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_app *)
    | id_app : symb
     (* id_last *)
    | id_last : symb
     (* id_nil *)
    | id_nil : symb
     (* id_reverse *)
    | id_reverse : symb
     (* id_hd *)
    | id_hd : symb
     (* id_compose *)
    | id_compose : symb
     (* id_init *)
    | id_init : symb
     (* id_cons *)
    | id_cons : symb
     (* id_reverse2 *)
    | id_reverse2 : symb
     (* id_tl *)
    | id_tl : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_app,id_app => true
      | id_last,id_last => true
      | id_nil,id_nil => true
      | id_reverse,id_reverse => true
      | id_hd,id_hd => true
      | id_compose,id_compose => true
      | id_init,id_init => true
      | id_cons,id_cons => true
      | id_reverse2,id_reverse2 => true
      | id_tl,id_tl => 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_app,id_app => refl_equal _
             | id_last,id_last => refl_equal _
             | id_nil,id_nil => refl_equal _
             | id_reverse,id_reverse => refl_equal _
             | id_hd,id_hd => refl_equal _
             | id_compose,id_compose => refl_equal _
             | id_init,id_init => refl_equal _
             | id_cons,id_cons => refl_equal _
             | id_reverse2,id_reverse2 => refl_equal _
             | id_tl,id_tl => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_app => term.Free 2
      | id_last => term.Free 0
      | id_nil => term.Free 0
      | id_reverse => term.Free 0
      | id_hd => term.Free 0
      | id_compose => term.Free 0
      | id_init => term.Free 0
      | id_cons => term.Free 0
      | id_reverse2 => term.Free 0
      | id_tl => term.Free 0
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_app,id_app => true
      | id_app,id_last => false
      | id_app,id_nil => false
      | id_app,id_reverse => false
      | id_app,id_hd => false
      | id_app,id_compose => false
      | id_app,id_init => false
      | id_app,id_cons => false
      | id_app,id_reverse2 => false
      | id_app,id_tl => false
      | id_last,id_app => true
      | id_last,id_last => true
      | id_last,id_nil => false
      | id_last,id_reverse => false
      | id_last,id_hd => false
      | id_last,id_compose => false
      | id_last,id_init => false
      | id_last,id_cons => false
      | id_last,id_reverse2 => false
      | id_last,id_tl => false
      | id_nil,id_app => true
      | id_nil,id_last => true
      | id_nil,id_nil => true
      | id_nil,id_reverse => false
      | id_nil,id_hd => false
      | id_nil,id_compose => false
      | id_nil,id_init => false
      | id_nil,id_cons => false
      | id_nil,id_reverse2 => false
      | id_nil,id_tl => false
      | id_reverse,id_app => true
      | id_reverse,id_last => true
      | id_reverse,id_nil => true
      | id_reverse,id_reverse => true
      | id_reverse,id_hd => false
      | id_reverse,id_compose => false
      | id_reverse,id_init => false
      | id_reverse,id_cons => false
      | id_reverse,id_reverse2 => false
      | id_reverse,id_tl => false
      | id_hd,id_app => true
      | id_hd,id_last => true
      | id_hd,id_nil => true
      | id_hd,id_reverse => true
      | id_hd,id_hd => true
      | id_hd,id_compose => false
      | id_hd,id_init => false
      | id_hd,id_cons => false
      | id_hd,id_reverse2 => false
      | id_hd,id_tl => false
      | id_compose,id_app => true
      | id_compose,id_last => true
      | id_compose,id_nil => true
      | id_compose,id_reverse => true
      | id_compose,id_hd => true
      | id_compose,id_compose => true
      | id_compose,id_init => false
      | id_compose,id_cons => false
      | id_compose,id_reverse2 => false
      | id_compose,id_tl => false
      | id_init,id_app => true
      | id_init,id_last => true
      | id_init,id_nil => true
      | id_init,id_reverse => true
      | id_init,id_hd => true
      | id_init,id_compose => true
      | id_init,id_init => true
      | id_init,id_cons => false
      | id_init,id_reverse2 => false
      | id_init,id_tl => false
      | id_cons,id_app => true
      | id_cons,id_last => true
      | id_cons,id_nil => true
      | id_cons,id_reverse => true
      | id_cons,id_hd => true
      | id_cons,id_compose => true
      | id_cons,id_init => true
      | id_cons,id_cons => true
      | id_cons,id_reverse2 => false
      | id_cons,id_tl => false
      | id_reverse2,id_app => true
      | id_reverse2,id_last => true
      | id_reverse2,id_nil => true
      | id_reverse2,id_reverse => true
      | id_reverse2,id_hd => true
      | id_reverse2,id_compose => true
      | id_reverse2,id_init => true
      | id_reverse2,id_cons => true
      | id_reverse2,id_reverse2 => true
      | id_reverse2,id_tl => false
      | id_tl,id_app => true
      | id_tl,id_last => true
      | id_tl,id_nil => true
      | id_tl,id_reverse => true
      | id_tl,id_hd => true
      | id_tl,id_compose => true
      | id_tl,id_init => true
      | id_tl,id_cons => true
      | id_tl,id_reverse2 => true
      | id_tl,id_tl => 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 := 
    (* app(app(app(compose,f_),g_),x_) -> app(g_,app(f_,x_)) *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Var 2)::
                   (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 3)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
      ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
      algebra.F.id_compose nil)::(algebra.Alg.Var 1)::nil))::
      (algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 3)::nil))
    (* app(reverse,l_) -> app(app(reverse2,l_),nil) *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_reverse2 
                   nil)::(algebra.Alg.Var 4)::nil))::(algebra.Alg.Term 
                   algebra.F.id_nil nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
      algebra.F.id_reverse nil)::(algebra.Alg.Var 4)::nil))
    (* app(app(reverse2,nil),l_) -> l_ *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Var 4) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
      ((algebra.Alg.Term algebra.F.id_reverse2 nil)::(algebra.Alg.Term 
      algebra.F.id_nil nil)::nil))::(algebra.Alg.Var 4)::nil))
   
    (* app(app(reverse2,app(app(cons,x_),xs_)),l_) -> app(app(reverse2,xs_),app(app(cons,x_),l_)) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_reverse2 
                   nil)::(algebra.Alg.Var 5)::nil))::(algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
                   ((algebra.Alg.Term algebra.F.id_cons nil)::
                   (algebra.Alg.Var 3)::nil))::
                   (algebra.Alg.Var 4)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
      ((algebra.Alg.Term algebra.F.id_reverse2 nil)::(algebra.Alg.Term 
      algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
      ((algebra.Alg.Term algebra.F.id_cons nil)::(algebra.Alg.Var 3)::nil))::
      (algebra.Alg.Var 5)::nil))::nil))::(algebra.Alg.Var 4)::nil))
    (* app(hd,app(app(cons,x_),xs_)) -> x_ *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Var 3) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_hd 
      nil)::(algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
      algebra.F.id_app ((algebra.Alg.Term algebra.F.id_cons nil)::
      (algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 5)::nil))::nil))
    (* app(tl,app(app(cons,x_),xs_)) -> xs_ *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Var 5) 
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_tl 
      nil)::(algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
      algebra.F.id_app ((algebra.Alg.Term algebra.F.id_cons nil)::
      (algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 5)::nil))::nil))
    (* last -> app(app(compose,hd),reverse) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                   nil)::(algebra.Alg.Term algebra.F.id_hd nil)::nil))::
                   (algebra.Alg.Term algebra.F.id_reverse nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_last nil)
   
