_PLUS x9 x11.
   
   Hypothesis P_id_U11_hat_1_monotonic :
    forall x12 x10 x9 x11, 
     (A0 <= x12)/\ (x12 <= x11) ->
      (A0 <= x10)/\ (x10 <= x9) ->
       P_id_U11_hat_1 x10 x12 <= P_id_U11_hat_1 x9 x11.
   
   Hypothesis P_id_ISNAT_monotonic :
    forall x10 x9, 
     (A0 <= x10)/\ (x10 <= x9) ->P_id_ISNAT x10 <= P_id_ISNAT x9.
   
   Hypothesis P_id_U21_hat_1_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (A0 <= x14)/\ (x14 <= x13) ->
      (A0 <= x12)/\ (x12 <= x11) ->
       (A0 <= x10)/\ (x10 <= x9) ->
        P_id_U21_hat_1 x10 x12 x14 <= P_id_U21_hat_1 x9 x11 x13.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_s (x9::nil)) => P_id_S (measure x9)
       | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0_hat_1 
       | (algebra.Alg.Term algebra.F.id_and (x10::x9::nil)) =>
        P_id_AND (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
        P_id_ACTIVATE (measure x9)
       | (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil)) =>
        P_id_PLUS (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil)) =>
        P_id_U11_hat_1 (measure x10) (measure x9)
       | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
        P_id_ISNAT (measure x9)
       | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil)) =>
        P_id_U21_hat_1 (measure x11) (measure x10) (measure x9)
       | _ => 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 x9;apply (P_id_ISNAT x9).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x11 x10 x9;apply (P_id_U21_hat_1 x11 x10 x9).
     
     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 x10 x9;apply (P_id_PLUS x10 x9).
     
     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 x10 x9;apply (P_id_AND x10 x9).
     
     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 x9;apply (P_id_ACTIVATE x9).
     
     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_0_hat_1 ).
     
     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 x9;apply (P_id_S x9).
     
     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 x10 x9;apply (P_id_U11_hat_1 x10 x9).
     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_U11 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_n__0 => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_0 => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_activate =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_n__isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_tt => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_n__plus =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_plus =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_U21 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_n__s =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_isNat =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                           P_id_S (measure x9)
                          | (algebra.Alg.Term algebra.F.id_0 nil) =>
                           P_id_0_hat_1 
                          | (algebra.Alg.Term algebra.F.id_and (x10::
                             x9::nil)) =>
                           P_id_AND (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_activate 
                             (x9::nil)) =>
                           P_id_ACTIVATE (measure x9)
                          | (algebra.Alg.Term algebra.F.id_plus (x10::
                             x9::nil)) =>
                           P_id_PLUS (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U11 (x10::
                             x9::nil)) =>
                           P_id_U11_hat_1 (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                           P_id_ISNAT (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::
                             x9::nil)) =>
                           P_id_U21_hat_1 (measure x11) (measure x10) 
                            (measure x9)
                          | _ => 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_U11_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_activate_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n__isNat_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_n__plus_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_plus_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n__s_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__isNat_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n__s_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_isNat_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     clear f.
     intros f.
     clear H.
     intros H.
     generalize H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_ISNAT_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_U21_hat_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_PLUS_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_AND_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_ACTIVATE_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply (Aop.(le_refl)).
     
     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_S_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_U11_hat_1_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_U11 : Z ->Z ->Z.
   
   Hypothesis P_id_n__0 : Z.
   
   Hypothesis P_id_s : Z ->Z.
   
   Hypothesis P_id_0 : Z.
   
   Hypothesis P_id_activate : Z ->Z.
   
   Hypothesis P_id_n__isNat : Z ->Z.
   
   Hypothesis P_id_and : Z ->Z ->Z.
   
   Hypothesis P_id_tt : Z.
   
   Hypothesis P_id_n__plus : Z ->Z ->Z.
   
   Hypothesis P_id_plus : Z ->Z ->Z.
   
   Hypothesis P_id_U21 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_n__s : Z ->Z.
   
   Hypothesis P_id_isNat : Z ->Z.
   
   Hypothesis P_id_U11_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->P_id_U11 x10 x12 <= P_id_U11 x9 x11.
   
   Hypothesis P_id_s_monotonic :
    forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
   
   Hypothesis P_id_activate_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
   
   Hypothesis P_id_n__isNat_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_n__isNat x10 <= P_id_n__isNat x9.
   
   Hypothesis P_id_and_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->P_id_and x10 x12 <= P_id_and x9 x11.
   
   Hypothesis P_id_n__plus_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->
       P_id_n__plus x10 x12 <= P_id_n__plus x9 x11.
   
   Hypothesis P_id_plus_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->
       P_id_plus x10 x12 <= P_id_plus x9 x11.
   
   Hypothesis P_id_U21_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       (min_value <= x10)/\ (x10 <= x9) ->
        P_id_U21 x10 x12 x14 <= P_id_U21 x9 x11 x13.
   
   Hypothesis P_id_n__s_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_n__s x10 <= P_id_n__s x9.
   
   Hypothesis P_id_isNat_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_isNat x10 <= P_id_isNat x9.
   
   Hypothesis P_id_U11_bounded :
    forall x10 x9, 
     (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_U11 x10 x9.
   
