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_active *)
    | id_active : symb
     (* id_X *)
    | id_X : symb
     (* id_c *)
    | id_c : symb
     (* id_mark *)
    | id_mark : symb
     (* id_ok *)
    | id_ok : symb
     (* id_start *)
    | id_start : symb
     (* id_f *)
    | id_f : symb
     (* id_proper *)
    | id_proper : symb
     (* id_check *)
    | id_check : symb
     (* id_top *)
    | id_top : symb
     (* id_found *)
    | id_found : symb
     (* id_match *)
    | id_match : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_active,id_active => true
      | id_X,id_X => true
      | id_c,id_c => true
      | id_mark,id_mark => true
      | id_ok,id_ok => true
      | id_start,id_start => true
      | id_f,id_f => true
      | id_proper,id_proper => true
      | id_check,id_check => true
      | id_top,id_top => true
      | id_found,id_found => true
      | id_match,id_match => 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_active,id_active => refl_equal _
             | id_X,id_X => refl_equal _
             | id_c,id_c => refl_equal _
             | id_mark,id_mark => refl_equal _
             | id_ok,id_ok => refl_equal _
             | id_start,id_start => refl_equal _
             | id_f,id_f => refl_equal _
             | id_proper,id_proper => refl_equal _
             | id_check,id_check => refl_equal _
             | id_top,id_top => refl_equal _
             | id_found,id_found => refl_equal _
             | id_match,id_match => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_active => term.Free 1
      | id_X => term.Free 0
      | id_c => term.Free 0
      | id_mark => term.Free 1
      | id_ok => term.Free 1
      | id_start => term.Free 1
      | id_f => term.Free 1
      | id_proper => term.Free 1
      | id_check => term.Free 1
      | id_top => term.Free 1
      | id_found => term.Free 1
      | id_match => term.Free 2
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_active,id_active => true
      | id_active,id_X => false
      | id_active,id_c => false
      | id_active,id_mark => false
      | id_active,id_ok => false
      | id_active,id_start => false
      | id_active,id_f => false
      | id_active,id_proper => false
      | id_active,id_check => false
      | id_active,id_top => false
      | id_active,id_found => false
      | id_active,id_match => false
      | id_X,id_active => true
      | id_X,id_X => true
      | id_X,id_c => false
      | id_X,id_mark => false
      | id_X,id_ok => false
      | id_X,id_start => false
      | id_X,id_f => false
      | id_X,id_proper => false
      | id_X,id_check => false
      | id_X,id_top => false
      | id_X,id_found => false
      | id_X,id_match => false
      | id_c,id_active => true
      | id_c,id_X => true
      | id_c,id_c => true
      | id_c,id_mark => false
      | id_c,id_ok => false
      | id_c,id_start => false
      | id_c,id_f => false
      | id_c,id_proper => false
      | id_c,id_check => false
      | id_c,id_top => false
      | id_c,id_found => false
      | id_c,id_match => false
      | id_mark,id_active => true
      | id_mark,id_X => true
      | id_mark,id_c => true
      | id_mark,id_mark => true
      | id_mark,id_ok => false
      | id_mark,id_start => false
      | id_mark,id_f => false
      | id_mark,id_proper => false
      | id_mark,id_check => false
      | id_mark,id_top => false
      | id_mark,id_found => false
      | id_mark,id_match => false
      | id_ok,id_active => true
      | id_ok,id_X => true
      | id_ok,id_c => true
      | id_ok,id_mark => true
      | id_ok,id_ok => true
      | id_ok,id_start => false
      | id_ok,id_f => false
      | id_ok,id_proper => false
      | id_ok,id_check => false
      | id_ok,id_top => false
      | id_ok,id_found => false
      | id_ok,id_match => false
      | id_start,id_active => true
      | id_start,id_X => true
      | id_start,id_c => true
      | id_start,id_mark => true
      | id_start,id_ok => true
      | id_start,id_start => true
      | id_start,id_f => false
      | id_start,id_proper => false
      | id_start,id_check => false
      | id_start,id_top => false
      | id_start,id_found => false
      | id_start,id_match => false
      | id_f,id_active => true
      | id_f,id_X => true
      | id_f,id_c => true
      | id_f,id_mark => true
      | id_f,id_ok => true
      | id_f,id_start => true
      | id_f,id_f => true
      | id_f,id_proper => false
      | id_f,id_check => false
      | id_f,id_top => false
      | id_f,id_found => false
      | id_f,id_match => false
      | id_proper,id_active => true
      | id_proper,id_X => true
      | id_proper,id_c => true
      | id_proper,id_mark => true
      | id_proper,id_ok => true
      | id_proper,id_start => true
      | id_proper,id_f => true
      | id_proper,id_proper => true
      | id_proper,id_check => false
      | id_proper,id_top => false
      | id_proper,id_found => false
      | id_proper,id_match => false
      | id_check,id_active => true
      | id_check,id_X => true
      | id_check,id_c => true
      | id_check,id_mark => true
      | id_check,id_ok => true
      | id_check,id_start => true
      | id_check,id_f => true
      | id_check,id_proper => true
      | id_check,id_check => true
      | id_check,id_top => false
      | id_check,id_found => false
      | id_check,id_match => false
      | id_top,id_active => true
      | id_top,id_X => true
      | id_top,id_c => true
      | id_top,id_mark => true
      | id_top,id_ok => true
      | id_top,id_start => true
      | id_top,id_f => true
      | id_top,id_proper => true
      | id_top,id_check => true
      | id_top,id_top => true
      | id_top,id_found => false
      | id_top,id_match => false
      | id_found,id_active => true
      | id_found,id_X => true
      | id_found,id_c => true
      | id_found,id_mark => true
      | id_found,id_ok => true
      | id_found,id_start => true
      | id_found,id_f => true
      | id_found,id_proper => true
      | id_found,id_check => true
      | id_found,id_top => true
      | id_found,id_found => true
      | id_found,id_match => false
      | id_match,id_active => true
      | id_match,id_X => true
      | id_match,id_c => true
      | id_match,id_mark => true
      | id_match,id_ok => true
      | id_match,id_start => true
      | id_match,id_f => true
      | id_match,id_proper => true
      | id_match,id_check => true
      | id_match,id_top => true
      | id_match,id_found => true
      | id_match,id_match => 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 := 
    (* active(f(x_)) -> mark(x_) *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* top(active(c)) -> top(mark(c)) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                   algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_c 
                   nil)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
      algebra.F.id_active ((algebra.Alg.Term algebra.F.id_c 
      nil)::nil))::nil))
    (* top(mark(x_)) -> top(check(x_)) *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                   algebra.F.id_check ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term algebra.F.id_mark 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* check(f(x_)) -> f(check(x_)) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                   algebra.F.id_check ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_check ((algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* check(x_) -> start(match(f(X),x_)) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_start ((algebra.Alg.Term 
                   algebra.F.id_match ((algebra.Alg.Term algebra.F.id_f 
                   ((algebra.Alg.Term algebra.F.id_X nil)::nil))::
                   (algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_check ((algebra.Alg.Var 1)::nil))
    (* match(f(x_),f(y_)) -> f(match(x_,y_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                   algebra.F.id_match ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_match ((algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* match(X,x_) -> proper(x_) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_proper 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_match ((algebra.Alg.Term algebra.F.id_X 
      nil)::(algebra.Alg.Var 1)::nil))
    (* proper(c) -> ok(c) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ok ((algebra.Alg.Term 
                   algebra.F.id_c nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_proper ((algebra.Alg.Term algebra.F.id_c 
      nil)::nil))
    (* proper(f(x_)) -> f(proper(x_)) *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                   algebra.F.id_proper ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_proper ((algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* f(ok(x_)) -> ok(f(x_)) *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ok ((algebra.Alg.Term 
                   algebra.F.id_f ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_ok 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* start(ok(x_)) -> found(x_) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_found 
                   ((algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_start ((algebra.Alg.Term algebra.F.id_ok 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* f(found(x_)) -> found(f(x_)) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_found ((algebra.Alg.Term 
                   algebra.F.id_f ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_found 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* top(found(x_)) -> top(active(x_)) *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                   algebra.F.id_active ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
      algebra.F.id_found ((algebra.Alg.Var 1)::nil))::nil))
    (* active(f(x_)) -> f(active(x_)) *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                   algebra.F.id_active ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term algebra.F.id_f 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* f(mark(x_)) -> mark(f(x_)) *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term 
                   algebra.F.id_f ((algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_mark 
      ((algebra.Alg.Var 1)::nil))::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)))::
    nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
     algebra.F.id_active ((algebra.Alg.Term algebra.F.id_c nil)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Term algebra.F.id_c nil)::nil))::nil)))::
    R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term algebra.F.id_check 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_check ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_check 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_check ((algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_start ((algebra.Alg.Term 
     algebra.F.id_match ((algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
     algebra.F.id_X nil)::nil))::(algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_match ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_match 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_match ((algebra.Alg.Term algebra.F.id_X 
     nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id_proper ((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_proper ((algebra.Alg.Term algebra.F.id_c 
     nil)::nil)),
    (algebra.Alg.Term algebra.F.id_ok ((algebra.Alg.Term algebra.F.id_c 
     nil)::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id_proper ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_proper 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_ok 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_ok ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id_start ((algebra.Alg.Term algebra.F.id_ok 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_found ((algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_found 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_found ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term algebra.F.id_found 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
     algebra.F.id_active ((algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_active 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term algebra.F.id_mark 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_f 
     ((algebra.Alg.Var 1)::nil))::nil)))::R_xml_0_rule_as_list_13.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_14.
 