    (* init -> app(app(compose,reverse),app(app(compose,tl),reverse)) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                   nil)::(algebra.Alg.Term algebra.F.id_reverse nil)::nil))::
                   (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                   nil)::(algebra.Alg.Term algebra.F.id_tl nil)::nil))::
                   (algebra.Alg.Term algebra.F.id_reverse nil)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_init nil)
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_compose nil)::(algebra.Alg.Var 1)::nil))::
     (algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 3)::nil)),
    (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Var 2)::
     (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 3)::nil))::nil)))::nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_reverse nil)::(algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
     (algebra.Alg.Var 4)::nil))::(algebra.Alg.Term algebra.F.id_nil 
     nil)::nil)))::R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_reverse2 nil)::(algebra.Alg.Term 
     algebra.F.id_nil nil)::nil))::(algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Var 4))::R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_reverse2 nil)::(algebra.Alg.Term 
     algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_cons nil)::(algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 5)::nil))::nil))::(algebra.Alg.Var 4)::nil)),
    (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
     (algebra.Alg.Var 5)::nil))::(algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_cons nil)::(algebra.Alg.Var 3)::nil))::
     (algebra.Alg.Var 4)::nil))::nil)))::R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_hd 
     nil)::(algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_app ((algebra.Alg.Term algebra.F.id_cons nil)::
     (algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Var 3))::R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_tl 
     nil)::(algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_app ((algebra.Alg.Term algebra.F.id_cons nil)::
     (algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 5)::nil))::nil)),
    (algebra.Alg.Var 5))::R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id_last nil),
    (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_compose nil)::(algebra.Alg.Term 
     algebra.F.id_hd nil)::nil))::(algebra.Alg.Term algebra.F.id_reverse 
     nil)::nil)))::R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id_init nil),
    (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_compose nil)::(algebra.Alg.Term 
     algebra.F.id_reverse nil)::nil))::(algebra.Alg.Term algebra.F.id_app 
     ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
     algebra.F.id_compose nil)::(algebra.Alg.Term algebra.F.id_tl 
     nil)::nil))::(algebra.Alg.Term algebra.F.id_reverse nil)::nil))::nil)))::
    R_xml_0_rule_as_list_6.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_7.
 
 
 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.or9_equiv in H|-.
   case H;clear H;intros H.
   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_7 (x7 x8 x9 x10 x11 x12 x13:Prop) :
  Prop := 
   | conj_7 : x7->x8->x9->x10->x11->x12->x13->and_7 x7 x8 x9 x10 x11 x12 x13
 .
 
 
 Lemma are_constuctors_of_R_xml_0 :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    and_7 (t = (algebra.Alg.Term algebra.F.id_nil nil) ->
           t' = (algebra.Alg.Term algebra.F.id_nil nil)) 
     (t = (algebra.Alg.Term algebra.F.id_reverse nil) ->
      t' = (algebra.Alg.Term algebra.F.id_reverse nil)) 
     (t = (algebra.Alg.Term algebra.F.id_hd nil) ->
      t' = (algebra.Alg.Term algebra.F.id_hd nil)) 
     (t = (algebra.Alg.Term algebra.F.id_compose nil) ->
      t' = (algebra.Alg.Term algebra.F.id_compose nil)) 
     (t = (algebra.Alg.Term algebra.F.id_cons nil) ->
      t' = (algebra.Alg.Term algebra.F.id_cons nil)) 
     (t = (algebra.Alg.Term algebra.F.id_reverse2 nil) ->
      t' = (algebra.Alg.Term algebra.F.id_reverse2 nil)) 
     (t = (algebra.Alg.Term algebra.F.id_tl nil) ->
      t' = (algebra.Alg.Term algebra.F.id_tl nil)).
 Proof.
   intros t t' H.
   
   induction H as [|y IH z z_to_y] using 
   closure_extension.refl_trans_clos_ind2.
   constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   inversion z_to_y as [t1 t2 H H0 H1|f l1 l2 H0 H H2];clear z_to_y;subst.
   
   inversion H as [t1 t2 sigma H2 H1 H0];clear H IH;subst;inversion H2;
    clear ;constructor;try (intros until 0 );clear ;intros abs;
    discriminate abs.
   
   destruct IH as 
   [H_id_nil H_id_reverse H_id_hd H_id_compose H_id_cons H_id_reverse2 
    H_id_tl].
   constructor.
   
   clear H_id_reverse H_id_hd H_id_compose H_id_cons H_id_reverse2 H_id_tl;
    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_nil H_id_hd H_id_compose H_id_cons H_id_reverse2 H_id_tl;
    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_nil H_id_reverse H_id_compose H_id_cons H_id_reverse2 H_id_tl;
    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_nil H_id_reverse H_id_hd H_id_cons H_id_reverse2 H_id_tl;
    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_nil H_id_reverse H_id_hd H_id_compose H_id_reverse2 H_id_tl;
    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_nil H_id_reverse H_id_hd H_id_compose H_id_cons H_id_tl;
    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_nil H_id_reverse H_id_hd H_id_compose H_id_cons H_id_reverse2;
    intros H;injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
 Qed.
 
 
 Lemma id_nil_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_nil nil) ->
     t' = (algebra.Alg.Term algebra.F.id_nil nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_reverse_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_reverse nil) ->
     t' = (algebra.Alg.Term algebra.F.id_reverse nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_hd_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_hd nil) ->
     t' = (algebra.Alg.Term algebra.F.id_hd nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_compose_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_compose nil) ->
     t' = (algebra.Alg.Term algebra.F.id_compose nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_cons_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_cons nil) ->
     t' = (algebra.Alg.Term algebra.F.id_cons nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_reverse2_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_reverse2 nil) ->
     t' = (algebra.Alg.Term algebra.F.id_reverse2 nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_tl_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_tl nil) ->
     t' = (algebra.Alg.Term algebra.F.id_tl nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Ltac impossible_star_reduction_R_xml_0  :=
  match goal with
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_reverse nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reverse_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_hd nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_hd_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_compose nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_compose_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_cons nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_cons_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_reverse2 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reverse2_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_tl nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tl_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    end
  .
 
 
 Ltac simplify_star_reduction_R_xml_0  :=
  match goal with
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_reverse nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reverse_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_hd nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_hd_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_compose nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_compose_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_cons nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_cons_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_reverse2 nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_reverse2_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_tl nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_tl_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    end
  .
End R_xml_0_deep_rew.

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

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

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

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

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

Module Interp.
 Section S.
   Require Import interp.
   
   Hypothesis A : Type.
   
   Hypothesis Ale Alt Aeq : A -> A -> Prop.
   
   Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
   
   Hypothesis A0 : A.
   
   Notation Local "a <= b" := (Ale a b).
   
   Hypothesis P_id_app : A ->A ->A.
   
   Hypothesis P_id_last : A.
   
   Hypothesis P_id_nil : A.
   
   Hypothesis P_id_reverse : A.
   
   Hypothesis P_id_hd : A.
   
   Hypothesis P_id_compose : A.
   