   Hypothesis P_id_n__0_bounded : min_value <= P_id_n__0 .
   
   Hypothesis P_id_s_bounded :
    forall x9, (min_value <= x9) ->min_value <= P_id_s x9.
   
   Hypothesis P_id_0_bounded : min_value <= P_id_0 .
   
   Hypothesis P_id_activate_bounded :
    forall x9, (min_value <= x9) ->min_value <= P_id_activate x9.
   
   Hypothesis P_id_n__isNat_bounded :
    forall x9, (min_value <= x9) ->min_value <= P_id_n__isNat x9.
   
   Hypothesis P_id_and_bounded :
    forall x10 x9, 
     (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_and x10 x9.
   
   Hypothesis P_id_tt_bounded : min_value <= P_id_tt .
   
   Hypothesis P_id_n__plus_bounded :
    forall x10 x9, 
     (min_value <= x9) ->
      (min_value <= x10) ->min_value <= P_id_n__plus x10 x9.
   
   Hypothesis P_id_plus_bounded :
    forall x10 x9, 
     (min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_plus x10 x9.
   
   Hypothesis P_id_U21_bounded :
    forall x10 x9 x11, 
     (min_value <= x9) ->
      (min_value <= x10) ->
       (min_value <= x11) ->min_value <= P_id_U21 x11 x10 x9.
   
   Hypothesis P_id_n__s_bounded :
    forall x9, (min_value <= x9) ->min_value <= P_id_n__s x9.
   
   Hypothesis P_id_isNat_bounded :
    forall x9, (min_value <= x9) ->min_value <= P_id_isNat x9.
   
   Definition measure  := 
     Interp.measure min_value P_id_U11 P_id_n__0 P_id_s P_id_0 P_id_activate 
      P_id_n__isNat P_id_and P_id_tt P_id_n__plus P_id_plus P_id_U21 
      P_id_n__s P_id_isNat.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil)) =>
                    P_id_U11 (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__0 nil) => P_id_n__0 
                   | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                    P_id_s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
                    P_id_activate (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil)) =>
                    P_id_n__isNat (measure x9)
                   | (algebra.Alg.Term algebra.F.id_and (x10::x9::nil)) =>
                    P_id_and (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_n__plus (x10::x9::nil)) =>
                    P_id_n__plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil)) =>
                    P_id_plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil)) =>
                    P_id_U21 (measure x11) (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__s (x9::nil)) =>
                    P_id_n__s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                    P_id_isNat (measure x9)
                   | _ => 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_U11_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_activate_monotonic;assumption.
     intros ;apply P_id_n__isNat_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_n__plus_monotonic;assumption.
     intros ;apply P_id_plus_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_n__s_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_n__0_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_activate_bounded;assumption.
     intros ;apply P_id_n__isNat_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_n__plus_bounded;assumption.
     intros ;apply P_id_plus_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_n__s_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_S : Z ->Z.
   
   Hypothesis P_id_0_hat_1 : Z.
   
   Hypothesis P_id_AND : Z ->Z ->Z.
   
   Hypothesis P_id_ACTIVATE : Z ->Z.
   
   Hypothesis P_id_PLUS : Z ->Z ->Z.
   
   Hypothesis P_id_U11_hat_1 : Z ->Z ->Z.
   
   Hypothesis P_id_ISNAT : Z ->Z.
   
   Hypothesis P_id_U21_hat_1 : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_S_monotonic :
    forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_S x10 <= P_id_S x9.
   
   Hypothesis P_id_AND_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->P_id_AND x10 x12 <= P_id_AND x9 x11.
   
   Hypothesis P_id_ACTIVATE_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
   
   Hypothesis P_id_PLUS_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->
       P_id_PLUS x10 x12 <= P_id_PLUS x9 x11.
   
   Hypothesis P_id_U11_hat_1_monotonic :
    forall x12 x10 x9 x11, 
     (min_value <= x12)/\ (x12 <= x11) ->
      (min_value <= x10)/\ (x10 <= x9) ->
       P_id_U11_hat_1 x10 x12 <= P_id_U11_hat_1 x9 x11.
   
   Hypothesis P_id_ISNAT_monotonic :
    forall x10 x9, 
     (min_value <= x10)/\ (x10 <= x9) ->P_id_ISNAT x10 <= P_id_ISNAT x9.
   