 
 Lemma R_xml_0_rules_included :
  forall l r, R_xml_0_rules r l <-> In (l,r) R_xml_0_rule_as_list.
 Proof.
   intros l r.
   constructor.
   intros H.
   
   case H;clear H;
    (apply (more_list.mem_impl_in (@eq (algebra.Alg.term*algebra.Alg.term)));
     [tauto|idtac]);
    match goal with
      |  |- _ _ _ ?t ?l =>
       let u := fresh "u" in 
        (generalize (more_list.mem_bool_ok _ _ 
                      algebra.Alg_ext.eq_term_term_bool_ok t l);
          set (u:=more_list.mem_bool algebra.Alg_ext.eq_term_term_bool t l) in *;
          vm_compute in u|-;unfold u in *;clear u;intros H;refine H)
      end
    .
   intros H.
   vm_compute in H|-.
   rewrite  <- or_ext_generated.or16_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 15.
   injection H;intros ;subst;constructor 14.
   injection H;intros ;subst;constructor 13.
   injection H;intros ;subst;constructor 12.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   injection H;intros ;subst;constructor 3.
   injection H;intros ;subst;constructor 2.
   injection H;intros ;subst;constructor 1.
   elim H.
 Qed.
 
 
 Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x).
 Proof.
   intros x t H.
   inversion H.
 Qed.
 
 
 Lemma R_xml_0_reg :
  forall s t, 
   (R_xml_0_rules s t) ->
    forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t).
 Proof.
   intros s t H.
   
   inversion H;intros x Hx;
    (apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]);
    apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx;
    vm_compute in Hx|-*;tauto.
 Qed.
 
 
 Inductive and_5 (x4 x5 x6 x7 x8:Prop) :
  Prop := 
   | conj_5 : x4->x5->x6->x7->x8->and_5 x4 x5 x6 x7 x8
 .
 
 
 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_X nil) ->
           t' = (algebra.Alg.Term algebra.F.id_X nil)) 
     (t = (algebra.Alg.Term algebra.F.id_c nil) ->
      t' = (algebra.Alg.Term algebra.F.id_c nil)) 
     (forall x5, 
      t = (algebra.Alg.Term algebra.F.id_mark (x5::nil)) ->
       exists x4,
         t' = (algebra.Alg.Term algebra.F.id_mark (x4::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5)) 
     
     (forall x5, 
      t = (algebra.Alg.Term algebra.F.id_ok (x5::nil)) ->
       exists x4,
         t' = (algebra.Alg.Term algebra.F.id_ok (x4::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5)) 
     
     (forall x5, 
      t = (algebra.Alg.Term algebra.F.id_found (x5::nil)) ->
       exists x4,
         t' = (algebra.Alg.Term algebra.F.id_found (x4::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5)).
 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 H;intuition;constructor 1.
   intros x5 H;exists x5;intuition;constructor 1.
   intros x5 H;exists x5;intuition;constructor 1.
   intros x5 H;exists x5;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_X H_id_c H_id_mark H_id_ok H_id_found].
   constructor.
   
   clear H_id_c H_id_mark H_id_ok H_id_found;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_X H_id_mark H_id_ok H_id_found;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_X H_id_c H_id_ok H_id_found;intros x5 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 x5 |- _ =>
      destruct (H_id_mark y (refl_equal _)) as [x4];intros ;intuition;
       exists x4;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_X H_id_c H_id_mark H_id_found;intros x5 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 x5 |- _ =>
      destruct (H_id_ok y (refl_equal _)) as [x4];intros ;intuition;
       exists x4;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
   
   clear H_id_X H_id_c H_id_mark H_id_ok;intros x5 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 x5 |- _ =>
      destruct (H_id_found y (refl_equal _)) as [x4];intros ;intuition;
       exists x4;intuition;eapply closure_extension.refl_trans_clos_R;
       eassumption
     end
   .
 Qed.
 
 
 Lemma id_X_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_X nil) ->
     t' = (algebra.Alg.Term algebra.F.id_X nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_c_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_c nil) ->
     t' = (algebra.Alg.Term algebra.F.id_c nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_mark_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x5, 
     t = (algebra.Alg.Term algebra.F.id_mark (x5::nil)) ->
      exists x4,
        t' = (algebra.Alg.Term algebra.F.id_mark (x4::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_ok_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x5, 
     t = (algebra.Alg.Term algebra.F.id_ok (x5::nil)) ->
      exists x4,
        t' = (algebra.Alg.Term algebra.F.id_ok (x4::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_found_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x5, 
     t = (algebra.Alg.Term algebra.F.id_found (x5::nil)) ->
      exists x4,
        t' = (algebra.Alg.Term algebra.F.id_found (x4::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x4 x5).
 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_X nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_X_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_c nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_c_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_mark (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_mark_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_ok (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_ok_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_found (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_found_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_X nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_X_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_c nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_c_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_mark (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_mark_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_ok (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_ok_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_found (?x4::nil)) |- _ =>
     let x4 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_found_is_R_xml_0_constructor H (refl_equal _)) as 
           [x4 [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_active : A ->A.
   
   Hypothesis P_id_X : A.
   
   Hypothesis P_id_c : A.
   
   Hypothesis P_id_mark : A ->A.
   
   Hypothesis P_id_ok : A ->A.
   
   Hypothesis P_id_start : A ->A.
   
   Hypothesis P_id_f : A ->A.
   
   Hypothesis P_id_proper : A ->A.
   
   Hypothesis P_id_check : A ->A.
   
   Hypothesis P_id_top : A ->A.
   
   Hypothesis P_id_found : A ->A.
   
   Hypothesis P_id_match : A ->A ->A.
   
   Hypothesis P_id_active_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   
   Hypothesis P_id_mark_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   
   Hypothesis P_id_ok_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   
   Hypothesis P_id_start_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   
   Hypothesis P_id_f_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   
   Hypothesis P_id_proper_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   
   Hypothesis P_id_check_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   
   Hypothesis P_id_top_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   
   Hypothesis P_id_found_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   
   Hypothesis P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (A0 <= x7)/\ (x7 <= x6) ->
      (A0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   
   Hypothesis P_id_active_bounded :
    forall x4, (A0 <= x4) ->A0 <= P_id_active x4.
   
   Hypothesis P_id_X_bounded : A0 <= P_id_X .
   
   Hypothesis P_id_c_bounded : A0 <= P_id_c .
   
   Hypothesis P_id_mark_bounded : forall x4, (A0 <= x4) ->A0 <= P_id_mark x4.
   
   Hypothesis P_id_ok_bounded : forall x4, (A0 <= x4) ->A0 <= P_id_ok x4.
   
   Hypothesis P_id_start_bounded :
    forall x4, (A0 <= x4) ->A0 <= P_id_start x4.
   
   Hypothesis P_id_f_bounded : forall x4, (A0 <= x4) ->A0 <= P_id_f x4.
   
   Hypothesis P_id_proper_bounded :
    forall x4, (A0 <= x4) ->A0 <= P_id_proper x4.
   
   Hypothesis P_id_check_bounded :
    forall x4, (A0 <= x4) ->A0 <= P_id_check x4.
   
   Hypothesis P_id_top_bounded : forall x4, (A0 <= x4) ->A0 <= P_id_top x4.
   
   Hypothesis P_id_found_bounded :
    forall x4, (A0 <= x4) ->A0 <= P_id_found x4.
   
   Hypothesis P_id_match_bounded :
    forall x4 x5, (A0 <= x4) ->(A0 <= x5) ->A0 <= P_id_match x5 x4.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
        P_id_active (measure x4)
       | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
       | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
       | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
        P_id_mark (measure x4)
       | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) => P_id_ok (measure x4)
       | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
        P_id_start (measure x4)
       | (algebra.Alg.Term algebra.F.id_f (x4::nil)) => P_id_f (measure x4)
       | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
        P_id_proper (measure x4)
       | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
        P_id_check (measure x4)
       | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
        P_id_top (measure x4)
       | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
        P_id_found (measure x4)
       | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
        P_id_match (measure x5) (measure x4)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 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_active => P_id_active
       | algebra.F.id_X => P_id_X
       | algebra.F.id_c => P_id_c
       | algebra.F.id_mark => P_id_mark
       | algebra.F.id_ok => P_id_ok
       | algebra.F.id_start => P_id_start
       | algebra.F.id_f => P_id_f
       | algebra.F.id_proper => P_id_proper
       | algebra.F.id_check => P_id_check
       | algebra.F.id_top => P_id_top
       | algebra.F.id_found => P_id_found
       | algebra.F.id_match => P_id_match
       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_active =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_X => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_c => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_ok =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_start =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_f =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_proper =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_check =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_top =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_found =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_match =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_active_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_X_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_c_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ok_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_start_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_f_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_proper_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_check_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_top_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_found_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_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_active_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_ok_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_start_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_f_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_proper_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_check_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_top_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_found_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_active_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_X_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_c_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ok_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_start_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_f_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_proper_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_check_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_top_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_found_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_MATCH : A ->A ->A.
   
   Hypothesis P_id_TOP : A ->A.
   
   Hypothesis P_id_ACTIVE : A ->A.
   
   Hypothesis P_id_F : A ->A.
   
   Hypothesis P_id_PROPER : A ->A.
   
   Hypothesis P_id_CHECK : A ->A.
   
   Hypothesis P_id_START : A ->A.
   
   Hypothesis P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (A0 <= x7)/\ (x7 <= x6) ->
      (A0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   
   Hypothesis P_id_TOP_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   
   Hypothesis P_id_ACTIVE_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   
   Hypothesis P_id_F_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   
   Hypothesis P_id_PROPER_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   
   Hypothesis P_id_CHECK_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   
   Hypothesis P_id_START_monotonic :
    forall x4 x5, (A0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
        P_id_MATCH (measure x5) (measure x4)
       | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
        P_id_TOP (measure x4)
       | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
        P_id_ACTIVE (measure x4)
       | (algebra.Alg.Term algebra.F.id_f (x4::nil)) => P_id_F (measure x4)
       | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
        P_id_PROPER (measure x4)
       | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
        P_id_CHECK (measure x4)
       | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
        P_id_START (measure x4)
       | _ => 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 x5 x4;apply (P_id_MATCH x5 x4).
     
     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 x4;apply (P_id_TOP x4).
     
     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 x4;apply (P_id_CHECK x4).
     
     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 x4;apply (P_id_PROPER x4).
     
     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 x4;apply (P_id_F x4).
     
     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 x4;apply (P_id_START x4).
     