   Hypothesis P_id_init : A.
   
   Hypothesis P_id_cons : A.
   
   Hypothesis P_id_reverse2 : A.
   
   Hypothesis P_id_tl : A.
   
   Hypothesis P_id_app_monotonic :
    forall x8 x10 x9 x7, 
     (A0 <= x10)/\ (x10 <= x9) ->
      (A0 <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
   
   Hypothesis P_id_app_bounded :
    forall x8 x7, (A0 <= x7) ->(A0 <= x8) ->A0 <= P_id_app x8 x7.
   
   Hypothesis P_id_last_bounded : A0 <= P_id_last .
   
   Hypothesis P_id_nil_bounded : A0 <= P_id_nil .
   
   Hypothesis P_id_reverse_bounded : A0 <= P_id_reverse .
   
   Hypothesis P_id_hd_bounded : A0 <= P_id_hd .
   
   Hypothesis P_id_compose_bounded : A0 <= P_id_compose .
   
   Hypothesis P_id_init_bounded : A0 <= P_id_init .
   
   Hypothesis P_id_cons_bounded : A0 <= P_id_cons .
   
   Hypothesis P_id_reverse2_bounded : A0 <= P_id_reverse2 .
   
   Hypothesis P_id_tl_bounded : A0 <= P_id_tl .
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
        P_id_app (measure x8) (measure x7)
       | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
       | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
       | (algebra.Alg.Term algebra.F.id_reverse nil) => P_id_reverse 
       | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
       | (algebra.Alg.Term algebra.F.id_compose nil) => P_id_compose 
       | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
       | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
       | (algebra.Alg.Term algebra.F.id_reverse2 nil) => P_id_reverse2 
       | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                    P_id_app (measure x8) (measure x7)
                   | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                    P_id_reverse 
                   | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                   | (algebra.Alg.Term algebra.F.id_compose nil) =>
                    P_id_compose 
                   | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
                   | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
                   | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                    P_id_reverse2 
                   | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                   | _ => 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_app => P_id_app
       | algebra.F.id_last => P_id_last
       | algebra.F.id_nil => P_id_nil
       | algebra.F.id_reverse => P_id_reverse
       | algebra.F.id_hd => P_id_hd
       | algebra.F.id_compose => P_id_compose
       | algebra.F.id_init => P_id_init
       | algebra.F.id_cons => P_id_cons
       | algebra.F.id_reverse2 => P_id_reverse2
       | algebra.F.id_tl => P_id_tl
       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_app =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_last => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_reverse => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_hd => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_compose => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_init => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_cons => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_reverse2 => match l with
                                           | nil => _
                                           | _::_ => _
                                           end
              | algebra.F.id_tl => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_app_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_last_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_hd_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_compose_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_init_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tl_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_app_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_app_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_last_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_hd_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_compose_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_init_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tl_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_INIT : A.
   
   Hypothesis P_id_APP : A ->A ->A.
   
   Hypothesis P_id_LAST : A.
   
   Hypothesis P_id_APP_monotonic :
    forall x8 x10 x9 x7, 
     (A0 <= x10)/\ (x10 <= x9) ->
      (A0 <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_init nil) => P_id_INIT 
       | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
        P_id_APP (measure x8) (measure x7)
       | (algebra.Alg.Term algebra.F.id_last nil) => P_id_LAST 
       | _ => 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 ;apply (P_id_INIT ).
     
     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_LAST ).
     
     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 x8 x7;apply (P_id_APP x8 x7).
     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_app =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_last => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_nil => match l with
                                      | nil => _
                                      | _::_ => _
                                      end
              | algebra.F.id_reverse => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_hd => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              | algebra.F.id_compose => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_init => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_cons => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_reverse2 => match l with
                                           | nil => _
                                           | _::_ => _
                                           end
              | algebra.F.id_tl => match l with
                                     | nil => _
                                     | _::_ => _
                                     end
              end.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_init nil) =>
                           P_id_INIT 
                          | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                           P_id_APP (measure x8) (measure x7)
                          | (algebra.Alg.Term algebra.F.id_last nil) =>
                           P_id_LAST 
                          | _ => 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_app_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;apply (Aop.(le_refl)).
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_app_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_last_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_hd_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_compose_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_init_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_reverse2_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_tl_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 (Aop.(le_refl)).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply (Aop.(le_refl)).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_APP_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_app : Z ->Z ->Z.
   
   Hypothesis P_id_last : Z.
   
   Hypothesis P_id_nil : Z.
   
   Hypothesis P_id_reverse : Z.
   
   Hypothesis P_id_hd : Z.
   
   Hypothesis P_id_compose : Z.
   
   Hypothesis P_id_init : Z.
   
   Hypothesis P_id_cons : Z.
   
   Hypothesis P_id_reverse2 : Z.
   
   Hypothesis P_id_tl : Z.
   
   Hypothesis P_id_app_monotonic :
    forall x8 x10 x9 x7, 
     (min_value <= x10)/\ (x10 <= x9) ->
      (min_value <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
   
   Hypothesis P_id_app_bounded :
    forall x8 x7, 
     (min_value <= x7) ->(min_value <= x8) ->min_value <= P_id_app x8 x7.
   
   Hypothesis P_id_last_bounded : min_value <= P_id_last .
   
   Hypothesis P_id_nil_bounded : min_value <= P_id_nil .
   
   Hypothesis P_id_reverse_bounded : min_value <= P_id_reverse .
   
   Hypothesis P_id_hd_bounded : min_value <= P_id_hd .
   
   Hypothesis P_id_compose_bounded : min_value <= P_id_compose .
   
   Hypothesis P_id_init_bounded : min_value <= P_id_init .
   
   Hypothesis P_id_cons_bounded : min_value <= P_id_cons .
   
   Hypothesis P_id_reverse2_bounded : min_value <= P_id_reverse2 .
   
   Hypothesis P_id_tl_bounded : min_value <= P_id_tl .
   
   Definition measure  := 
     Interp.measure min_value P_id_app P_id_last P_id_nil P_id_reverse 
      P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                    P_id_app (measure x8) (measure x7)
                   | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                    P_id_reverse 
                   | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                   | (algebra.Alg.Term algebra.F.id_compose nil) =>
                    P_id_compose 
                   | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
                   | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
                   | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                    P_id_reverse2 
                   | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                   | _ => 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_app_monotonic;assumption.
     intros ;apply P_id_app_bounded;assumption.
     intros ;apply P_id_last_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_reverse_bounded;assumption.
     intros ;apply P_id_hd_bounded;assumption.
     intros ;apply P_id_compose_bounded;assumption.
     intros ;apply P_id_init_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_reverse2_bounded;assumption.
     intros ;apply P_id_tl_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_INIT : Z.
   
   Hypothesis P_id_APP : Z ->Z ->Z.
   
   Hypothesis P_id_LAST : Z.
   