   Hypothesis P_id_U21_hat_1_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (min_value <= x14)/\ (x14 <= x13) ->
      (min_value <= x12)/\ (x12 <= x11) ->
       (min_value <= x10)/\ (x10 <= x9) ->
        P_id_U21_hat_1 x10 x12 x14 <= P_id_U21_hat_1 x9 x11 x13.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_U11 P_id_n__0 P_id_s P_id_0 
      P_id_activate P_id_n__isNat P_id_and P_id_tt P_id_n__plus P_id_plus 
      P_id_U21 P_id_n__s P_id_isNat P_id_S P_id_0_hat_1 P_id_AND 
      P_id_ACTIVATE P_id_PLUS P_id_U11_hat_1 P_id_ISNAT P_id_U21_hat_1.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                           P_id_S (measure x9)
                          | (algebra.Alg.Term algebra.F.id_0 nil) =>
                           P_id_0_hat_1 
                          | (algebra.Alg.Term algebra.F.id_and (x10::
                             x9::nil)) =>
                           P_id_AND (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_activate 
                             (x9::nil)) =>
                           P_id_ACTIVATE (measure x9)
                          | (algebra.Alg.Term algebra.F.id_plus (x10::
                             x9::nil)) =>
                           P_id_PLUS (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U11 (x10::
                             x9::nil)) =>
                           P_id_U11_hat_1 (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                           P_id_ISNAT (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::
                             x9::nil)) =>
                           P_id_U21_hat_1 (measure x11) (measure x10) 
                            (measure x9)
                          | _ => 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_U11_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_activate_monotonic;assumption.
     intros ;apply P_id_n__isNat_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_n__plus_monotonic;assumption.
     intros ;apply P_id_plus_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_n__s_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_n__0_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_activate_bounded;assumption.
     intros ;apply P_id_n__isNat_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_n__plus_bounded;assumption.
     intros ;apply P_id_plus_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_n__s_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_S_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_ACTIVATE_monotonic;assumption.
     intros ;apply P_id_PLUS_monotonic;assumption.
     intros ;apply P_id_U11_hat_1_monotonic;assumption.
     intros ;apply P_id_ISNAT_monotonic;assumption.
     intros ;apply P_id_U21_hat_1_monotonic;assumption.
   Qed.
   
   End S.
End InterpZ.

Module WF_R_xml_0_deep_rew.
 Inductive DP_R_xml_0  :
  algebra.Alg.term ->algebra.Alg.term ->Prop := 
    (* <U11(tt,N_),activate(N_)> *)
   | DP_R_xml_0_0 :
    forall x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil))
    (* <U21(tt,M_,N_),s(plus(activate(N_),activate(M_)))> *)
   | DP_R_xml_0_1 :
    forall x2 x10 x1 x9 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x11) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x9) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                    algebra.F.id_plus ((algebra.Alg.Term 
                    algebra.F.id_activate (x1::nil))::(algebra.Alg.Term 
                    algebra.F.id_activate (x2::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
    (* <U21(tt,M_,N_),plus(activate(N_),activate(M_))> *)
   | DP_R_xml_0_2 :
    forall x2 x10 x1 x9 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x11) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x9) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Term 
                    algebra.F.id_activate (x1::nil))::(algebra.Alg.Term 
                    algebra.F.id_activate (x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
    (* <U21(tt,M_,N_),activate(N_)> *)
   | DP_R_xml_0_3 :
    forall x2 x10 x1 x9 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x11) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x9) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
    (* <U21(tt,M_,N_),activate(M_)> *)
   | DP_R_xml_0_4 :
    forall x2 x10 x1 x9 x11, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x11) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x2 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x9) ->
        DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
    (* <and(tt,X_),activate(X_)> *)
   | DP_R_xml_0_5 :
    forall x10 x9 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_tt nil) 
       x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_and (x10::x9::nil))
   
    (* <isNat(n__plus(V1_,V2_)),and(isNat(activate(V1_)),n__isNat(activate(V2_)))> *)
   | DP_R_xml_0_6 :
    forall x4 x9 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_isNat ((algebra.Alg.Term 
                  algebra.F.id_activate (x4::nil))::nil))::(algebra.Alg.Term 
                  algebra.F.id_n__isNat ((algebra.Alg.Term 
                  algebra.F.id_activate (x5::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <isNat(n__plus(V1_,V2_)),isNat(activate(V1_))> *)
   | DP_R_xml_0_7 :
    forall x4 x9 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
                  algebra.F.id_activate (x4::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <isNat(n__plus(V1_,V2_)),activate(V1_)> *)
   | DP_R_xml_0_8 :
    forall x4 x9 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <isNat(n__plus(V1_,V2_)),activate(V2_)> *)
   | DP_R_xml_0_9 :
    forall x4 x9 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x5::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <isNat(n__s(V1_)),isNat(activate(V1_))> *)
   | DP_R_xml_0_10 :
    forall x4 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_isNat ((algebra.Alg.Term 
                  algebra.F.id_activate (x4::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <isNat(n__s(V1_)),activate(V1_)> *)
   | DP_R_xml_0_11 :
    forall x4 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x4::nil)) 
       (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    (* <plus(N_,0),U11(isNat(N_),N_)> *)
   | DP_R_xml_0_12 :
    forall x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 nil) 
        x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Term 
                   algebra.F.id_isNat (x1::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    (* <plus(N_,0),isNat(N_)> *)
   | DP_R_xml_0_13 :
    forall x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 nil) 
        x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_isNat (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
   
    (* <plus(N_,s(M_)),U21(and(isNat(M_),n__isNat(N_)),M_,N_)> *)
   | DP_R_xml_0_14 :
    forall x2 x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x2::nil)) 
        x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Term 
                   algebra.F.id_and ((algebra.Alg.Term algebra.F.id_isNat 
                   (x2::nil))::(algebra.Alg.Term algebra.F.id_n__isNat 
                   (x1::nil))::nil))::x2::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    (* <plus(N_,s(M_)),and(isNat(M_),n__isNat(N_))> *)
   | DP_R_xml_0_15 :
    forall x2 x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x2::nil)) 
        x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                   algebra.F.id_isNat (x2::nil))::(algebra.Alg.Term 
                   algebra.F.id_n__isNat (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    (* <plus(N_,s(M_)),isNat(M_)> *)
   | DP_R_xml_0_16 :
    forall x2 x10 x1 x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x10) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_s (x2::nil)) 
        x9) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_isNat (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    (* <activate(n__0),0> *)
   | DP_R_xml_0_17 :
    forall x9, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_n__0 nil) 
       x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_0 nil) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
   