     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 x4;apply (P_id_ACTIVE x4).
     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_active =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_X => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_c => match l with
                                    | nil => _
                                    | _::_ => _
                                    end
              | algebra.F.id_mark =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_ok =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_start =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_f =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_proper =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_check =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_top =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_found =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_match =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              end.
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_ok_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_start_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_f_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_proper_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_check_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_top_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_found_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_match_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_active_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_X_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_c_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ok_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_start_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_f_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_proper_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_check_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_top_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_found_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_match_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_MATCH_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_TOP_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_CHECK_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_PROPER_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_F_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_START_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_ACTIVE_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_active : Z ->Z.
   
   Hypothesis P_id_X : Z.
   
   Hypothesis P_id_c : Z.
   
   Hypothesis P_id_mark : Z ->Z.
   
   Hypothesis P_id_ok : Z ->Z.
   
   Hypothesis P_id_start : Z ->Z.
   
   Hypothesis P_id_f : Z ->Z.
   
   Hypothesis P_id_proper : Z ->Z.
   
   Hypothesis P_id_check : Z ->Z.
   
   Hypothesis P_id_top : Z ->Z.
   
   Hypothesis P_id_found : Z ->Z.
   
   Hypothesis P_id_match : Z ->Z ->Z.
   
   Hypothesis P_id_active_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   
   Hypothesis P_id_mark_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   
   Hypothesis P_id_ok_monotonic :
    forall x4 x5, (min_value <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   
   Hypothesis P_id_start_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   
   Hypothesis P_id_f_monotonic :
    forall x4 x5, (min_value <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   
   Hypothesis P_id_proper_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   
   Hypothesis P_id_check_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   
   Hypothesis P_id_top_monotonic :
    forall x4 x5, (min_value <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   
   Hypothesis P_id_found_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   
   Hypothesis P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (min_value <= x7)/\ (x7 <= x6) ->
      (min_value <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   
   Hypothesis P_id_active_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_active x4.
   
   Hypothesis P_id_X_bounded : min_value <= P_id_X .
   
   Hypothesis P_id_c_bounded : min_value <= P_id_c .
   
   Hypothesis P_id_mark_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_mark x4.
   
   Hypothesis P_id_ok_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_ok x4.
   
   Hypothesis P_id_start_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_start x4.
   
   Hypothesis P_id_f_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_f x4.
   
   Hypothesis P_id_proper_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_proper x4.
   
   Hypothesis P_id_check_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_check x4.
   
   Hypothesis P_id_top_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_top x4.
   
   Hypothesis P_id_found_bounded :
    forall x4, (min_value <= x4) ->min_value <= P_id_found x4.
   
   Hypothesis P_id_match_bounded :
    forall x4 x5, 
     (min_value <= x4) ->(min_value <= x5) ->min_value <= P_id_match x5 x4.
   
   Definition measure  := 
     Interp.measure min_value P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_MATCH : Z ->Z ->Z.
   
   Hypothesis P_id_TOP : Z ->Z.
   
   Hypothesis P_id_ACTIVE : Z ->Z.
   
   Hypothesis P_id_F : Z ->Z.
   
   Hypothesis P_id_PROPER : Z ->Z.
   
   Hypothesis P_id_CHECK : Z ->Z.
   
   Hypothesis P_id_START : Z ->Z.
   
   Hypothesis P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (min_value <= x7)/\ (x7 <= x6) ->
      (min_value <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   
   Hypothesis P_id_TOP_monotonic :
    forall x4 x5, (min_value <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   
   Hypothesis P_id_ACTIVE_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   
   Hypothesis P_id_F_monotonic :
    forall x4 x5, (min_value <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   
   Hypothesis P_id_PROPER_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   
   Hypothesis P_id_CHECK_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   
   Hypothesis P_id_START_monotonic :
    forall x4 x5, 
     (min_value <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_active P_id_X P_id_c P_id_mark 
      P_id_ok P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_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 := 
    (* <top(active(c)),top(mark(c))> *)
   | DP_R_xml_0_0 :
    forall x4, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term 
        algebra.F.id_c nil)::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                  algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_c 
                  nil)::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_top (x4::nil))
    (* <top(mark(x_)),top(check(x_))> *)
   | DP_R_xml_0_1 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                  algebra.F.id_check (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_top (x4::nil))
    (* <top(mark(x_)),check(x_)> *)
   | DP_R_xml_0_2 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_check (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_top (x4::nil))
    (* <check(f(x_)),f(check(x_))> *)
   | DP_R_xml_0_3 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                  algebra.F.id_check (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_check (x4::nil))
    (* <check(f(x_)),check(x_)> *)
   | DP_R_xml_0_4 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_check (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_check (x4::nil))
    (* <check(x_),start(match(f(X),x_))> *)
   | DP_R_xml_0_5 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_start ((algebra.Alg.Term 
                  algebra.F.id_match ((algebra.Alg.Term algebra.F.id_f 
                  ((algebra.Alg.Term algebra.F.id_X nil)::nil))::
                  x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_check (x4::nil))
    (* <check(x_),match(f(X),x_)> *)
   | DP_R_xml_0_6 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match ((algebra.Alg.Term 
                  algebra.F.id_f ((algebra.Alg.Term algebra.F.id_X 
                  nil)::nil))::x1::nil)) 
       (algebra.Alg.Term algebra.F.id_check (x4::nil))
    (* <check(x_),f(X)> *)
   | DP_R_xml_0_7 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                  algebra.F.id_X nil)::nil)) 
       (algebra.Alg.Term algebra.F.id_check (x4::nil))
    (* <match(f(x_),f(y_)),f(match(x_,y_))> *)
   | DP_R_xml_0_8 :
    forall x4 x2 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x5) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x2::nil)) 
        x4) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                   algebra.F.id_match (x1::x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
    (* <match(f(x_),f(y_)),match(x_,y_)> *)
   | DP_R_xml_0_9 :
    forall x4 x2 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x5) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x2::nil)) 
        x4) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_match (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
    (* <match(X,x_),proper(x_)> *)
   | DP_R_xml_0_10 :
    forall x4 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_X nil) 
       x5) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x4) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_proper (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
    (* <proper(f(x_)),f(proper(x_))> *)
   | DP_R_xml_0_11 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                  algebra.F.id_proper (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_proper (x4::nil))
    (* <proper(f(x_)),proper(x_)> *)
   | DP_R_xml_0_12 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_proper (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_proper (x4::nil))
    (* <f(ok(x_)),f(x_)> *)
   | DP_R_xml_0_13 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_ok (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_f (x4::nil))
    (* <f(found(x_)),f(x_)> *)
   | DP_R_xml_0_14 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_f (x4::nil))
    (* <top(found(x_)),top(active(x_))> *)
   | DP_R_xml_0_15 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_top ((algebra.Alg.Term 
                  algebra.F.id_active (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_top (x4::nil))
    (* <top(found(x_)),active(x_)> *)
   | DP_R_xml_0_16 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_active (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_top (x4::nil))
    (* <active(f(x_)),f(active(x_))> *)
   | DP_R_xml_0_17 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f ((algebra.Alg.Term 
                  algebra.F.id_active (x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_active (x4::nil))
    (* <active(f(x_)),active(x_)> *)
   | DP_R_xml_0_18 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_active (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_active (x4::nil))
    (* <f(mark(x_)),f(x_)> *)
   | DP_R_xml_0_19 :
    forall x4 x1, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_f (x4::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_scc_1  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <f(found(x_)),f(x_)> *)
    | DP_R_xml_0_scc_1_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
       DP_R_xml_0_scc_1 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_f (x4::nil))
     (* <f(ok(x_)),f(x_)> *)
    | DP_R_xml_0_scc_1_1 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_ok (x1::nil)) x4) ->
       DP_R_xml_0_scc_1 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_f (x4::nil))
     (* <f(mark(x_)),f(x_)> *)
    | DP_R_xml_0_scc_1_2 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
       DP_R_xml_0_scc_1 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_f (x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_1.
   Inductive DP_R_xml_0_scc_1_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <f(found(x_)),f(x_)> *)
     | DP_R_xml_0_scc_1_large_0 :
      forall x4 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
        DP_R_xml_0_scc_1_large (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_f (x4::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_1_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <f(ok(x_)),f(x_)> *)
     | DP_R_xml_0_scc_1_strict_0 :
      forall x4 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_ok (x1::nil)) x4) ->
        DP_R_xml_0_scc_1_strict (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_f (x4::nil))
      (* <f(mark(x_)),f(x_)> *)
     | DP_R_xml_0_scc_1_strict_1 :
      forall x4 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
        DP_R_xml_0_scc_1_strict (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_f (x4::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_1_large.
    Open Scope Z_scope.
    
    Import ring_extention.
    
    Notation Local "a <= b" := (Zle a b).
    
    Notation Local "a < b" := (Zlt a b).
    
    Definition P_id_active (x4:Z) := 1* x4.
    
    Definition P_id_X  := 0.
    
    Definition P_id_c  := 0.
    
    Definition P_id_mark (x4:Z) := 1* x4.
    
    Definition P_id_ok (x4:Z) := 2* x4.
    
    Definition P_id_start (x4:Z) := 1 + 1* x4.
    
    Definition P_id_f (x4:Z) := 1* x4.
    
    Definition P_id_proper (x4:Z) := 0.
    
    Definition P_id_check (x4:Z) := 2 + 1* x4.
    
    Definition P_id_top (x4:Z) := 0.
    
    Definition P_id_found (x4:Z) := 1 + 1* x4.
    
    Definition P_id_match (x4:Z) (x5:Z) := 0.
    
    Lemma P_id_active_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_mark_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ok_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_start_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_f_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_proper_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_check_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_top_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_found_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_match_monotonic :
     forall x4 x6 x5 x7, 
      (0 <= x7)/\ (x7 <= x6) ->
       (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
    Proof.
      intros x7 x6 x5 x4.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_X_bounded : 0 <= P_id_X .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_c_bounded : 0 <= P_id_c .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_match_bounded :
     forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition measure  := 
      InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
       P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
       P_id_match.
    