   Hypothesis P_id_APP_monotonic :
    forall x8 x10 x9 x7, 
     (min_value <= x10)/\ (x10 <= x9) ->
      (min_value <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_app P_id_last P_id_nil 
      P_id_reverse P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 
      P_id_tl P_id_INIT P_id_APP P_id_LAST.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_init nil) =>
                           P_id_INIT 
                          | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                           P_id_APP (measure x8) (measure x7)
                          | (algebra.Alg.Term algebra.F.id_last nil) =>
                           P_id_LAST 
                          | _ => 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_app_monotonic;assumption.
     intros ;apply P_id_app_bounded;assumption.
     intros ;apply P_id_last_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_reverse_bounded;assumption.
     intros ;apply P_id_hd_bounded;assumption.
     intros ;apply P_id_compose_bounded;assumption.
     intros ;apply P_id_init_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_reverse2_bounded;assumption.
     intros ;apply P_id_tl_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_APP_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 := 
    (* <app(app(app(compose,f_),g_),x_),app(g_,app(f_,x_))> *)
   | DP_R_xml_0_0 :
    forall x8 x2 x1 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
        x1::nil))::x2::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app (x2::(algebra.Alg.Term 
                   algebra.F.id_app (x1::x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    (* <app(app(app(compose,f_),g_),x_),app(f_,x_)> *)
   | DP_R_xml_0_1 :
    forall x8 x2 x1 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
        x1::nil))::x2::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x3 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app (x1::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    (* <app(reverse,l_),app(app(reverse2,l_),nil)> *)
   | DP_R_xml_0_2 :
    forall x8 x4 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_reverse nil) 
       x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_reverse2 
                   nil)::x4::nil))::(algebra.Alg.Term algebra.F.id_nil 
                   nil)::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    (* <app(reverse,l_),app(reverse2,l_)> *)
   | DP_R_xml_0_3 :
    forall x8 x4 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_reverse nil) 
       x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_reverse2 nil)::x4::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   
    (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(reverse2,xs_),app(app(cons,x_),l_))> *)
   | DP_R_xml_0_4 :
    forall x8 x4 x5 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
        ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_reverse2 
                   nil)::x5::nil))::(algebra.Alg.Term algebra.F.id_app 
                   ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_cons nil)::x3::nil))::x4::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   
    (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(reverse2,xs_)> *)
   | DP_R_xml_0_5 :
    forall x8 x4 x5 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
        ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_reverse2 nil)::x5::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   
    (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(cons,x_),l_)> *)
   | DP_R_xml_0_6 :
    forall x8 x4 x5 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
        ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_app ((algebra.Alg.Term algebra.F.id_cons 
                   nil)::x3::nil))::x4::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   
    (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(cons,x_)> *)
   | DP_R_xml_0_7 :
    forall x8 x4 x5 x3 x7, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
        ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
        algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x4 x7) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                   algebra.F.id_cons nil)::x3::nil)) 
        (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    (* <last,app(app(compose,hd),reverse)> *)
   | DP_R_xml_0_8 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                nil)::(algebra.Alg.Term algebra.F.id_hd nil)::nil))::
                (algebra.Alg.Term algebra.F.id_reverse nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_last nil)
    (* <last,app(compose,hd)> *)
   | DP_R_xml_0_9 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_compose nil)::(algebra.Alg.Term algebra.F.id_hd 
                nil)::nil)) (algebra.Alg.Term algebra.F.id_last nil)
   
    (* <init,app(app(compose,reverse),app(app(compose,tl),reverse))> *)
   | DP_R_xml_0_10 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                nil)::(algebra.Alg.Term algebra.F.id_reverse nil)::nil))::
                (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                nil)::(algebra.Alg.Term algebra.F.id_tl nil)::nil))::
                (algebra.Alg.Term algebra.F.id_reverse nil)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_init nil)
    (* <init,app(compose,reverse)> *)
   | DP_R_xml_0_11 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_compose nil)::(algebra.Alg.Term 
                algebra.F.id_reverse nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_init nil)
    (* <init,app(app(compose,tl),reverse)> *)
   | DP_R_xml_0_12 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose 
                nil)::(algebra.Alg.Term algebra.F.id_tl nil)::nil))::
                (algebra.Alg.Term algebra.F.id_reverse nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_init nil)
    (* <init,app(compose,tl)> *)
   | DP_R_xml_0_13 :
    DP_R_xml_0 (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
                algebra.F.id_compose nil)::(algebra.Alg.Term algebra.F.id_tl 
                nil)::nil)) (algebra.Alg.Term algebra.F.id_init nil)
 .
 
 Module ddp := dp.MakeDP(algebra.EQT).
 
 
 Lemma R_xml_0_dp_step_spec :
  forall x y, 
   (ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) ->
    exists f,
      exists l1,
        exists l2,
          y = algebra.Alg.Term f l2/\ 
          (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step 
                                                            R_xml_0_deep_rew.R_xml_0_rules)
                                                           ) l1 l2)/\ 
          (ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)).
 Proof.
   intros x y H.
   induction H.
   inversion H.
   subst.
   destruct t0.
   refine ((False_ind) _ _).
   refine (R_xml_0_deep_rew.R_xml_0_non_var H0).
   simpl in H|-*.
   exists a.
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   exists ((List.map) (algebra.Alg.apply_subst sigma) l).
   repeat (constructor).
   assumption.
   exists f.
   exists l2.
   exists l1.
   constructor.
   constructor.
   constructor.
   constructor.
   rewrite  <- closure.rwr_list_trans_clos_one_step_list.
   assumption.
   assumption.
 Qed.
 
 
 Ltac included_dp_tac H :=
  injection H;clear H;intros;subst;
  repeat (match goal with 
  | H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=>           
  let x := fresh "x" in 
  let l := fresh "l" in 
  let h1 := fresh "h" in 
  let h2 := fresh "h" in 
  let h3 := fresh "h" in 
  destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _  H) as [x [l[h1[h2 h3]]]];clear H;subst
  | H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ => 
  rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H
  end
  );simpl;
  econstructor eassumption
 .
 
 
 Ltac dp_concl_tac h2 h cont_tac 
  t :=
  match t with
    | False => let h' := fresh "a" in 
                (set (h':=t) in *;cont_tac h';
                  repeat (
                  let e := type of h in 
                   (match e with
                      | ?t => unfold t in h|-;
                               (case h;
                                [abstract (clear h;intros h;injection h;
                                            clear h;intros ;subst;
                                            included_dp_tac h2)|
                                clear h;intros h;clear t])
                      | ?t => unfold t in h|-;elim h
                      end
                    )
                  ))
    | or ?a ?b => let cont_tac 
                   h' := let h'' := fresh "a" in 
                          (set (h'':=or a h') in *;cont_tac h'') in 
                   (dp_concl_tac h2 h cont_tac b)
    end
  .
 