    (* <activate(n__plus(X1_,X2_)),plus(activate(X1_),activate(X2_))> *)
   | DP_R_xml_0_18 :
    forall x6 x9 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Term 
                  algebra.F.id_activate (x6::nil))::(algebra.Alg.Term 
                  algebra.F.id_activate (x7::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    (* <activate(n__plus(X1_,X2_)),activate(X1_)> *)
   | DP_R_xml_0_19 :
    forall x6 x9 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x6::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    (* <activate(n__plus(X1_,X2_)),activate(X2_)> *)
   | DP_R_xml_0_20 :
    forall x6 x9 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x7::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    (* <activate(n__isNat(X_)),isNat(X_)> *)
   | DP_R_xml_0_21 :
    forall x9 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__isNat (x3::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_isNat (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    (* <activate(n__s(X_)),s(activate(X_))> *)
   | DP_R_xml_0_22 :
    forall x9 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__s (x3::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term 
                  algebra.F.id_activate (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    (* <activate(n__s(X_)),activate(X_)> *)
   | DP_R_xml_0_23 :
    forall x9 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n__s (x3::nil)) x9) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_activate (x9::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 := 
     (* <activate(n__s(X_)),s(activate(X_))> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x9 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__s (x3::nil)) x9) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_s 
                             ((algebra.Alg.Term algebra.F.id_activate 
                             (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::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 := 
     (* <activate(n__0),0> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_n__0 nil) 
        x9) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_0 nil) 
        (algebra.Alg.Term algebra.F.id_activate (x9::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 := 
     (* <U21(tt,M_,N_),s(plus(activate(N_),activate(M_)))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x2 x10 x1 x9 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x11) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x9) ->
         DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_s 
                               ((algebra.Alg.Term algebra.F.id_plus 
                               ((algebra.Alg.Term algebra.F.id_activate 
                               (x1::nil))::(algebra.Alg.Term 
                               algebra.F.id_activate (x2::nil))::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_3 :
   forall x y, 
    (DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <activate(n__plus(X1_,X2_)),plus(activate(X1_),activate(X2_))> *)
    | DP_R_xml_0_scc_4_0 :
     forall x6 x9 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_plus 
                         ((algebra.Alg.Term algebra.F.id_activate 
                         (x6::nil))::(algebra.Alg.Term algebra.F.id_activate 
                         (x7::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     (* <plus(N_,0),U11(isNat(N_),N_)> *)
    | DP_R_xml_0_scc_4_1 :
     forall x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_0 nil) 
         x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_U11 
                          ((algebra.Alg.Term algebra.F.id_isNat (x1::nil))::
                          x1::nil)) 
         (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     (* <U11(tt,N_),activate(N_)> *)
    | DP_R_xml_0_scc_4_2 :
     forall x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil))
     (* <activate(n__plus(X1_,X2_)),activate(X1_)> *)
    | DP_R_xml_0_scc_4_3 :
     forall x6 x9 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x6::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     (* <activate(n__plus(X1_,X2_)),activate(X2_)> *)
    | DP_R_xml_0_scc_4_4 :
     forall x6 x9 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x7::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     (* <activate(n__isNat(X_)),isNat(X_)> *)
    | DP_R_xml_0_scc_4_5 :
     forall x9 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__isNat (x3::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_isNat (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    
     (* <isNat(n__plus(V1_,V2_)),and(isNat(activate(V1_)),n__isNat(activate(V2_)))> *)
    | DP_R_xml_0_scc_4_6 :
     forall x4 x9 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_and 
                         ((algebra.Alg.Term algebra.F.id_isNat 
                         ((algebra.Alg.Term algebra.F.id_activate 
                         (x4::nil))::nil))::(algebra.Alg.Term 
                         algebra.F.id_n__isNat ((algebra.Alg.Term 
                         algebra.F.id_activate (x5::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <and(tt,X_),activate(X_)> *)
    | DP_R_xml_0_scc_4_7 :
     forall x10 x9 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_and (x10::x9::nil))
     (* <activate(n__s(X_)),activate(X_)> *)
    | DP_R_xml_0_scc_4_8 :
     forall x9 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__s (x3::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_activate (x9::nil))
    