    Lemma measure_equation :
     forall t, 
      measure t = match t with
                    | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                     P_id_active (measure x4)
                    | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                    | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                    | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                     P_id_mark (measure x4)
                    | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                     P_id_ok (measure x4)
                    | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                     P_id_start (measure x4)
                    | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                     P_id_f (measure x4)
                    | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                     P_id_proper (measure x4)
                    | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                     P_id_check (measure x4)
                    | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                     P_id_top (measure x4)
                    | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                     P_id_found (measure x4)
                    | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                     P_id_match (measure x5) (measure x4)
                    | _ => 0
                    end.
    Proof.
      intros t;case t;intros ;apply refl_equal.
    Qed.
    
    Lemma measure_bounded : forall t, 0 <= measure t.
    Proof.
      unfold measure in |-*.
      
      apply InterpZ.measure_bounded;
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Ltac generate_pos_hyp  :=
     match goal with
       | H:context [measure ?x] |- _ =>
        let v := fresh "v" in 
         (let H := fresh "h" in 
           (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
             clearbody H;clearbody v))
       |  |- context [measure ?x] =>
        let v := fresh "v" in 
         (let H := fresh "h" in 
           (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
             clearbody H;clearbody v))
       end
     .
    
    Lemma rules_monotonic :
     forall l r, 
      (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
       measure r <= measure l.
    Proof.
      intros l r H.
      fold measure in |-*.
      
      inversion H;clear H;subst;inversion H0;clear H0;subst;
       simpl algebra.EQT.T.apply_subst in |-*;
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         end
       );repeat (generate_pos_hyp );
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma measure_star_monotonic :
     forall l r, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 r l) ->measure r <= measure l.
    Proof.
      unfold measure in *.
      apply InterpZ.measure_star_monotonic.
      intros ;apply P_id_active_monotonic;assumption.
      intros ;apply P_id_mark_monotonic;assumption.
      intros ;apply P_id_ok_monotonic;assumption.
      intros ;apply P_id_start_monotonic;assumption.
      intros ;apply P_id_f_monotonic;assumption.
      intros ;apply P_id_proper_monotonic;assumption.
      intros ;apply P_id_check_monotonic;assumption.
      intros ;apply P_id_top_monotonic;assumption.
      intros ;apply P_id_found_monotonic;assumption.
      intros ;apply P_id_match_monotonic;assumption.
      intros ;apply P_id_active_bounded;assumption.
      intros ;apply P_id_X_bounded;assumption.
      intros ;apply P_id_c_bounded;assumption.
      intros ;apply P_id_mark_bounded;assumption.
      intros ;apply P_id_ok_bounded;assumption.
      intros ;apply P_id_start_bounded;assumption.
      intros ;apply P_id_f_bounded;assumption.
      intros ;apply P_id_proper_bounded;assumption.
      intros ;apply P_id_check_bounded;assumption.
      intros ;apply P_id_top_bounded;assumption.
      intros ;apply P_id_found_bounded;assumption.
      intros ;apply P_id_match_bounded;assumption.
      apply rules_monotonic.
    Qed.
    
    Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
    
    Definition P_id_TOP (x4:Z) := 0.
    
    Definition P_id_ACTIVE (x4:Z) := 0.
    
    Definition P_id_F (x4:Z) := 1* x4.
    
    Definition P_id_PROPER (x4:Z) := 0.
    
    Definition P_id_CHECK (x4:Z) := 0.
    
    Definition P_id_START (x4:Z) := 0.
    
    Lemma P_id_MATCH_monotonic :
     forall x4 x6 x5 x7, 
      (0 <= x7)/\ (x7 <= x6) ->
       (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
    Proof.
      intros x7 x6 x5 x4.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_TOP_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ACTIVE_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_F_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_PROPER_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_CHECK_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_START_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition marked_measure  := 
      InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
       P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
       P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
       P_id_CHECK P_id_START.
    
    Lemma marked_measure_equation :
     forall t, 
      marked_measure t = match t with
                           | (algebra.Alg.Term algebra.F.id_match (x5::
                              x4::nil)) =>
                            P_id_MATCH (measure x5) (measure x4)
                           | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                            P_id_TOP (measure x4)
                           | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                            P_id_ACTIVE (measure x4)
                           | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                            P_id_F (measure x4)
                           | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                            P_id_PROPER (measure x4)
                           | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                            P_id_CHECK (measure x4)
                           | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                            P_id_START (measure x4)
                           | _ => 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_active_monotonic;assumption.
      intros ;apply P_id_mark_monotonic;assumption.
      intros ;apply P_id_ok_monotonic;assumption.
      intros ;apply P_id_start_monotonic;assumption.
      intros ;apply P_id_f_monotonic;assumption.
      intros ;apply P_id_proper_monotonic;assumption.
      intros ;apply P_id_check_monotonic;assumption.
      intros ;apply P_id_top_monotonic;assumption.
      intros ;apply P_id_found_monotonic;assumption.
      intros ;apply P_id_match_monotonic;assumption.
      intros ;apply P_id_active_bounded;assumption.
      intros ;apply P_id_X_bounded;assumption.
      intros ;apply P_id_c_bounded;assumption.
      intros ;apply P_id_mark_bounded;assumption.
      intros ;apply P_id_ok_bounded;assumption.
      intros ;apply P_id_start_bounded;assumption.
      intros ;apply P_id_f_bounded;assumption.
      intros ;apply P_id_proper_bounded;assumption.
      intros ;apply P_id_check_bounded;assumption.
      intros ;apply P_id_top_bounded;assumption.
      intros ;apply P_id_found_bounded;assumption.
      intros ;apply P_id_match_bounded;assumption.
      apply rules_monotonic.
      intros ;apply P_id_MATCH_monotonic;assumption.
      intros ;apply P_id_TOP_monotonic;assumption.
      intros ;apply P_id_ACTIVE_monotonic;assumption.
      intros ;apply P_id_F_monotonic;assumption.
      intros ;apply P_id_PROPER_monotonic;assumption.
      intros ;apply P_id_CHECK_monotonic;assumption.
      intros ;apply P_id_START_monotonic;assumption.
    Qed.
    
    Ltac rewrite_and_unfold  :=
     do 2 (rewrite marked_measure_equation);
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
         rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
        end
      ).
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_1.DP_R_xml_0_scc_1_large.
    Proof.
      intros x.
      
      apply well_founded_ind with
        (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
      apply Inverse_Image.wf_inverse_image with  (B:=Z).
      apply Zwf.Zwf_well_founded.
      clear x.
      intros x IHx.
      
      repeat (
      constructor;inversion 1;subst;
       full_prove_ineq algebra.Alg.Term 
       ltac:(algebra.Alg_ext.find_replacement ) 
       algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
       marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
       ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
       ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
       ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                       try (constructor))
        IHx
      ).
    Qed.
   End WF_DP_R_xml_0_scc_1_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_active (x4:Z) := 2* x4.
   
   Definition P_id_X  := 0.
   
   Definition P_id_c  := 3.
   
   Definition P_id_mark (x4:Z) := 1 + 1* x4.
   
   Definition P_id_ok (x4:Z) := 3 + 1* x4.
   
   Definition P_id_start (x4:Z) := 1* x4.
   
   Definition P_id_f (x4:Z) := 1 + 1* x4.
   
   Definition P_id_proper (x4:Z) := 3* x4.
   
   Definition P_id_check (x4:Z) := 1 + 3* x4.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 1* x4.
   
   Definition P_id_match (x4:Z) (x5:Z) := 1 + 3* x5.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
   
   Definition P_id_TOP (x4:Z) := 0.
   
   Definition P_id_ACTIVE (x4:Z) := 0.
   
   Definition P_id_F (x4:Z) := 2* x4.
   
   Definition P_id_PROPER (x4:Z) := 0.
   
   Definition P_id_CHECK (x4:Z) := 0.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_1_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_1_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_1_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_1_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_1_large  := WF_DP_R_xml_0_scc_1_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_1.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_1_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_1_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_1_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_1_large.
   Qed.
  End WF_DP_R_xml_0_scc_1.
  
  Definition wf_DP_R_xml_0_scc_1  := WF_DP_R_xml_0_scc_1.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_1 :
   forall x y, (DP_R_xml_0_scc_1 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_1).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    apply wf_DP_R_xml_0_scc_1.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_2  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <active(f(x_)),f(active(x_))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_f 
                             ((algebra.Alg.Term algebra.F.id_active 
                             (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_active (x4::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;
      (eapply acc_DP_R_xml_0_scc_1;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_3  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <active(f(x_)),active(x_)> *)
    | DP_R_xml_0_scc_3_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_active (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_active (x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_3.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_active (x4:Z) := 1* x4.
   
   Definition P_id_X  := 0.
   
   Definition P_id_c  := 1.
   
   Definition P_id_mark (x4:Z) := 1* x4.
   
   Definition P_id_ok (x4:Z) := 1.
   
   Definition P_id_start (x4:Z) := 1 + 1* x4.
   
   Definition P_id_f (x4:Z) := 1 + 2* x4.
   
   Definition P_id_proper (x4:Z) := 2* x4.
   
   Definition P_id_check (x4:Z) := 1 + 3* x4.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 0.
   
   Definition P_id_match (x4:Z) (x5:Z) := 3* x5.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
   
   Definition P_id_TOP (x4:Z) := 0.
   
   Definition P_id_ACTIVE (x4:Z) := 1* x4.
   
   Definition P_id_F (x4:Z) := 0.
   
   Definition P_id_PROPER (x4:Z) := 0.
   
   Definition P_id_CHECK (x4:Z) := 0.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_3.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_3.
  