 
 Module WF_DP_R_xml_0.
  Inductive DP_R_xml_0_non_scc_1  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(cons,x_)> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x8 x4 x5 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
         ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_app 
                              ((algebra.Alg.Term algebra.F.id_cons nil)::
                              x3::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_1 :
   forall x y, 
    (DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_2  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(reverse2,xs_)> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x8 x4 x5 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
         ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_app 
                              ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
                              x5::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_2 :
   forall x y, 
    (DP_R_xml_0_non_scc_2 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_3  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <app(reverse,l_),app(reverse2,l_)> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x8 x4 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_reverse nil) 
        x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_app 
                              ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
                              x4::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_3 :
   forall x y, 
    (DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <app(app(app(compose,f_),g_),x_),app(f_,x_)> *)
    | DP_R_xml_0_scc_4_0 :
     forall x8 x2 x1 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
         x1::nil))::x2::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x7) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_app (x1::x3::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    
     (* <app(app(app(compose,f_),g_),x_),app(g_,app(f_,x_))> *)
    | DP_R_xml_0_scc_4_1 :
     forall x8 x2 x1 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
         x1::nil))::x2::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x3 x7) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_app (x2::
                          (algebra.Alg.Term algebra.F.id_app (x1::
                          x3::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     (* <app(reverse,l_),app(app(reverse2,l_),nil)> *)
    | DP_R_xml_0_scc_4_2 :
     forall x8 x4 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_reverse nil) 
        x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
                          x4::nil))::(algebra.Alg.Term algebra.F.id_nil 
                          nil)::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    
     (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(reverse2,xs_),app(app(cons,x_),l_))> *)
    | DP_R_xml_0_scc_4_3 :
     forall x8 x4 x5 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
         ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_reverse2 nil)::
                          x5::nil))::(algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_cons nil)::
                          x3::nil))::x4::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    
     (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(cons,x_),l_)> *)
    | DP_R_xml_0_scc_4_4 :
     forall x8 x4 x5 x3 x7, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
         ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
         algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x4 x7) ->
        DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_app 
                          ((algebra.Alg.Term algebra.F.id_cons nil)::
                          x3::nil))::x4::nil)) 
         (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_4.
   Inductive DP_R_xml_0_scc_4_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <app(app(app(compose,f_),g_),x_),app(f_,x_)> *)
     | DP_R_xml_0_scc_4_large_0 :
      forall x8 x2 x1 x3 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
          x1::nil))::x2::nil)) x8) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x7) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_app (x1::
                                 x3::nil)) 
          (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     
      (* <app(app(app(compose,f_),g_),x_),app(g_,app(f_,x_))> *)
     | DP_R_xml_0_scc_4_large_1 :
      forall x8 x2 x1 x3 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
          x1::nil))::x2::nil)) x8) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x3 x7) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_app (x2::
                                 (algebra.Alg.Term algebra.F.id_app (x1::
                                 x3::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     
      (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(reverse2,xs_),app(app(cons,x_),l_))> *)
     | DP_R_xml_0_scc_4_large_2 :
      forall x8 x4 x5 x3 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
          ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x7) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_app 
                                 ((algebra.Alg.Term algebra.F.id_app 
                                 ((algebra.Alg.Term algebra.F.id_reverse2 
                                 nil)::x5::nil))::(algebra.Alg.Term 
                                 algebra.F.id_app ((algebra.Alg.Term 
                                 algebra.F.id_app ((algebra.Alg.Term 
                                 algebra.F.id_cons nil)::x3::nil))::
                                 x4::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     
      (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(cons,x_),l_)> *)
     | DP_R_xml_0_scc_4_large_3 :
      forall x8 x4 x5 x3 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
          ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
          algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x7) ->
         DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_app 
                                 ((algebra.Alg.Term algebra.F.id_app 
                                 ((algebra.Alg.Term algebra.F.id_cons nil)::
                                 x3::nil))::x4::nil)) 
          (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_4_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <app(reverse,l_),app(app(reverse2,l_),nil)> *)
     | DP_R_xml_0_scc_4_strict_0 :
      forall x8 x4 x7, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_reverse nil) x8) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x4 x7) ->
         DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_app 
                                  ((algebra.Alg.Term algebra.F.id_app 
                                  ((algebra.Alg.Term algebra.F.id_reverse2 
                                  nil)::x4::nil))::(algebra.Alg.Term 
                                  algebra.F.id_nil nil)::nil)) 
          (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_4_large.
    Inductive DP_R_xml_0_scc_4_large_large  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(reverse2,xs_),app(app(cons,x_),l_))> *)
      | DP_R_xml_0_scc_4_large_large_0 :
       forall x8 x4 x5 x3 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
           ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x7) ->
          DP_R_xml_0_scc_4_large_large (algebra.Alg.Term algebra.F.id_app 
                                        ((algebra.Alg.Term algebra.F.id_app 
                                        ((algebra.Alg.Term 
                                        algebra.F.id_reverse2 nil)::
                                        x5::nil))::(algebra.Alg.Term 
                                        algebra.F.id_app ((algebra.Alg.Term 
                                        algebra.F.id_app ((algebra.Alg.Term 
                                        algebra.F.id_cons nil)::x3::nil))::
                                        x4::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
      
       (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(cons,x_),l_)> *)
      | DP_R_xml_0_scc_4_large_large_1 :
       forall x8 x4 x5 x3 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
           ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x4 x7) ->
          DP_R_xml_0_scc_4_large_large (algebra.Alg.Term algebra.F.id_app 
                                        ((algebra.Alg.Term algebra.F.id_app 
                                        ((algebra.Alg.Term algebra.F.id_cons 
                                        nil)::x3::nil))::x4::nil)) 
           (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    .
    
    
    Inductive DP_R_xml_0_scc_4_large_strict  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <app(app(app(compose,f_),g_),x_),app(f_,x_)> *)
      | DP_R_xml_0_scc_4_large_strict_0 :
       forall x8 x2 x1 x3 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
           x1::nil))::x2::nil)) x8) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x7) ->
          DP_R_xml_0_scc_4_large_strict (algebra.Alg.Term algebra.F.id_app 
                                         (x1::x3::nil)) 
           (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
      
       (* <app(app(app(compose,f_),g_),x_),app(g_,app(f_,x_))> *)
      | DP_R_xml_0_scc_4_large_strict_1 :
       forall x8 x2 x1 x3 x7, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
           algebra.F.id_app ((algebra.Alg.Term algebra.F.id_compose nil)::
           x1::nil))::x2::nil)) x8) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    x3 x7) ->
          DP_R_xml_0_scc_4_large_strict (algebra.Alg.Term algebra.F.id_app 
                                         (x2::(algebra.Alg.Term 
                                         algebra.F.id_app (x1::
                                         x3::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_4_large_large.
     Inductive DP_R_xml_0_scc_4_large_large_large  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(cons,x_),l_)> *)
       | DP_R_xml_0_scc_4_large_large_large_0 :
        forall x8 x4 x5 x3 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
            algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
            ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
            algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x7) ->
           DP_R_xml_0_scc_4_large_large_large (algebra.Alg.Term 
                                               algebra.F.id_app 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_app 
                                               ((algebra.Alg.Term 
                                               algebra.F.id_cons nil)::
                                               x3::nil))::x4::nil)) 
            (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     .
     
     
     Inductive DP_R_xml_0_scc_4_large_large_strict  :
      algebra.Alg.term ->algebra.Alg.term ->Prop := 
        (* <app(app(reverse2,app(app(cons,x_),xs_)),l_),app(app(reverse2,xs_),app(app(cons,x_),l_))> *)
       | DP_R_xml_0_scc_4_large_large_strict_0 :
        forall x8 x4 x5 x3 x7, 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
            algebra.F.id_reverse2 nil)::(algebra.Alg.Term algebra.F.id_app 
            ((algebra.Alg.Term algebra.F.id_app ((algebra.Alg.Term 
            algebra.F.id_cons nil)::x3::nil))::x5::nil))::nil)) x8) ->
          (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                     x4 x7) ->
           DP_R_xml_0_scc_4_large_large_strict (algebra.Alg.Term 
                                                algebra.F.id_app 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_app 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_reverse2 nil)::
                                                x5::nil))::(algebra.Alg.Term 
                                                algebra.F.id_app 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_app 
                                                ((algebra.Alg.Term 
                                                algebra.F.id_cons nil)::
                                                x3::nil))::x4::nil))::nil)) 
            (algebra.Alg.Term algebra.F.id_app (x8::x7::nil))
     .
     