     (* <isNat(n__plus(V1_,V2_)),isNat(activate(V1_))> *)
    | DP_R_xml_0_scc_4_9 :
     forall x4 x9 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_isNat 
                         ((algebra.Alg.Term algebra.F.id_activate 
                         (x4::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <isNat(n__plus(V1_,V2_)),activate(V1_)> *)
    | DP_R_xml_0_scc_4_10 :
     forall x4 x9 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <isNat(n__plus(V1_,V2_)),activate(V2_)> *)
    | DP_R_xml_0_scc_4_11 :
     forall x4 x9 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x5::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <isNat(n__s(V1_)),isNat(activate(V1_))> *)
    | DP_R_xml_0_scc_4_12 :
     forall x4 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_isNat 
                         ((algebra.Alg.Term algebra.F.id_activate 
                         (x4::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <isNat(n__s(V1_)),activate(V1_)> *)
    | DP_R_xml_0_scc_4_13 :
     forall x4 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
       DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x4::nil)) 
        (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     (* <plus(N_,0),isNat(N_)> *)
    | DP_R_xml_0_scc_4_14 :
     forall x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_0 nil) 
         x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_isNat (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    
     (* <plus(N_,s(M_)),U21(and(isNat(M_),n__isNat(N_)),M_,N_)> *)
    | DP_R_xml_0_scc_4_15 :
     forall x2 x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_U21 
                          ((algebra.Alg.Term algebra.F.id_and 
                          ((algebra.Alg.Term algebra.F.id_isNat (x2::nil))::
                          (algebra.Alg.Term algebra.F.id_n__isNat 
                          (x1::nil))::nil))::x2::x1::nil)) 
         (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
    
     (* <U21(tt,M_,N_),plus(activate(N_),activate(M_))> *)
    | DP_R_xml_0_scc_4_16 :
     forall x2 x10 x1 x9 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x11) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x9) ->
         DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_plus 
                           ((algebra.Alg.Term algebra.F.id_activate 
                           (x1::nil))::(algebra.Alg.Term 
                           algebra.F.id_activate (x2::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
    
     (* <plus(N_,s(M_)),and(isNat(M_),n__isNat(N_))> *)
    | DP_R_xml_0_scc_4_17 :
     forall x2 x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_and 
                          ((algebra.Alg.Term algebra.F.id_isNat (x2::nil))::
                          (algebra.Alg.Term algebra.F.id_n__isNat 
                          (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     (* <plus(N_,s(M_)),isNat(M_)> *)
    | DP_R_xml_0_scc_4_18 :
     forall x2 x10 x1 x9, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x10) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_isNat (x2::nil)) 
         (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     (* <U21(tt,M_,N_),activate(N_)> *)
    | DP_R_xml_0_scc_4_19 :
     forall x2 x10 x1 x9 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x11) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x9) ->
         DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
     (* <U21(tt,M_,N_),activate(M_)> *)
    | DP_R_xml_0_scc_4_20 :
     forall x2 x10 x1 x9 x11, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_tt nil) 
        x11) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x2 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x9) ->
         DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_activate (x2::nil)) 
          (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_4.
   Inductive DP_R_xml_0_scc_4_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <U11(tt,N_),activate(N_)> *)
     | DP_R_xml_0_scc_4_large_0 :
      forall x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x9) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_activate 
                                 (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil))
      (* <and(tt,X_),activate(X_)> *)
     | DP_R_xml_0_scc_4_large_1 :
      forall x10 x9 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x9) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_activate 
                                 (x3::nil)) 
          (algebra.Alg.Term algebra.F.id_and (x10::x9::nil))
      (* <isNat(n__s(V1_)),activate(V1_)> *)
     | DP_R_xml_0_scc_4_large_2 :
      forall x4 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
        DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_activate 
                                (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_4_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <activate(n__plus(X1_,X2_)),plus(activate(X1_),activate(X2_))> *)
     | DP_R_xml_0_scc_4_strict_0 :
      forall x6 x9 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_plus 
                                 ((algebra.Alg.Term algebra.F.id_activate 
                                 (x6::nil))::(algebra.Alg.Term 
                                 algebra.F.id_activate (x7::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_activate (x9::nil))
      (* <plus(N_,0),U11(isNat(N_),N_)> *)
     | DP_R_xml_0_scc_4_strict_1 :
      forall x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_0 nil) 
          x9) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_U11 
                                  ((algebra.Alg.Term algebra.F.id_isNat 
                                  (x1::nil))::x1::nil)) 
          (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     
      (* <activate(n__plus(X1_,X2_)),activate(X1_)> *)
     | DP_R_xml_0_scc_4_strict_2 :
      forall x6 x9 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                 (x6::nil)) 
         (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     
      (* <activate(n__plus(X1_,X2_)),activate(X2_)> *)
     | DP_R_xml_0_scc_4_strict_3 :
      forall x6 x9 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x6::x7::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                 (x7::nil)) 
         (algebra.Alg.Term algebra.F.id_activate (x9::nil))
      (* <activate(n__isNat(X_)),isNat(X_)> *)
     | DP_R_xml_0_scc_4_strict_4 :
      forall x9 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__isNat (x3::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_isNat 
                                 (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     
      (* <isNat(n__plus(V1_,V2_)),and(isNat(activate(V1_)),n__isNat(activate(V2_)))> *)
     | DP_R_xml_0_scc_4_strict_5 :
      forall x4 x9 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_and 
                                 ((algebra.Alg.Term algebra.F.id_isNat 
                                 ((algebra.Alg.Term algebra.F.id_activate 
                                 (x4::nil))::nil))::(algebra.Alg.Term 
                                 algebra.F.id_n__isNat ((algebra.Alg.Term 
                                 algebra.F.id_activate 
                                 (x5::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
      (* <activate(n__s(X_)),activate(X_)> *)
     | DP_R_xml_0_scc_4_strict_6 :
      forall x9 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__s (x3::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                 (x3::nil)) 
         (algebra.Alg.Term algebra.F.id_activate (x9::nil))
     