  Definition wf_DP_R_xml_0_scc_3  := WF_DP_R_xml_0_scc_3.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_3 :
   forall x y, (DP_R_xml_0_scc_3 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_3).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
    apply wf_DP_R_xml_0_scc_3.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <top(found(x_)),active(x_)> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
       DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_active (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_top (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_4 :
   forall x y, 
    (DP_R_xml_0_non_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_3;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <proper(f(x_)),f(proper(x_))> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_f 
                             ((algebra.Alg.Term algebra.F.id_proper 
                             (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_proper (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_5 :
   forall x y, 
    (DP_R_xml_0_non_scc_5 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_1;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_6  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <proper(f(x_)),proper(x_)> *)
    | DP_R_xml_0_scc_6_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_scc_6 (algebra.Alg.Term algebra.F.id_proper (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_proper (x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_6.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_active (x4:Z) := 1* x4.
   
   Definition P_id_X  := 0.
   
   Definition P_id_c  := 1.
   
   Definition P_id_mark (x4:Z) := 1* x4.
   
   Definition P_id_ok (x4:Z) := 1.
   
   Definition P_id_start (x4:Z) := 1 + 1* x4.
   
   Definition P_id_f (x4:Z) := 1 + 2* x4.
   
   Definition P_id_proper (x4:Z) := 2* x4.
   
   Definition P_id_check (x4:Z) := 1 + 3* x4.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 0.
   
   Definition P_id_match (x4:Z) (x5:Z) := 3* x5.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
   
   Definition P_id_TOP (x4:Z) := 0.
   
   Definition P_id_ACTIVE (x4:Z) := 0.
   
   Definition P_id_F (x4:Z) := 0.
   
   Definition P_id_PROPER (x4:Z) := 1* x4.
   
   Definition P_id_CHECK (x4:Z) := 0.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_6.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_6.
  
  Definition wf_DP_R_xml_0_scc_6  := WF_DP_R_xml_0_scc_6.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_6 :
   forall x y, (DP_R_xml_0_scc_6 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_6).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_5;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
    apply wf_DP_R_xml_0_scc_6.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_7  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match(X,x_),proper(x_)> *)
    | DP_R_xml_0_non_scc_7_0 :
     forall x4 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_X nil) 
        x5) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x4) ->
        DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_proper (x1::nil)) 
         (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_7 :
   forall x y, 
    (DP_R_xml_0_non_scc_7 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_6;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_5;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
        (eapply Hrec;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_8  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match(f(x_),f(y_)),f(match(x_,y_))> *)
    | DP_R_xml_0_non_scc_8_0 :
     forall x4 x2 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x5) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_f (x2::nil)) x4) ->
        DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_f 
                              ((algebra.Alg.Term algebra.F.id_match (x1::
                              x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_8 :
   forall x y, 
    (DP_R_xml_0_non_scc_8 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_1;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_9  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <match(f(x_),f(y_)),match(x_,y_)> *)
    | DP_R_xml_0_scc_9_0 :
     forall x4 x2 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x5) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_f (x2::nil)) x4) ->
        DP_R_xml_0_scc_9 (algebra.Alg.Term algebra.F.id_match (x1::x2::nil)) 
         (algebra.Alg.Term algebra.F.id_match (x5::x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_9.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_active (x4:Z) := 1* x4.
   
   Definition P_id_X  := 0.
   
   Definition P_id_c  := 1.
   
   Definition P_id_mark (x4:Z) := 0.
   
   Definition P_id_ok (x4:Z) := 0.
   
   Definition P_id_start (x4:Z) := 0.
   
   Definition P_id_f (x4:Z) := 1 + 2* x4.
   
   Definition P_id_proper (x4:Z) := 2* x4.
   
   Definition P_id_check (x4:Z) := 2* x4.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 0.
   
   Definition P_id_match (x4:Z) (x5:Z) := 2* x5.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 1* x4.
   
   Definition P_id_TOP (x4:Z) := 0.
   
   Definition P_id_ACTIVE (x4:Z) := 0.
   
   Definition P_id_F (x4:Z) := 0.
   
   Definition P_id_PROPER (x4:Z) := 0.
   
   Definition P_id_CHECK (x4:Z) := 0.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_9.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_9.
  
  Definition wf_DP_R_xml_0_scc_9  := WF_DP_R_xml_0_scc_9.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_9 :
   forall x y, (DP_R_xml_0_scc_9 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_9).
    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_8;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_7;
          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_9.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_10  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <check(x_),f(X)> *)
    | DP_R_xml_0_non_scc_10_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x4) ->
       DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_f 
                              ((algebra.Alg.Term algebra.F.id_X nil)::nil)) 
        (algebra.Alg.Term algebra.F.id_check (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_10 :
   forall x y, 
    (DP_R_xml_0_non_scc_10 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_1;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_11  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <check(x_),match(f(X),x_)> *)
    | DP_R_xml_0_non_scc_11_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x4) ->
       DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_match 
                              ((algebra.Alg.Term algebra.F.id_f 
                              ((algebra.Alg.Term algebra.F.id_X nil)::nil))::
                              x1::nil)) 
        (algebra.Alg.Term algebra.F.id_check (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_11 :
   forall x y, 
    (DP_R_xml_0_non_scc_11 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_9;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_8;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_7;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_12  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <check(x_),start(match(f(X),x_))> *)
    | DP_R_xml_0_non_scc_12_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x4) ->
       DP_R_xml_0_non_scc_12 (algebra.Alg.Term algebra.F.id_start 
                              ((algebra.Alg.Term algebra.F.id_match 
                              ((algebra.Alg.Term algebra.F.id_f 
                              ((algebra.Alg.Term algebra.F.id_X nil)::nil))::
                              x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_check (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_12 :
   forall x y, 
    (DP_R_xml_0_non_scc_12 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_13  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <check(f(x_)),f(check(x_))> *)
    | DP_R_xml_0_non_scc_13_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_non_scc_13 (algebra.Alg.Term algebra.F.id_f 
                              ((algebra.Alg.Term algebra.F.id_check 
                              (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_check (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_13 :
   forall x y, 
    (DP_R_xml_0_non_scc_13 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_1;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_14  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <check(f(x_)),check(x_)> *)
    | DP_R_xml_0_scc_14_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_f (x1::nil)) 
        x4) ->
       DP_R_xml_0_scc_14 (algebra.Alg.Term algebra.F.id_check (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_check (x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_14.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_active (x4:Z) := 1* x4.
   
   Definition P_id_X  := 0.
   
   Definition P_id_c  := 1.
   
   Definition P_id_mark (x4:Z) := 1* x4.
   
   Definition P_id_ok (x4:Z) := 1.
   
   Definition P_id_start (x4:Z) := 1 + 1* x4.
   
   Definition P_id_f (x4:Z) := 1 + 2* x4.
   
   Definition P_id_proper (x4:Z) := 2* x4.
   
   Definition P_id_check (x4:Z) := 1 + 3* x4.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 0.
   
   Definition P_id_match (x4:Z) (x5:Z) := 3* x5.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
   
   Definition P_id_TOP (x4:Z) := 0.
   
   Definition P_id_ACTIVE (x4:Z) := 0.
   
   Definition P_id_F (x4:Z) := 0.
   
   Definition P_id_PROPER (x4:Z) := 0.
   
   Definition P_id_CHECK (x4:Z) := 1* x4.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_14.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_14.
  
  Definition wf_DP_R_xml_0_scc_14  := WF_DP_R_xml_0_scc_14.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_14 :
   forall x y, (DP_R_xml_0_scc_14 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_14).
    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_13;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_12;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_11;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_10;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((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_14.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_15  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <top(mark(x_)),check(x_)> *)
    | DP_R_xml_0_non_scc_15_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
       DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_check (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_top (x4::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_15 :
   forall x y, 
    (DP_R_xml_0_non_scc_15 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_13;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_12;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_11;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_10;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((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_16  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <top(mark(x_)),top(check(x_))> *)
    | DP_R_xml_0_scc_16_0 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
       DP_R_xml_0_scc_16 (algebra.Alg.Term algebra.F.id_top 
                          ((algebra.Alg.Term algebra.F.id_check 
                          (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_top (x4::nil))
     (* <top(active(c)),top(mark(c))> *)
    | DP_R_xml_0_scc_16_1 :
     forall x4, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term 
         algebra.F.id_c nil)::nil)) x4) ->
       DP_R_xml_0_scc_16 (algebra.Alg.Term algebra.F.id_top 
                          ((algebra.Alg.Term algebra.F.id_mark 
                          ((algebra.Alg.Term algebra.F.id_c 
                          nil)::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_top (x4::nil))
     (* <top(found(x_)),top(active(x_))> *)
    | DP_R_xml_0_scc_16_2 :
     forall x4 x1, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
       DP_R_xml_0_scc_16 (algebra.Alg.Term algebra.F.id_top 
                          ((algebra.Alg.Term algebra.F.id_active 
                          (x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_top (x4::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_16.
   Inductive DP_R_xml_0_scc_16_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <top(mark(x_)),top(check(x_))> *)
     | DP_R_xml_0_scc_16_large_0 :
      forall x4 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
        DP_R_xml_0_scc_16_large (algebra.Alg.Term algebra.F.id_top 
                                 ((algebra.Alg.Term algebra.F.id_check 
                                 (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_top (x4::nil))
      (* <top(found(x_)),top(active(x_))> *)
     | DP_R_xml_0_scc_16_large_1 :
      forall x4 x1, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
        DP_R_xml_0_scc_16_large (algebra.Alg.Term algebra.F.id_top 
                                 ((algebra.Alg.Term algebra.F.id_active 
                                 (x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_top (x4::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_16_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <top(active(c)),top(mark(c))> *)
     | DP_R_xml_0_scc_16_strict_0 :
      forall x4, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_active ((algebra.Alg.Term 
          algebra.F.id_c nil)::nil)) x4) ->
        DP_R_xml_0_scc_16_strict (algebra.Alg.Term algebra.F.id_top 
                                  ((algebra.Alg.Term algebra.F.id_mark 
                                  ((algebra.Alg.Term algebra.F.id_c 
                                  nil)::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_top (x4::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_16_large.
    Inductive DP_R_xml_0_scc_16_large_large  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <top(mark(x_)),top(check(x_))> *)
      | DP_R_xml_0_scc_16_large_large_0 :
       forall x4 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_mark (x1::nil)) x4) ->
         DP_R_xml_0_scc_16_large_large (algebra.Alg.Term algebra.F.id_top 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_check (x1::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_top (x4::nil))
    .
    