     
     Module WF_DP_R_xml_0_scc_4_large_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_app (x7:Z) (x8:Z) := 1* x7 + 1* x8.
      
      Definition P_id_last  := 2.
      
      Definition P_id_nil  := 0.
      
      Definition P_id_reverse  := 1.
      
      Definition P_id_hd  := 0.
      
      Definition P_id_compose  := 0.
      
      Definition P_id_init  := 3.
      
      Definition P_id_cons  := 0.
      
      Definition P_id_reverse2  := 1.
      
      Definition P_id_tl  := 1.
      
      Lemma P_id_app_monotonic :
       forall x8 x10 x9 x7, 
        (0 <= x10)/\ (x10 <= x9) ->
         (0 <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
      Proof.
        intros x10 x9 x8 x7.
        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_app_bounded :
       forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_app x8 x7.
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_last_bounded : 0 <= P_id_last .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_nil_bounded : 0 <= P_id_nil .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_reverse_bounded : 0 <= P_id_reverse .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_hd_bounded : 0 <= P_id_hd .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_compose_bounded : 0 <= P_id_compose .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_init_bounded : 0 <= P_id_init .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_cons_bounded : 0 <= P_id_cons .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_reverse2_bounded : 0 <= P_id_reverse2 .
      Proof.
        intros .
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Lemma P_id_tl_bounded : 0 <= P_id_tl .
      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_app P_id_last P_id_nil P_id_reverse P_id_hd 
         P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl.
      
      Lemma measure_equation :
       forall t, 
        measure t = match t with
                      | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                       P_id_app (measure x8) (measure x7)
                      | (algebra.Alg.Term algebra.F.id_last nil) =>
                       P_id_last 
                      | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                      | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                       P_id_reverse 
                      | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                      | (algebra.Alg.Term algebra.F.id_compose nil) =>
                       P_id_compose 
                      | (algebra.Alg.Term algebra.F.id_init nil) =>
                       P_id_init 
                      | (algebra.Alg.Term algebra.F.id_cons nil) =>
                       P_id_cons 
                      | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                       P_id_reverse2 
                      | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                      | _ => 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_app_monotonic;assumption.
        intros ;apply P_id_app_bounded;assumption.
        intros ;apply P_id_last_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_reverse_bounded;assumption.
        intros ;apply P_id_hd_bounded;assumption.
        intros ;apply P_id_compose_bounded;assumption.
        intros ;apply P_id_init_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_reverse2_bounded;assumption.
        intros ;apply P_id_tl_bounded;assumption.
        apply rules_monotonic.
      Qed.
      
      Definition P_id_INIT  := 0.
      
      Definition P_id_APP (x7:Z) (x8:Z) := 2* x7.
      
      Definition P_id_LAST  := 0.
      
      Lemma P_id_APP_monotonic :
       forall x8 x10 x9 x7, 
        (0 <= x10)/\ (x10 <= x9) ->
         (0 <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
      Proof.
        intros x10 x9 x8 x7.
        intros [H_1 H_0].
        intros [H_3 H_2].
        
        cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
         (auto with zarith)||(repeat (translate_vars );prove_ineq ).
      Qed.
      
      Definition marked_measure  := 
        InterpZ.marked_measure 0 P_id_app P_id_last P_id_nil P_id_reverse 
         P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl 
         P_id_INIT P_id_APP P_id_LAST.
      
      Lemma marked_measure_equation :
       forall t, 
        marked_measure t = match t with
                             | (algebra.Alg.Term algebra.F.id_init nil) =>
                              P_id_INIT 
                             | (algebra.Alg.Term algebra.F.id_app (x8::
                                x7::nil)) =>
                              P_id_APP (measure x8) (measure x7)
                             | (algebra.Alg.Term algebra.F.id_last nil) =>
                              P_id_LAST 
                             | _ => 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_app_monotonic;assumption.
        intros ;apply P_id_app_bounded;assumption.
        intros ;apply P_id_last_bounded;assumption.
        intros ;apply P_id_nil_bounded;assumption.
        intros ;apply P_id_reverse_bounded;assumption.
        intros ;apply P_id_hd_bounded;assumption.
        intros ;apply P_id_compose_bounded;assumption.
        intros ;apply P_id_init_bounded;assumption.
        intros ;apply P_id_cons_bounded;assumption.
        intros ;apply P_id_reverse2_bounded;assumption.
        intros ;apply P_id_tl_bounded;assumption.
        apply rules_monotonic.
        intros ;apply P_id_APP_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_4_large_large.DP_R_xml_0_scc_4_large_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_4_large_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_app (x7:Z) (x8:Z) := 1* x7 + 1* x8.
     
     Definition P_id_last  := 3.
     
     Definition P_id_nil  := 0.
     
     Definition P_id_reverse  := 0.
     
     Definition P_id_hd  := 0.
     
     Definition P_id_compose  := 0.
     
     Definition P_id_init  := 2.
     
     Definition P_id_cons  := 1.
     
     Definition P_id_reverse2  := 0.
     
     Definition P_id_tl  := 1.
     
     Lemma P_id_app_monotonic :
      forall x8 x10 x9 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
     Proof.
       intros x10 x9 x8 x7.
       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_app_bounded :
      forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_app x8 x7.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_last_bounded : 0 <= P_id_last .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_nil_bounded : 0 <= P_id_nil .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_reverse_bounded : 0 <= P_id_reverse .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_hd_bounded : 0 <= P_id_hd .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_compose_bounded : 0 <= P_id_compose .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_init_bounded : 0 <= P_id_init .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_cons_bounded : 0 <= P_id_cons .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_reverse2_bounded : 0 <= P_id_reverse2 .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_tl_bounded : 0 <= P_id_tl .
     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_app P_id_last P_id_nil P_id_reverse P_id_hd 
        P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                      P_id_app (measure x8) (measure x7)
                     | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
                     | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                     | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                      P_id_reverse 
                     | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                     | (algebra.Alg.Term algebra.F.id_compose nil) =>
                      P_id_compose 
                     | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
                     | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
                     | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                      P_id_reverse2 
                     | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                     | _ => 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_app_monotonic;assumption.
       intros ;apply P_id_app_bounded;assumption.
       intros ;apply P_id_last_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_reverse_bounded;assumption.
       intros ;apply P_id_hd_bounded;assumption.
       intros ;apply P_id_compose_bounded;assumption.
       intros ;apply P_id_init_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_reverse2_bounded;assumption.
       intros ;apply P_id_tl_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_INIT  := 0.
     
     Definition P_id_APP (x7:Z) (x8:Z) := 2* x7 + 1* x8.
     
     Definition P_id_LAST  := 0.
     