      (* <isNat(n__plus(V1_,V2_)),isNat(activate(V1_))> *)
     | DP_R_xml_0_scc_4_strict_7 :
      forall x4 x9 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_isNat 
                                 ((algebra.Alg.Term algebra.F.id_activate 
                                 (x4::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     
      (* <isNat(n__plus(V1_,V2_)),activate(V1_)> *)
     | DP_R_xml_0_scc_4_strict_8 :
      forall x4 x9 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                 (x4::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     
      (* <isNat(n__plus(V1_,V2_)),activate(V2_)> *)
     | DP_R_xml_0_scc_4_strict_9 :
      forall x4 x9 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__plus (x4::x5::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                 (x5::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
     
      (* <isNat(n__s(V1_)),isNat(activate(V1_))> *)
     | DP_R_xml_0_scc_4_strict_10 :
      forall x4 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
        DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_isNat 
                                 ((algebra.Alg.Term algebra.F.id_activate 
                                 (x4::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
      (* <plus(N_,0),isNat(N_)> *)
     | DP_R_xml_0_scc_4_strict_11 :
      forall x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_0 nil) 
          x9) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_isNat 
                                  (x1::nil)) 
          (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     
      (* <plus(N_,s(M_)),U21(and(isNat(M_),n__isNat(N_)),M_,N_)> *)
     | DP_R_xml_0_scc_4_strict_12 :
      forall x2 x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_U21 
                                  ((algebra.Alg.Term algebra.F.id_and 
                                  ((algebra.Alg.Term algebra.F.id_isNat 
                                  (x2::nil))::(algebra.Alg.Term 
                                  algebra.F.id_n__isNat (x1::nil))::nil))::
                                  x2::x1::nil)) 
          (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
     
      (* <U21(tt,M_,N_),plus(activate(N_),activate(M_))> *)
     | DP_R_xml_0_scc_4_strict_13 :
      forall x2 x10 x1 x9 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x11) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x2 x10) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x9) ->
          DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_plus 
                                   ((algebra.Alg.Term algebra.F.id_activate 
                                   (x1::nil))::(algebra.Alg.Term 
                                   algebra.F.id_activate (x2::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
     
      (* <plus(N_,s(M_)),and(isNat(M_),n__isNat(N_))> *)
     | DP_R_xml_0_scc_4_strict_14 :
      forall x2 x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_and 
                                  ((algebra.Alg.Term algebra.F.id_isNat 
                                  (x2::nil))::(algebra.Alg.Term 
                                  algebra.F.id_n__isNat (x1::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
      (* <plus(N_,s(M_)),isNat(M_)> *)
     | DP_R_xml_0_scc_4_strict_15 :
      forall x2 x10 x1 x9, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x10) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_s (x2::nil)) x9) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_isNat 
                                  (x2::nil)) 
          (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil))
      (* <U21(tt,M_,N_),activate(N_)> *)
     | DP_R_xml_0_scc_4_strict_16 :
      forall x2 x10 x1 x9 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x11) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x2 x10) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x9) ->
          DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                   (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
      (* <U21(tt,M_,N_),activate(M_)> *)
     | DP_R_xml_0_scc_4_strict_17 :
      forall x2 x10 x1 x9 x11, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  (algebra.Alg.Term algebra.F.id_tt nil) 
         x11) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x2 x10) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x9) ->
          DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_activate 
                                   (x2::nil)) 
           (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_4_large.
    Inductive DP_R_xml_0_scc_4_large_non_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <isNat(n__s(V1_)),activate(V1_)> *)
      | DP_R_xml_0_scc_4_large_non_scc_1_0 :
       forall x4 x9, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_n__s (x4::nil)) x9) ->
         DP_R_xml_0_scc_4_large_non_scc_1 (algebra.Alg.Term 
                                           algebra.F.id_activate (x4::nil)) 
          (algebra.Alg.Term algebra.F.id_isNat (x9::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_4_large_non_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_4_large_non_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_4_large_non_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <and(tt,X_),activate(X_)> *)
      | DP_R_xml_0_scc_4_large_non_scc_2_0 :
       forall x10 x9 x3, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x10) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x9) ->
          DP_R_xml_0_scc_4_large_non_scc_2 (algebra.Alg.Term 
                                            algebra.F.id_activate (x3::nil)) 
           (algebra.Alg.Term algebra.F.id_and (x10::x9::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_4_large_non_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_4_large_non_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Inductive DP_R_xml_0_scc_4_large_non_scc_3  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <U11(tt,N_),activate(N_)> *)
      | DP_R_xml_0_scc_4_large_non_scc_3_0 :
       forall x10 x1 x9, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   (algebra.Alg.Term algebra.F.id_tt nil) 
          x10) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x1 x9) ->
          DP_R_xml_0_scc_4_large_non_scc_3 (algebra.Alg.Term 
                                            algebra.F.id_activate (x1::nil)) 
           (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil))
    .
    