    
    Inductive DP_R_xml_0_scc_16_large_strict  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <top(found(x_)),top(active(x_))> *)
      | DP_R_xml_0_scc_16_large_strict_0 :
       forall x4 x1, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_found (x1::nil)) x4) ->
         DP_R_xml_0_scc_16_large_strict (algebra.Alg.Term algebra.F.id_top 
                                         ((algebra.Alg.Term 
                                         algebra.F.id_active 
                                         (x1::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_top (x4::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_16_large_large.
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_active (x4:Z) := 1* x4.
     
     Definition P_id_X  := 3.
     
     Definition P_id_c  := 0.
     
     Definition P_id_mark (x4:Z) := 2 + 2* x4.
     
     Definition P_id_ok (x4:Z) := 1.
     
     Definition P_id_start (x4:Z) := 0.
     
     Definition P_id_f (x4:Z) := 2 + 2* x4.
     
     Definition P_id_proper (x4:Z) := 1 + 2* x4.
     
     Definition P_id_check (x4:Z) := 1* x4.
     
     Definition P_id_top (x4:Z) := 0.
     
     Definition P_id_found (x4:Z) := 0.
     
     Definition P_id_match (x4:Z) (x5:Z) := 3 + 3* x4 + 3* x5.
     
     Lemma P_id_active_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ok_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_start_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_f_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_proper_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_check_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_top_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_found_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_match_monotonic :
      forall x4 x6 x5 x7, 
       (0 <= x7)/\ (x7 <= x6) ->
        (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
     Proof.
       intros x7 x6 x5 x4.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_X_bounded : 0 <= P_id_X .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_c_bounded : 0 <= P_id_c .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_match_bounded :
      forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition measure  := 
       InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
        P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
        P_id_match.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                      P_id_active (measure x4)
                     | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                     | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                     | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                      P_id_mark (measure x4)
                     | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                      P_id_ok (measure x4)
                     | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                      P_id_start (measure x4)
                     | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                      P_id_f (measure x4)
                     | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                      P_id_proper (measure x4)
                     | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                      P_id_check (measure x4)
                     | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                      P_id_top (measure x4)
                     | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                      P_id_found (measure x4)
                     | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                      P_id_match (measure x5) (measure x4)
                     | _ => 0
                     end.
     Proof.
       intros t;case t;intros ;apply refl_equal.
     Qed.
     
     Lemma measure_bounded : forall t, 0 <= measure t.
     Proof.
       unfold measure in |-*.
       
       apply InterpZ.measure_bounded;
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Ltac generate_pos_hyp  :=
      match goal with
        | H:context [measure ?x] |- _ =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        |  |- context [measure ?x] =>
         let v := fresh "v" in 
          (let H := fresh "h" in 
            (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
              clearbody H;clearbody v))
        end
      .
     
     Lemma rules_monotonic :
      forall l r, 
       (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
        measure r <= measure l.
     Proof.
       intros l r H.
       fold measure in |-*.
       
       inversion H;clear H;subst;inversion H0;clear H0;subst;
        simpl algebra.EQT.T.apply_subst in |-*;
        repeat (
        match goal with
          |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
           rewrite (measure_equation (algebra.Alg.Term f t))
          end
        );repeat (generate_pos_hyp );
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma measure_star_monotonic :
      forall l r, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  r l) ->measure r <= measure l.
     Proof.
       unfold measure in *.
       apply InterpZ.measure_star_monotonic.
       intros ;apply P_id_active_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_ok_monotonic;assumption.
       intros ;apply P_id_start_monotonic;assumption.
       intros ;apply P_id_f_monotonic;assumption.
       intros ;apply P_id_proper_monotonic;assumption.
       intros ;apply P_id_check_monotonic;assumption.
       intros ;apply P_id_top_monotonic;assumption.
       intros ;apply P_id_found_monotonic;assumption.
       intros ;apply P_id_match_monotonic;assumption.
       intros ;apply P_id_active_bounded;assumption.
       intros ;apply P_id_X_bounded;assumption.
       intros ;apply P_id_c_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_ok_bounded;assumption.
       intros ;apply P_id_start_bounded;assumption.
       intros ;apply P_id_f_bounded;assumption.
       intros ;apply P_id_proper_bounded;assumption.
       intros ;apply P_id_check_bounded;assumption.
       intros ;apply P_id_top_bounded;assumption.
       intros ;apply P_id_found_bounded;assumption.
       intros ;apply P_id_match_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
     
     Definition P_id_TOP (x4:Z) := 3* x4.
     
     Definition P_id_ACTIVE (x4:Z) := 0.
     
     Definition P_id_F (x4:Z) := 0.
     
     Definition P_id_PROPER (x4:Z) := 0.
     
     Definition P_id_CHECK (x4:Z) := 0.
     
     Definition P_id_START (x4:Z) := 0.
     
     Lemma P_id_MATCH_monotonic :
      forall x4 x6 x5 x7, 
       (0 <= x7)/\ (x7 <= x6) ->
        (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
     Proof.
       intros x7 x6 x5 x4.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_TOP_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ACTIVE_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_F_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_PROPER_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_CHECK_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_START_monotonic :
      forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
     Proof.
       intros x5 x4.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
        P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
        P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
        P_id_CHECK P_id_START.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_match (x5::
                               x4::nil)) =>
                             P_id_MATCH (measure x5) (measure x4)
                            | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                             P_id_TOP (measure x4)
                            | (algebra.Alg.Term algebra.F.id_active 
                               (x4::nil)) =>
                             P_id_ACTIVE (measure x4)
                            | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                             P_id_F (measure x4)
                            | (algebra.Alg.Term algebra.F.id_proper 
                               (x4::nil)) =>
                             P_id_PROPER (measure x4)
                            | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                             P_id_CHECK (measure x4)
                            | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                             P_id_START (measure x4)
                            | _ => 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_active_monotonic;assumption.
       intros ;apply P_id_mark_monotonic;assumption.
       intros ;apply P_id_ok_monotonic;assumption.
       intros ;apply P_id_start_monotonic;assumption.
       intros ;apply P_id_f_monotonic;assumption.
       intros ;apply P_id_proper_monotonic;assumption.
       intros ;apply P_id_check_monotonic;assumption.
       intros ;apply P_id_top_monotonic;assumption.
       intros ;apply P_id_found_monotonic;assumption.
       intros ;apply P_id_match_monotonic;assumption.
       intros ;apply P_id_active_bounded;assumption.
       intros ;apply P_id_X_bounded;assumption.
       intros ;apply P_id_c_bounded;assumption.
       intros ;apply P_id_mark_bounded;assumption.
       intros ;apply P_id_ok_bounded;assumption.
       intros ;apply P_id_start_bounded;assumption.
       intros ;apply P_id_f_bounded;assumption.
       intros ;apply P_id_proper_bounded;assumption.
       intros ;apply P_id_check_bounded;assumption.
       intros ;apply P_id_top_bounded;assumption.
       intros ;apply P_id_found_bounded;assumption.
       intros ;apply P_id_match_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_MATCH_monotonic;assumption.
       intros ;apply P_id_TOP_monotonic;assumption.
       intros ;apply P_id_ACTIVE_monotonic;assumption.
       intros ;apply P_id_F_monotonic;assumption.
       intros ;apply P_id_PROPER_monotonic;assumption.
       intros ;apply P_id_CHECK_monotonic;assumption.
       intros ;apply P_id_START_monotonic;assumption.
     Qed.
     
     Ltac rewrite_and_unfold  :=
      do 2 (rewrite marked_measure_equation);
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
          rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
         end
       ).
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_16_large.DP_R_xml_0_scc_16_large_large.
     Proof.
       intros x.
       
       apply well_founded_ind with
         (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
       apply Inverse_Image.wf_inverse_image with  (B:=Z).
       apply Zwf.Zwf_well_founded.
       clear x.
       intros x IHx.
       
       repeat (
       constructor;inversion 1;subst;
        full_prove_ineq algebra.Alg.Term 
        ltac:(algebra.Alg_ext.find_replacement ) 
        algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
        marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
        ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
        ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp )
         
        ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                        try (constructor))
         IHx
       ).
     Qed.
    End WF_DP_R_xml_0_scc_16_large_large.
    
    Open Scope Z_scope.
    
    Import ring_extention.
    
    Notation Local "a <= b" := (Zle a b).
    
    Notation Local "a < b" := (Zlt a b).
    
    Definition P_id_active (x4:Z) := 2* x4.
    
    Definition P_id_X  := 0.
    
    Definition P_id_c  := 0.
    
    Definition P_id_mark (x4:Z) := 1 + 2* x4.
    
    Definition P_id_ok (x4:Z) := 1* x4.
    
    Definition P_id_start (x4:Z) := 1 + 2* x4.
    
    Definition P_id_f (x4:Z) := 2 + 3* x4.
    
    Definition P_id_proper (x4:Z) := 1* x4.
    
    Definition P_id_check (x4:Z) := 1 + 2* x4.
    
    Definition P_id_top (x4:Z) := 0.
    
    Definition P_id_found (x4:Z) := 1 + 2* x4.
    
    Definition P_id_match (x4:Z) (x5:Z) := 1* x5.
    
    Lemma P_id_active_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_mark_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ok_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_start_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_f_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_proper_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_check_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_top_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_found_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_match_monotonic :
     forall x4 x6 x5 x7, 
      (0 <= x7)/\ (x7 <= x6) ->
       (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
    Proof.
      intros x7 x6 x5 x4.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_X_bounded : 0 <= P_id_X .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_c_bounded : 0 <= P_id_c .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_match_bounded :
     forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition measure  := 
      InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
       P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
       P_id_match.
    
    Lemma measure_equation :
     forall t, 
      measure t = match t with
                    | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                     P_id_active (measure x4)
                    | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                    | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                    | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                     P_id_mark (measure x4)
                    | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                     P_id_ok (measure x4)
                    | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                     P_id_start (measure x4)
                    | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                     P_id_f (measure x4)
                    | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                     P_id_proper (measure x4)
                    | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                     P_id_check (measure x4)
                    | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                     P_id_top (measure x4)
                    | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                     P_id_found (measure x4)
                    | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                     P_id_match (measure x5) (measure x4)
                    | _ => 0
                    end.
    Proof.
      intros t;case t;intros ;apply refl_equal.
    Qed.
    