     Lemma P_id_APP_monotonic :
      forall x8 x10 x9 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
     Proof.
       intros x10 x9 x8 x7.
       intros [H_1 H_0].
       intros [H_3 H_2].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_app P_id_last P_id_nil P_id_reverse 
        P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl 
        P_id_INIT P_id_APP P_id_LAST.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_init nil) =>
                             P_id_INIT 
                            | (algebra.Alg.Term algebra.F.id_app (x8::
                               x7::nil)) =>
                             P_id_APP (measure x8) (measure x7)
                            | (algebra.Alg.Term algebra.F.id_last nil) =>
                             P_id_LAST 
                            | _ => 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_app_monotonic;assumption.
       intros ;apply P_id_app_bounded;assumption.
       intros ;apply P_id_last_bounded;assumption.
       intros ;apply P_id_nil_bounded;assumption.
       intros ;apply P_id_reverse_bounded;assumption.
       intros ;apply P_id_hd_bounded;assumption.
       intros ;apply P_id_compose_bounded;assumption.
       intros ;apply P_id_init_bounded;assumption.
       intros ;apply P_id_cons_bounded;assumption.
       intros ;apply P_id_reverse2_bounded;assumption.
       intros ;apply P_id_tl_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_APP_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_4_large_large_strict_in_lt :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large_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_4_large_large_large_in_le :
      Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large_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_4_large_large_large  := 
       WF_DP_R_xml_0_scc_4_large_large_large.wf.
     
     
     Lemma wf :
      well_founded WF_DP_R_xml_0_scc_4_large.DP_R_xml_0_scc_4_large_large.
     Proof.
       intros x.
       apply (well_founded_ind wf_lt).
       clear x.
       intros x.
       pattern x.
       apply (@Acc_ind _ DP_R_xml_0_scc_4_large_large_large).
       clear x.
       intros x _ IHx IHx'.
       constructor.
       intros y H.
       
       destruct H;
        (apply IHx';apply DP_R_xml_0_scc_4_large_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_4_large_large_large_in_le;
           econstructor eassumption])).
       apply wf_DP_R_xml_0_scc_4_large_large_large.
     Qed.
    End WF_DP_R_xml_0_scc_4_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_app (x7:Z) (x8:Z) := 1* x7 + 1* x8.
    
    Definition P_id_last  := 1.
    
    Definition P_id_nil  := 0.
    
    Definition P_id_reverse  := 0.
    
    Definition P_id_hd  := 0.
    
    Definition P_id_compose  := 1.
    
    Definition P_id_init  := 2.
    
    Definition P_id_cons  := 1.
    
    Definition P_id_reverse2  := 0.
    
    Definition P_id_tl  := 0.
    
    Lemma P_id_app_monotonic :
     forall x8 x10 x9 x7, 
      (0 <= x10)/\ (x10 <= x9) ->
       (0 <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
    Proof.
      intros x10 x9 x8 x7.
      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_app_bounded :
     forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_app x8 x7.
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_last_bounded : 0 <= P_id_last .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_nil_bounded : 0 <= P_id_nil .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_reverse_bounded : 0 <= P_id_reverse .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_hd_bounded : 0 <= P_id_hd .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_compose_bounded : 0 <= P_id_compose .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_init_bounded : 0 <= P_id_init .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_cons_bounded : 0 <= P_id_cons .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_reverse2_bounded : 0 <= P_id_reverse2 .
    Proof.
      intros .
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Lemma P_id_tl_bounded : 0 <= P_id_tl .
    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_app P_id_last P_id_nil P_id_reverse P_id_hd 
       P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl.
    
    Lemma measure_equation :
     forall t, 
      measure t = match t with
                    | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                     P_id_app (measure x8) (measure x7)
                    | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
                    | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                    | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                     P_id_reverse 
                    | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                    | (algebra.Alg.Term algebra.F.id_compose nil) =>
                     P_id_compose 
                    | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
                    | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
                    | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                     P_id_reverse2 
                    | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                    | _ => 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_app_monotonic;assumption.
      intros ;apply P_id_app_bounded;assumption.
      intros ;apply P_id_last_bounded;assumption.
      intros ;apply P_id_nil_bounded;assumption.
      intros ;apply P_id_reverse_bounded;assumption.
      intros ;apply P_id_hd_bounded;assumption.
      intros ;apply P_id_compose_bounded;assumption.
      intros ;apply P_id_init_bounded;assumption.
      intros ;apply P_id_cons_bounded;assumption.
      intros ;apply P_id_reverse2_bounded;assumption.
      intros ;apply P_id_tl_bounded;assumption.
      apply rules_monotonic.
    Qed.
    
    Definition P_id_INIT  := 0.
    
    Definition P_id_APP (x7:Z) (x8:Z) := 1* x7 + 1* x8.
    
    Definition P_id_LAST  := 0.
    
    Lemma P_id_APP_monotonic :
     forall x8 x10 x9 x7, 
      (0 <= x10)/\ (x10 <= x9) ->
       (0 <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
    Proof.
      intros x10 x9 x8 x7.
      intros [H_1 H_0].
      intros [H_3 H_2].
      
      cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
       (auto with zarith)||(repeat (translate_vars );prove_ineq ).
    Qed.
    
    Definition marked_measure  := 
      InterpZ.marked_measure 0 P_id_app P_id_last P_id_nil P_id_reverse 
       P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl 
       P_id_INIT P_id_APP P_id_LAST.
    
    Lemma marked_measure_equation :
     forall t, 
      marked_measure t = match t with
                           | (algebra.Alg.Term algebra.F.id_init nil) =>
                            P_id_INIT 
                           | (algebra.Alg.Term algebra.F.id_app (x8::
                              x7::nil)) =>
                            P_id_APP (measure x8) (measure x7)
                           | (algebra.Alg.Term algebra.F.id_last nil) =>
                            P_id_LAST 
                           | _ => 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_app_monotonic;assumption.
      intros ;apply P_id_app_bounded;assumption.
      intros ;apply P_id_last_bounded;assumption.
      intros ;apply P_id_nil_bounded;assumption.
      intros ;apply P_id_reverse_bounded;assumption.
      intros ;apply P_id_hd_bounded;assumption.
      intros ;apply P_id_compose_bounded;assumption.
      intros ;apply P_id_init_bounded;assumption.
      intros ;apply P_id_cons_bounded;assumption.
      intros ;apply P_id_reverse2_bounded;assumption.
      intros ;apply P_id_tl_bounded;assumption.
      apply rules_monotonic.
      intros ;apply P_id_APP_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_4_large_strict_in_lt :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_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_4_large_large_in_le :
     Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_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_4_large_large  := 
      WF_DP_R_xml_0_scc_4_large_large.wf.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large.
    Proof.
      intros x.
      apply (well_founded_ind wf_lt).
      clear x.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_4_large_large).
      clear x.
      intros x _ IHx IHx'.
      constructor.
      intros y H.
      
      destruct H;
       (apply IHx';apply DP_R_xml_0_scc_4_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_4_large_large_in_le;econstructor eassumption])).
      apply wf_DP_R_xml_0_scc_4_large_large.
    Qed.
   End WF_DP_R_xml_0_scc_4_large.
   
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_app (x7:Z) (x8:Z) := 1* x7 + 1* x8.
   
   Definition P_id_last  := 1.
   
   Definition P_id_nil  := 0.
   
   Definition P_id_reverse  := 1.
   
   Definition P_id_hd  := 0.
   