    
    Lemma acc_DP_R_xml_0_scc_4_large_non_scc_3 :
     forall x y, 
      (DP_R_xml_0_scc_4_large_non_scc_3 x y) ->
       Acc WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large x.
    Proof.
      intros x y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_4_large_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_4_large_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_4_large_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_4_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_4_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_U11 (x9:Z) (x10:Z) := 2 + 1* x10.
   
   Definition P_id_n__0  := 1.
   
   Definition P_id_s (x9:Z) := 1 + 1* x9.
   
   Definition P_id_0  := 1.
   
   Definition P_id_activate (x9:Z) := 1* x9.
   
   Definition P_id_n__isNat (x9:Z) := 1* x9.
   
   Definition P_id_and (x9:Z) (x10:Z) := 1* x10.
   
   Definition P_id_tt  := 1.
   
   Definition P_id_n__plus (x9:Z) (x10:Z) := 2 + 1* x9 + 2* x10.
   
   Definition P_id_plus (x9:Z) (x10:Z) := 2 + 1* x9 + 2* x10.
   
   Definition P_id_U21 (x9:Z) (x10:Z) (x11:Z) := 3 + 2* x10 + 1* x11.
   
   Definition P_id_n__s (x9:Z) := 1 + 1* x9.
   
   Definition P_id_isNat (x9:Z) := 1* x9.
   
   Lemma P_id_U11_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_U11 x10 x12 <= P_id_U11 x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_monotonic :
    forall x10 x9, (1 <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_activate_monotonic :
    forall x10 x9, 
     (1 <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__isNat_monotonic :
    forall x10 x9, 
     (1 <= x10)/\ (x10 <= x9) ->P_id_n__isNat x10 <= P_id_n__isNat x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_and x10 x12 <= P_id_and x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__plus_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_n__plus x10 x12 <= P_id_n__plus x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_plus_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_plus x10 x12 <= P_id_plus x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (1 <= x14)/\ (x14 <= x13) ->
      (1 <= x12)/\ (x12 <= x11) ->
       (1 <= x10)/\ (x10 <= x9) ->P_id_U21 x10 x12 x14 <= P_id_U21 x9 x11 x13.
   Proof.
     intros x14 x13 x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__s_monotonic :
    forall x10 x9, (1 <= x10)/\ (x10 <= x9) ->P_id_n__s x10 <= P_id_n__s x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_monotonic :
    forall x10 x9, (1 <= x10)/\ (x10 <= x9) ->P_id_isNat x10 <= P_id_isNat x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_bounded :
    forall x10 x9, (1 <= x9) ->(1 <= x10) ->1 <= P_id_U11 x10 x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__0_bounded : 1 <= P_id_n__0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_s_bounded : forall x9, (1 <= x9) ->1 <= P_id_s x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : 1 <= P_id_0 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_activate_bounded :
    forall x9, (1 <= x9) ->1 <= P_id_activate x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__isNat_bounded :
    forall x9, (1 <= x9) ->1 <= P_id_n__isNat x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x10 x9, (1 <= x9) ->(1 <= x10) ->1 <= P_id_and x10 x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tt_bounded : 1 <= P_id_tt .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__plus_bounded :
    forall x10 x9, (1 <= x9) ->(1 <= x10) ->1 <= P_id_n__plus x10 x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_plus_bounded :
    forall x10 x9, (1 <= x9) ->(1 <= x10) ->1 <= P_id_plus x10 x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_bounded :
    forall x10 x9 x11, 
     (1 <= x9) ->(1 <= x10) ->(1 <= x11) ->1 <= P_id_U21 x11 x10 x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n__s_bounded : forall x9, (1 <= x9) ->1 <= P_id_n__s x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_isNat_bounded : forall x9, (1 <= x9) ->1 <= P_id_isNat x9.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 1 P_id_U11 P_id_n__0 P_id_s P_id_0 P_id_activate 
      P_id_n__isNat P_id_and P_id_tt P_id_n__plus P_id_plus P_id_U21 
      P_id_n__s P_id_isNat.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_U11 (x10::x9::nil)) =>
                    P_id_U11 (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__0 nil) => P_id_n__0 
                   | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                    P_id_s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0 
                   | (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
                    P_id_activate (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__isNat (x9::nil)) =>
                    P_id_n__isNat (measure x9)
                   | (algebra.Alg.Term algebra.F.id_and (x10::x9::nil)) =>
                    P_id_and (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt 
                   | (algebra.Alg.Term algebra.F.id_n__plus (x10::x9::nil)) =>
                    P_id_n__plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_plus (x10::x9::nil)) =>
                    P_id_plus (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::x9::nil)) =>
                    P_id_U21 (measure x11) (measure x10) (measure x9)
                   | (algebra.Alg.Term algebra.F.id_n__s (x9::nil)) =>
                    P_id_n__s (measure x9)
                   | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                    P_id_isNat (measure x9)
                   | _ => 1
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 1 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_U11_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_activate_monotonic;assumption.
     intros ;apply P_id_n__isNat_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_n__plus_monotonic;assumption.
     intros ;apply P_id_plus_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_n__s_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_n__0_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_activate_bounded;assumption.
     intros ;apply P_id_n__isNat_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_n__plus_bounded;assumption.
     intros ;apply P_id_plus_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_n__s_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_S (x9:Z) := 0.
   