    Lemma measure_bounded : forall t, 0 <= measure t.
    Proof.
      unfold measure in |-*.
      
      apply InterpZ.measure_bounded;
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Ltac generate_pos_hyp  :=
     match goal with
       | H:context [measure ?x] |- _ =>
        let v := fresh "v" in 
         (let H := fresh "h" in 
           (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
             clearbody H;clearbody v))
       |  |- context [measure ?x] =>
        let v := fresh "v" in 
         (let H := fresh "h" in 
           (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
             clearbody H;clearbody v))
       end
     .
    
    Lemma rules_monotonic :
     forall l r, 
      (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
       measure r <= measure l.
    Proof.
      intros l r H.
      fold measure in |-*.
      
      inversion H;clear H;subst;inversion H0;clear H0;subst;
       simpl algebra.EQT.T.apply_subst in |-*;
       repeat (
       match goal with
         |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
          rewrite (measure_equation (algebra.Alg.Term f t))
         end
       );repeat (generate_pos_hyp );
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma measure_star_monotonic :
     forall l r, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 r l) ->measure r <= measure l.
    Proof.
      unfold measure in *.
      apply InterpZ.measure_star_monotonic.
      intros ;apply P_id_active_monotonic;assumption.
      intros ;apply P_id_mark_monotonic;assumption.
      intros ;apply P_id_ok_monotonic;assumption.
      intros ;apply P_id_start_monotonic;assumption.
      intros ;apply P_id_f_monotonic;assumption.
      intros ;apply P_id_proper_monotonic;assumption.
      intros ;apply P_id_check_monotonic;assumption.
      intros ;apply P_id_top_monotonic;assumption.
      intros ;apply P_id_found_monotonic;assumption.
      intros ;apply P_id_match_monotonic;assumption.
      intros ;apply P_id_active_bounded;assumption.
      intros ;apply P_id_X_bounded;assumption.
      intros ;apply P_id_c_bounded;assumption.
      intros ;apply P_id_mark_bounded;assumption.
      intros ;apply P_id_ok_bounded;assumption.
      intros ;apply P_id_start_bounded;assumption.
      intros ;apply P_id_f_bounded;assumption.
      intros ;apply P_id_proper_bounded;assumption.
      intros ;apply P_id_check_bounded;assumption.
      intros ;apply P_id_top_bounded;assumption.
      intros ;apply P_id_found_bounded;assumption.
      intros ;apply P_id_match_bounded;assumption.
      apply rules_monotonic.
    Qed.
    
    Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
    
    Definition P_id_TOP (x4:Z) := 1* x4.
    
    Definition P_id_ACTIVE (x4:Z) := 0.
    
    Definition P_id_F (x4:Z) := 0.
    
    Definition P_id_PROPER (x4:Z) := 0.
    
    Definition P_id_CHECK (x4:Z) := 0.
    
    Definition P_id_START (x4:Z) := 0.
    
    Lemma P_id_MATCH_monotonic :
     forall x4 x6 x5 x7, 
      (0 <= x7)/\ (x7 <= x6) ->
       (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
    Proof.
      intros x7 x6 x5 x4.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_TOP_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_ACTIVE_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_F_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_PROPER_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_CHECK_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_START_monotonic :
     forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
    Proof.
      intros x5 x4.
      intros [H_1 H_0].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition marked_measure  := 
      InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
       P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
       P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
       P_id_CHECK P_id_START.
    
    Lemma marked_measure_equation :
     forall t, 
      marked_measure t = match t with
                           | (algebra.Alg.Term algebra.F.id_match (x5::
                              x4::nil)) =>
                            P_id_MATCH (measure x5) (measure x4)
                           | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                            P_id_TOP (measure x4)
                           | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                            P_id_ACTIVE (measure x4)
                           | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                            P_id_F (measure x4)
                           | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                            P_id_PROPER (measure x4)
                           | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                            P_id_CHECK (measure x4)
                           | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                            P_id_START (measure x4)
                           | _ => 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_active_monotonic;assumption.
      intros ;apply P_id_mark_monotonic;assumption.
      intros ;apply P_id_ok_monotonic;assumption.
      intros ;apply P_id_start_monotonic;assumption.
      intros ;apply P_id_f_monotonic;assumption.
      intros ;apply P_id_proper_monotonic;assumption.
      intros ;apply P_id_check_monotonic;assumption.
      intros ;apply P_id_top_monotonic;assumption.
      intros ;apply P_id_found_monotonic;assumption.
      intros ;apply P_id_match_monotonic;assumption.
      intros ;apply P_id_active_bounded;assumption.
      intros ;apply P_id_X_bounded;assumption.
      intros ;apply P_id_c_bounded;assumption.
      intros ;apply P_id_mark_bounded;assumption.
      intros ;apply P_id_ok_bounded;assumption.
      intros ;apply P_id_start_bounded;assumption.
      intros ;apply P_id_f_bounded;assumption.
      intros ;apply P_id_proper_bounded;assumption.
      intros ;apply P_id_check_bounded;assumption.
      intros ;apply P_id_top_bounded;assumption.
      intros ;apply P_id_found_bounded;assumption.
      intros ;apply P_id_match_bounded;assumption.
      apply rules_monotonic.
      intros ;apply P_id_MATCH_monotonic;assumption.
      intros ;apply P_id_TOP_monotonic;assumption.
      intros ;apply P_id_ACTIVE_monotonic;assumption.
      intros ;apply P_id_F_monotonic;assumption.
      intros ;apply P_id_PROPER_monotonic;assumption.
      intros ;apply P_id_CHECK_monotonic;assumption.
      intros ;apply P_id_START_monotonic;assumption.
    Qed.
    
    Ltac rewrite_and_unfold  :=
     do 2 (rewrite marked_measure_equation);
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
         rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
        end
      ).
    
    Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
    
    Definition le a b := marked_measure a <= marked_measure b.
    
    Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
    Proof.
      unfold lt, le in *.
      intros a b c.
      apply (interp.le_lt_compat_right (interp.o_Z 0)).
    Qed.
    
    Lemma wf_lt : well_founded lt.
    Proof.
      unfold lt in *.
      apply Inverse_Image.wf_inverse_image with  (B:=Z).
      apply Zwf.Zwf_well_founded.
    Qed.
    
    Lemma DP_R_xml_0_scc_16_large_strict_in_lt :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_16_large_strict lt.
    Proof.
      unfold Relation_Definitions.inclusion, lt in *.
      
      intros a b H;destruct H;
       match goal with
         |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
          let l'' := algebra.Alg_ext.find_replacement l  in 
           ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
              (marked_measure (algebra.Alg.Term f l''));[idtac|
             apply marked_measure_star_monotonic;
              repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
              (assumption)||(constructor 1)]))
         end
       ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma DP_R_xml_0_scc_16_large_large_in_le :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_16_large_large le.
    Proof.
      unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
      
      intros a b H;destruct H;
       match goal with
         |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
          let l'' := algebra.Alg_ext.find_replacement l  in 
           ((apply (interp.le_trans (interp.o_Z 0)) with
              (marked_measure (algebra.Alg.Term f l''));[idtac|
             apply marked_measure_star_monotonic;
              repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
              (assumption)||(constructor 1)]))
         end
       ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition wf_DP_R_xml_0_scc_16_large_large  := 
      WF_DP_R_xml_0_scc_16_large_large.wf.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_16.DP_R_xml_0_scc_16_large.
    Proof.
      intros x.
      apply (well_founded_ind wf_lt).
      clear x.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_16_large_large).
      clear x.
      intros x _ IHx IHx'.
      constructor.
      intros y H.
      
      destruct H;
       (apply IHx';apply DP_R_xml_0_scc_16_large_strict_in_lt;
         econstructor eassumption)||
       ((apply IHx;[econstructor eassumption|
         intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
          apply DP_R_xml_0_scc_16_large_large_in_le;econstructor eassumption])).
      apply wf_DP_R_xml_0_scc_16_large_large.
    Qed.
   End WF_DP_R_xml_0_scc_16_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_active (x4:Z) := 2* x4.
   
   Definition P_id_X  := 2.
   
   Definition P_id_c  := 2.
   
   Definition P_id_mark (x4:Z) := 0.
   
   Definition P_id_ok (x4:Z) := 1* x4.
   
   Definition P_id_start (x4:Z) := 2* x4.
   
   Definition P_id_f (x4:Z) := 0.
   
   Definition P_id_proper (x4:Z) := 2.
   
   Definition P_id_check (x4:Z) := 0.
   
   Definition P_id_top (x4:Z) := 0.
   
   Definition P_id_found (x4:Z) := 2* x4.
   
   Definition P_id_match (x4:Z) (x5:Z) := 2* x4.
   