   Definition P_id_compose  := 0.
   
   Definition P_id_init  := 3.
   
   Definition P_id_cons  := 0.
   
   Definition P_id_reverse2  := 0.
   
   Definition P_id_tl  := 0.
   
   Lemma P_id_app_monotonic :
    forall x8 x10 x9 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->P_id_app x8 x10 <= P_id_app x7 x9.
   Proof.
     intros x10 x9 x8 x7.
     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_app_bounded :
    forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_app x8 x7.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_last_bounded : 0 <= P_id_last .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_nil_bounded : 0 <= P_id_nil .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reverse_bounded : 0 <= P_id_reverse .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_hd_bounded : 0 <= P_id_hd .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_compose_bounded : 0 <= P_id_compose .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_init_bounded : 0 <= P_id_init .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_cons_bounded : 0 <= P_id_cons .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_reverse2_bounded : 0 <= P_id_reverse2 .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_tl_bounded : 0 <= P_id_tl .
   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_app P_id_last P_id_nil P_id_reverse P_id_hd 
      P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                    P_id_app (measure x8) (measure x7)
                   | (algebra.Alg.Term algebra.F.id_last nil) => P_id_last 
                   | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil 
                   | (algebra.Alg.Term algebra.F.id_reverse nil) =>
                    P_id_reverse 
                   | (algebra.Alg.Term algebra.F.id_hd nil) => P_id_hd 
                   | (algebra.Alg.Term algebra.F.id_compose nil) =>
                    P_id_compose 
                   | (algebra.Alg.Term algebra.F.id_init nil) => P_id_init 
                   | (algebra.Alg.Term algebra.F.id_cons nil) => P_id_cons 
                   | (algebra.Alg.Term algebra.F.id_reverse2 nil) =>
                    P_id_reverse2 
                   | (algebra.Alg.Term algebra.F.id_tl nil) => P_id_tl 
                   | _ => 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_app_monotonic;assumption.
     intros ;apply P_id_app_bounded;assumption.
     intros ;apply P_id_last_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_reverse_bounded;assumption.
     intros ;apply P_id_hd_bounded;assumption.
     intros ;apply P_id_compose_bounded;assumption.
     intros ;apply P_id_init_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_reverse2_bounded;assumption.
     intros ;apply P_id_tl_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_INIT  := 0.
   
   Definition P_id_APP (x7:Z) (x8:Z) := 2* x7 + 2* x8.
   
   Definition P_id_LAST  := 0.
   
   Lemma P_id_APP_monotonic :
    forall x8 x10 x9 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->P_id_APP x8 x10 <= P_id_APP x7 x9.
   Proof.
     intros x10 x9 x8 x7.
     intros [H_1 H_0].
     intros [H_3 H_2].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_app P_id_last P_id_nil P_id_reverse 
      P_id_hd P_id_compose P_id_init P_id_cons P_id_reverse2 P_id_tl 
      P_id_INIT P_id_APP P_id_LAST.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_init nil) =>
                           P_id_INIT 
                          | (algebra.Alg.Term algebra.F.id_app (x8::x7::nil)) =>
                           P_id_APP (measure x8) (measure x7)
                          | (algebra.Alg.Term algebra.F.id_last nil) =>
                           P_id_LAST 
                          | _ => 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_app_monotonic;assumption.
     intros ;apply P_id_app_bounded;assumption.
     intros ;apply P_id_last_bounded;assumption.
     intros ;apply P_id_nil_bounded;assumption.
     intros ;apply P_id_reverse_bounded;assumption.
     intros ;apply P_id_hd_bounded;assumption.
     intros ;apply P_id_compose_bounded;assumption.
     intros ;apply P_id_init_bounded;assumption.
     intros ;apply P_id_cons_bounded;assumption.
     intros ;apply P_id_reverse2_bounded;assumption.
     intros ;apply P_id_tl_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_APP_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_4_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_strict lt.
   Proof.
     unfold Relation_Definitions.inclusion, lt in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- (Zwf.Zwf 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_4_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large le.
   Proof.
     unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
     
     intros a b H;destruct H;
      match goal with
        |  |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
         let l'' := algebra.Alg_ext.find_replacement l  in 
          ((apply (interp.le_trans (interp.o_Z 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_4_large  := WF_DP_R_xml_0_scc_4_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_4.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_4_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_4_strict_in_lt;
        econstructor eassumption)||
      ((apply IHx;[econstructor eassumption|
        intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
         apply DP_R_xml_0_scc_4_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_4_large.
   Qed.
  End WF_DP_R_xml_0_scc_4.
  
  Definition wf_DP_R_xml_0_scc_4  := WF_DP_R_xml_0_scc_4.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_4 :
   forall x y, (DP_R_xml_0_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_4).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
    apply wf_DP_R_xml_0_scc_4.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <init,app(compose,tl)> *)
    | DP_R_xml_0_non_scc_5_0 :
     DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_tl nil)::nil)) 
      (algebra.Alg.Term algebra.F.id_init nil)
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_5 :
   forall x y, 
    (DP_R_xml_0_non_scc_5 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_6  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <init,app(app(compose,tl),reverse)> *)
    | DP_R_xml_0_non_scc_6_0 :
     DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_tl nil)::nil))::
                           (algebra.Alg.Term algebra.F.id_reverse nil)::nil)) 
      (algebra.Alg.Term algebra.F.id_init nil)
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_6 :
   forall x y, 
    (DP_R_xml_0_non_scc_6 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_7  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <init,app(compose,reverse)> *)
    | DP_R_xml_0_non_scc_7_0 :
     DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_reverse nil)::nil)) 
      (algebra.Alg.Term algebra.F.id_init 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_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_8  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <init,app(app(compose,reverse),app(app(compose,tl),reverse))> *)
    | DP_R_xml_0_non_scc_8_0 :
     DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_reverse 
                           nil)::nil))::(algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_tl nil)::nil))::
                           (algebra.Alg.Term algebra.F.id_reverse 
                           nil)::nil))::nil)) 
      (algebra.Alg.Term algebra.F.id_init 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_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_9  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <last,app(compose,hd)> *)
    | DP_R_xml_0_non_scc_9_0 :
     DP_R_xml_0_non_scc_9 (algebra.Alg.Term algebra.F.id_app 
                           ((algebra.Alg.Term algebra.F.id_compose nil)::
                           (algebra.Alg.Term algebra.F.id_hd nil)::nil)) 
      (algebra.Alg.Term algebra.F.id_last nil)
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_9 :
   forall x y, 
    (DP_R_xml_0_non_scc_9 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_10  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <last,app(app(compose,hd),reverse)> *)
    | DP_R_xml_0_non_scc_10_0 :
     DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_app 
                            ((algebra.Alg.Term algebra.F.id_app 
                            ((algebra.Alg.Term algebra.F.id_compose nil)::
                            (algebra.Alg.Term algebra.F.id_hd nil)::nil))::
                            (algebra.Alg.Term algebra.F.id_reverse 
                            nil)::nil)) 
      (algebra.Alg.Term algebra.F.id_last 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_4;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_3;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_2;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_1;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
          (eapply Hrec;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
  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_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_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___higher-order___AProVE_HO___ReverseLastInit.trs/a3pat.v" ***
*** End: ***
 *)