   Definition P_id_0_hat_1  := 0.
   
   Definition P_id_AND (x9:Z) (x10:Z) := 1 + 1* x9 + 1* x10.
   
   Definition P_id_ACTIVATE (x9:Z) := 2 + 1* x9.
   
   Definition P_id_PLUS (x9:Z) (x10:Z) := 2 + 1* x9 + 2* x10.
   
   Definition P_id_U11_hat_1 (x9:Z) (x10:Z) := 2 + 1* x10.
   
   Definition P_id_ISNAT (x9:Z) := 1 + 1* x9.
   
   Definition P_id_U21_hat_1 (x9:Z) (x10:Z) (x11:Z) := 3 + 2* x10 + 1* x11.
   
   Lemma P_id_S_monotonic :
    forall x10 x9, (1 <= x10)/\ (x10 <= x9) ->P_id_S x10 <= P_id_S x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_AND_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_AND x10 x12 <= P_id_AND x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVATE_monotonic :
    forall x10 x9, 
     (1 <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PLUS_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->P_id_PLUS x10 x12 <= P_id_PLUS x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U11_hat_1_monotonic :
    forall x12 x10 x9 x11, 
     (1 <= x12)/\ (x12 <= x11) ->
      (1 <= x10)/\ (x10 <= x9) ->
       P_id_U11_hat_1 x10 x12 <= P_id_U11_hat_1 x9 x11.
   Proof.
     intros x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ISNAT_monotonic :
    forall x10 x9, (1 <= x10)/\ (x10 <= x9) ->P_id_ISNAT x10 <= P_id_ISNAT x9.
   Proof.
     intros x10 x9.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_U21_hat_1_monotonic :
    forall x12 x10 x14 x9 x13 x11, 
     (1 <= x14)/\ (x14 <= x13) ->
      (1 <= x12)/\ (x12 <= x11) ->
       (1 <= x10)/\ (x10 <= x9) ->
        P_id_U21_hat_1 x10 x12 x14 <= P_id_U21_hat_1 x9 x11 x13.
   Proof.
     intros x14 x13 x12 x11 x10 x9.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 1 P_id_U11 P_id_n__0 P_id_s P_id_0 P_id_activate 
      P_id_n__isNat P_id_and P_id_tt P_id_n__plus P_id_plus P_id_U21 
      P_id_n__s P_id_isNat P_id_S P_id_0_hat_1 P_id_AND P_id_ACTIVATE 
      P_id_PLUS P_id_U11_hat_1 P_id_ISNAT P_id_U21_hat_1.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
                           P_id_S (measure x9)
                          | (algebra.Alg.Term algebra.F.id_0 nil) =>
                           P_id_0_hat_1 
                          | (algebra.Alg.Term algebra.F.id_and (x10::
                             x9::nil)) =>
                           P_id_AND (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_activate 
                             (x9::nil)) =>
                           P_id_ACTIVATE (measure x9)
                          | (algebra.Alg.Term algebra.F.id_plus (x10::
                             x9::nil)) =>
                           P_id_PLUS (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U11 (x10::
                             x9::nil)) =>
                           P_id_U11_hat_1 (measure x10) (measure x9)
                          | (algebra.Alg.Term algebra.F.id_isNat (x9::nil)) =>
                           P_id_ISNAT (measure x9)
                          | (algebra.Alg.Term algebra.F.id_U21 (x11::x10::
                             x9::nil)) =>
                           P_id_U21_hat_1 (measure x11) (measure x10) 
                            (measure x9)
                          | _ => 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_U11_monotonic;assumption.
     intros ;apply P_id_s_monotonic;assumption.
     intros ;apply P_id_activate_monotonic;assumption.
     intros ;apply P_id_n__isNat_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id_n__plus_monotonic;assumption.
     intros ;apply P_id_plus_monotonic;assumption.
     intros ;apply P_id_U21_monotonic;assumption.
     intros ;apply P_id_n__s_monotonic;assumption.
     intros ;apply P_id_isNat_monotonic;assumption.
     intros ;apply P_id_U11_bounded;assumption.
     intros ;apply P_id_n__0_bounded;assumption.
     intros ;apply P_id_s_bounded;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_activate_bounded;assumption.
     intros ;apply P_id_n__isNat_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id_tt_bounded;assumption.
     intros ;apply P_id_n__plus_bounded;assumption.
     intros ;apply P_id_plus_bounded;assumption.
     intros ;apply P_id_U21_bounded;assumption.
     intros ;apply P_id_n__s_bounded;assumption.
     intros ;apply P_id_isNat_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_S_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_ACTIVATE_monotonic;assumption.
     intros ;apply P_id_PLUS_monotonic;assumption.
     intros ;apply P_id_U11_hat_1_monotonic;assumption.
     intros ;apply P_id_ISNAT_monotonic;assumption.
     intros ;apply P_id_U21_hat_1_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 1) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 1)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_4_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 1) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 1)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_4_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 1)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_4_large  := WF_DP_R_xml_0_scc_4_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_4.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_4_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_4_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_4_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_4_large.
   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' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
    apply wf_DP_R_xml_0_scc_4.
  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_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_4;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_scc_3;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_scc_2;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_scc_1;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_scc_0;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


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