   Lemma P_id_active_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_active x5 <= P_id_active x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_mark x5 <= P_id_mark x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ok x5 <= P_id_ok x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_start x5 <= P_id_start x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_f x5 <= P_id_f x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_proper x5 <= P_id_proper x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_check x5 <= P_id_check x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_top x5 <= P_id_top x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_found x5 <= P_id_found x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_match x5 x7 <= P_id_match x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_active_bounded : forall x4, (0 <= x4) ->0 <= P_id_active x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_X_bounded : 0 <= P_id_X .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_c_bounded : 0 <= P_id_c .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_mark_bounded : forall x4, (0 <= x4) ->0 <= P_id_mark x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ok_bounded : forall x4, (0 <= x4) ->0 <= P_id_ok x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_start_bounded : forall x4, (0 <= x4) ->0 <= P_id_start x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_f_bounded : forall x4, (0 <= x4) ->0 <= P_id_f x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_proper_bounded : forall x4, (0 <= x4) ->0 <= P_id_proper x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_check_bounded : forall x4, (0 <= x4) ->0 <= P_id_check x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_top_bounded : forall x4, (0 <= x4) ->0 <= P_id_top x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_found_bounded : forall x4, (0 <= x4) ->0 <= P_id_found x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_match_bounded :
    forall x4 x5, (0 <= x4) ->(0 <= x5) ->0 <= P_id_match x5 x4.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition measure  := 
     InterpZ.measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found P_id_match
     .
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                    P_id_active (measure x4)
                   | (algebra.Alg.Term algebra.F.id_X nil) => P_id_X 
                   | (algebra.Alg.Term algebra.F.id_c nil) => P_id_c 
                   | (algebra.Alg.Term algebra.F.id_mark (x4::nil)) =>
                    P_id_mark (measure x4)
                   | (algebra.Alg.Term algebra.F.id_ok (x4::nil)) =>
                    P_id_ok (measure x4)
                   | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                    P_id_start (measure x4)
                   | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                    P_id_f (measure x4)
                   | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                    P_id_proper (measure x4)
                   | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                    P_id_check (measure x4)
                   | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                    P_id_top (measure x4)
                   | (algebra.Alg.Term algebra.F.id_found (x4::nil)) =>
                    P_id_found (measure x4)
                   | (algebra.Alg.Term algebra.F.id_match (x5::x4::nil)) =>
                    P_id_match (measure x5) (measure x4)
                   | _ => 0
                   end.
   Proof.
     intros t;case t;intros ;apply refl_equal.
   Qed.
   
   Lemma measure_bounded : forall t, 0 <= measure t.
   Proof.
     unfold measure in |-*.
     
     apply InterpZ.measure_bounded;
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Ltac generate_pos_hyp  :=
    match goal with
      | H:context [measure ?x] |- _ =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      |  |- context [measure ?x] =>
       let v := fresh "v" in 
        (let H := fresh "h" in 
          (set (H:=measure_bounded x) in *;set (v:=measure x) in *;
            clearbody H;clearbody v))
      end
    .
   
   Lemma rules_monotonic :
    forall l r, 
     (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
      measure r <= measure l.
   Proof.
     intros l r H.
     fold measure in |-*.
     
     inversion H;clear H;subst;inversion H0;clear H0;subst;
      simpl algebra.EQT.T.apply_subst in |-*;
      repeat (
      match goal with
        |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
         rewrite (measure_equation (algebra.Alg.Term f t))
        end
      );repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma measure_star_monotonic :
    forall l r, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                r l) ->measure r <= measure l.
   Proof.
     unfold measure in *.
     apply InterpZ.measure_star_monotonic.
     intros ;apply P_id_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MATCH (x4:Z) (x5:Z) := 0.
   
   Definition P_id_TOP (x4:Z) := 1* x4.
   
   Definition P_id_ACTIVE (x4:Z) := 0.
   
   Definition P_id_F (x4:Z) := 0.
   
   Definition P_id_PROPER (x4:Z) := 0.
   
   Definition P_id_CHECK (x4:Z) := 0.
   
   Definition P_id_START (x4:Z) := 0.
   
   Lemma P_id_MATCH_monotonic :
    forall x4 x6 x5 x7, 
     (0 <= x7)/\ (x7 <= x6) ->
      (0 <= x5)/\ (x5 <= x4) ->P_id_MATCH x5 x7 <= P_id_MATCH x4 x6.
   Proof.
     intros x7 x6 x5 x4.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_TOP_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_TOP x5 <= P_id_TOP x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ACTIVE_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_ACTIVE x5 <= P_id_ACTIVE x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_F_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_F x5 <= P_id_F x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_PROPER_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_PROPER x5 <= P_id_PROPER x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_CHECK_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_CHECK x5 <= P_id_CHECK x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_START_monotonic :
    forall x4 x5, (0 <= x5)/\ (x5 <= x4) ->P_id_START x5 <= P_id_START x4.
   Proof.
     intros x5 x4.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_active P_id_X P_id_c P_id_mark P_id_ok 
      P_id_start P_id_f P_id_proper P_id_check P_id_top P_id_found 
      P_id_match P_id_MATCH P_id_TOP P_id_ACTIVE P_id_F P_id_PROPER 
      P_id_CHECK P_id_START.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_match (x5::
                             x4::nil)) =>
                           P_id_MATCH (measure x5) (measure x4)
                          | (algebra.Alg.Term algebra.F.id_top (x4::nil)) =>
                           P_id_TOP (measure x4)
                          | (algebra.Alg.Term algebra.F.id_active (x4::nil)) =>
                           P_id_ACTIVE (measure x4)
                          | (algebra.Alg.Term algebra.F.id_f (x4::nil)) =>
                           P_id_F (measure x4)
                          | (algebra.Alg.Term algebra.F.id_proper (x4::nil)) =>
                           P_id_PROPER (measure x4)
                          | (algebra.Alg.Term algebra.F.id_check (x4::nil)) =>
                           P_id_CHECK (measure x4)
                          | (algebra.Alg.Term algebra.F.id_start (x4::nil)) =>
                           P_id_START (measure x4)
                          | _ => 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_active_monotonic;assumption.
     intros ;apply P_id_mark_monotonic;assumption.
     intros ;apply P_id_ok_monotonic;assumption.
     intros ;apply P_id_start_monotonic;assumption.
     intros ;apply P_id_f_monotonic;assumption.
     intros ;apply P_id_proper_monotonic;assumption.
     intros ;apply P_id_check_monotonic;assumption.
     intros ;apply P_id_top_monotonic;assumption.
     intros ;apply P_id_found_monotonic;assumption.
     intros ;apply P_id_match_monotonic;assumption.
     intros ;apply P_id_active_bounded;assumption.
     intros ;apply P_id_X_bounded;assumption.
     intros ;apply P_id_c_bounded;assumption.
     intros ;apply P_id_mark_bounded;assumption.
     intros ;apply P_id_ok_bounded;assumption.
     intros ;apply P_id_start_bounded;assumption.
     intros ;apply P_id_f_bounded;assumption.
     intros ;apply P_id_proper_bounded;assumption.
     intros ;apply P_id_check_bounded;assumption.
     intros ;apply P_id_top_bounded;assumption.
     intros ;apply P_id_found_bounded;assumption.
     intros ;apply P_id_match_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MATCH_monotonic;assumption.
     intros ;apply P_id_TOP_monotonic;assumption.
     intros ;apply P_id_ACTIVE_monotonic;assumption.
     intros ;apply P_id_F_monotonic;assumption.
     intros ;apply P_id_PROPER_monotonic;assumption.
     intros ;apply P_id_CHECK_monotonic;assumption.
     intros ;apply P_id_START_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
   
   Definition le a b := marked_measure a <= marked_measure b.
   
   Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
   Proof.
     unfold lt, le in *.
     intros a b c.
     apply (interp.le_lt_compat_right (interp.o_Z 0)).
   Qed.
   
   Lemma wf_lt : well_founded lt.
   Proof.
     unfold lt in *.
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
   Qed.
   
   Lemma DP_R_xml_0_scc_16_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_16_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma DP_R_xml_0_scc_16_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_16_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 0)) with
             (marked_measure (algebra.Alg.Term f l''));[idtac|
            apply marked_measure_star_monotonic;
             repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
             (assumption)||(constructor 1)]))
        end
      ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition wf_DP_R_xml_0_scc_16_large  := WF_DP_R_xml_0_scc_16_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_16.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_16_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_16_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_16_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_16_large.
   Qed.
  End WF_DP_R_xml_0_scc_16.
  
  Definition wf_DP_R_xml_0_scc_16  := WF_DP_R_xml_0_scc_16.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_16 :
   forall x y, (DP_R_xml_0_scc_16 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_16).
    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_15;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_4;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
    apply wf_DP_R_xml_0_scc_16.
  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_15;
       econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
     ((eapply acc_DP_R_xml_0_non_scc_14;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_13;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_12;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_11;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_10;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_9;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_8;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_7;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_6;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_5;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_4;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_3;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_2;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_1;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_0;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_scc_16;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_scc_15;
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_scc_14;
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_scc_13;
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_scc_12;
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_scc_11;
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_scc_10;
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_scc_9;
                              econstructor 
                              (eassumption)||(algebra.Alg_ext.star_refl' ))||
                            ((eapply acc_DP_R_xml_0_scc_8;
                               econstructor 
                               (eassumption)||(algebra.Alg_ext.star_refl' ))||
                             ((eapply acc_DP_R_xml_0_scc_7;
                                econstructor 
                                (eassumption)||(algebra.Alg_ext.star_refl' ))||
                              ((eapply acc_DP_R_xml_0_scc_6;
                                 econstructor 
                                 (eassumption)||(algebra.Alg_ext.star_refl' ))||
                               ((eapply acc_DP_R_xml_0_scc_5;
                                  econstructor 
                                  (eassumption)||
                                  (algebra.Alg_ext.star_refl' ))||
                                ((eapply acc_DP_R_xml_0_scc_4;
                                   econstructor 
                                   (eassumption)||
                                   (algebra.Alg_ext.star_refl' ))||
                                 ((eapply acc_DP_R_xml_0_scc_3;
                                    econstructor 
                                    (eassumption)||
                                    (algebra.Alg_ext.star_refl' ))||
                                  ((eapply acc_DP_R_xml_0_scc_2;
                                     econstructor 
                                     (eassumption)||
                                     (algebra.Alg_ext.star_refl' ))||
                                   ((eapply acc_DP_R_xml_0_scc_1;
                                      econstructor 
                                      (eassumption)||
                                      (algebra.Alg_ext.star_refl' ))||
                                    ((eapply acc_DP_R_xml_0_scc_0;
                                       econstructor 
                                       (eassumption)||
                                       (algebra.Alg_ext.star_refl' ))||
                                     ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                                      (fail)))))))))))))))))))))))))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


(* 
*** Local Variables: ***
*** coq-prog-name: "coqtop" ***
*** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") ***
*** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/  c_output/strat/tpdb-5.0___TRS___AProVE___Liveness8.trs/a3pat.v" ***
*** End: ***
 *)