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_0 *)
    | id_0 : symb
     (* id_bs *)
    | id_bs : symb
     (* id_and *)
    | id_and : symb
     (* id__minus_ *)
    | id__minus_ : symb
     (* id_l *)
    | id_l : symb
     (* id__plus_ *)
    | id__plus_ : symb
     (* id_wb *)
    | id_wb : symb
     (* id_ge *)
    | id_ge : symb
     (* id_false *)
    | id_false : symb
     (* id_min *)
    | id_min : symb
     (* id__sharp_ *)
    | id__sharp_ : symb
     (* id_size *)
    | id_size : symb
     (* id_if *)
    | id_if : symb
     (* id_not *)
    | id_not : symb
     (* id_n *)
    | id_n : symb
     (* id_1 *)
    | id_1 : symb
     (* id_val *)
    | id_val : symb
     (* id_true *)
    | id_true : symb
     (* id_max *)
    | id_max : symb
  .
  
  
  Definition symb_eq_bool (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_0,id_0 => true
      | id_bs,id_bs => true
      | id_and,id_and => true
      | id__minus_,id__minus_ => true
      | id_l,id_l => true
      | id__plus_,id__plus_ => true
      | id_wb,id_wb => true
      | id_ge,id_ge => true
      | id_false,id_false => true
      | id_min,id_min => true
      | id__sharp_,id__sharp_ => true
      | id_size,id_size => true
      | id_if,id_if => true
      | id_not,id_not => true
      | id_n,id_n => true
      | id_1,id_1 => true
      | id_val,id_val => true
      | id_true,id_true => true
      | id_max,id_max => 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_0,id_0 => refl_equal _
             | id_bs,id_bs => refl_equal _
             | id_and,id_and => refl_equal _
             | id__minus_,id__minus_ => refl_equal _
             | id_l,id_l => refl_equal _
             | id__plus_,id__plus_ => refl_equal _
             | id_wb,id_wb => refl_equal _
             | id_ge,id_ge => refl_equal _
             | id_false,id_false => refl_equal _
             | id_min,id_min => refl_equal _
             | id__sharp_,id__sharp_ => refl_equal _
             | id_size,id_size => refl_equal _
             | id_if,id_if => refl_equal _
             | id_not,id_not => refl_equal _
             | id_n,id_n => refl_equal _
             | id_1,id_1 => refl_equal _
             | id_val,id_val => refl_equal _
             | id_true,id_true => refl_equal _
             | id_max,id_max => refl_equal _
             | _,_ => _
             end;intros abs;discriminate.
  Defined.
  
  
  Definition arity (f:symb) := 
    match f with
      | id_0 => term.Free 1
      | id_bs => term.Free 1
      | id_and => term.Free 2
      | id__minus_ => term.Free 2
      | id_l => term.Free 1
      | id__plus_ => term.Free 2
      | id_wb => term.Free 1
      | id_ge => term.Free 2
      | id_false => term.Free 0
      | id_min => term.Free 1
      | id__sharp_ => term.Free 0
      | id_size => term.Free 1
      | id_if => term.Free 3
      | id_not => term.Free 1
      | id_n => term.Free 3
      | id_1 => term.Free 1
      | id_val => term.Free 1
      | id_true => term.Free 0
      | id_max => term.Free 1
      end.
  
  
  Definition symb_order (f1 f2:symb) : bool := 
    match f1,f2 with
      | id_0,id_0 => true
      | id_0,id_bs => false
      | id_0,id_and => false
      | id_0,id__minus_ => false
      | id_0,id_l => false
      | id_0,id__plus_ => false
      | id_0,id_wb => false
      | id_0,id_ge => false
      | id_0,id_false => false
      | id_0,id_min => false
      | id_0,id__sharp_ => false
      | id_0,id_size => false
      | id_0,id_if => false
      | id_0,id_not => false
      | id_0,id_n => false
      | id_0,id_1 => false
      | id_0,id_val => false
      | id_0,id_true => false
      | id_0,id_max => false
      | id_bs,id_0 => true
      | id_bs,id_bs => true
      | id_bs,id_and => false
      | id_bs,id__minus_ => false
      | id_bs,id_l => false
      | id_bs,id__plus_ => false
      | id_bs,id_wb => false
      | id_bs,id_ge => false
      | id_bs,id_false => false
      | id_bs,id_min => false
      | id_bs,id__sharp_ => false
      | id_bs,id_size => false
      | id_bs,id_if => false
      | id_bs,id_not => false
      | id_bs,id_n => false
      | id_bs,id_1 => false
      | id_bs,id_val => false
      | id_bs,id_true => false
      | id_bs,id_max => false
      | id_and,id_0 => true
      | id_and,id_bs => true
      | id_and,id_and => true
      | id_and,id__minus_ => false
      | id_and,id_l => false
      | id_and,id__plus_ => false
      | id_and,id_wb => false
      | id_and,id_ge => false
      | id_and,id_false => false
      | id_and,id_min => false
      | id_and,id__sharp_ => false
      | id_and,id_size => false
      | id_and,id_if => false
      | id_and,id_not => false
      | id_and,id_n => false
      | id_and,id_1 => false
      | id_and,id_val => false
      | id_and,id_true => false
      | id_and,id_max => false
      | id__minus_,id_0 => true
      | id__minus_,id_bs => true
      | id__minus_,id_and => true
      | id__minus_,id__minus_ => true
      | id__minus_,id_l => false
      | id__minus_,id__plus_ => false
      | id__minus_,id_wb => false
      | id__minus_,id_ge => false
      | id__minus_,id_false => false
      | id__minus_,id_min => false
      | id__minus_,id__sharp_ => false
      | id__minus_,id_size => false
      | id__minus_,id_if => false
      | id__minus_,id_not => false
      | id__minus_,id_n => false
      | id__minus_,id_1 => false
      | id__minus_,id_val => false
      | id__minus_,id_true => false
      | id__minus_,id_max => false
      | id_l,id_0 => true
      | id_l,id_bs => true
      | id_l,id_and => true
      | id_l,id__minus_ => true
      | id_l,id_l => true
      | id_l,id__plus_ => false
      | id_l,id_wb => false
      | id_l,id_ge => false
      | id_l,id_false => false
      | id_l,id_min => false
      | id_l,id__sharp_ => false
      | id_l,id_size => false
      | id_l,id_if => false
      | id_l,id_not => false
      | id_l,id_n => false
      | id_l,id_1 => false
      | id_l,id_val => false
      | id_l,id_true => false
      | id_l,id_max => false
      | id__plus_,id_0 => true
      | id__plus_,id_bs => true
      | id__plus_,id_and => true
      | id__plus_,id__minus_ => true
      | id__plus_,id_l => true
      | id__plus_,id__plus_ => true
      | id__plus_,id_wb => false
      | id__plus_,id_ge => false
      | id__plus_,id_false => false
      | id__plus_,id_min => false
      | id__plus_,id__sharp_ => false
      | id__plus_,id_size => false
      | id__plus_,id_if => false
      | id__plus_,id_not => false
      | id__plus_,id_n => false
      | id__plus_,id_1 => false
      | id__plus_,id_val => false
      | id__plus_,id_true => false
      | id__plus_,id_max => false
      | id_wb,id_0 => true
      | id_wb,id_bs => true
      | id_wb,id_and => true
      | id_wb,id__minus_ => true
      | id_wb,id_l => true
      | id_wb,id__plus_ => true
      | id_wb,id_wb => true
      | id_wb,id_ge => false
      | id_wb,id_false => false
      | id_wb,id_min => false
      | id_wb,id__sharp_ => false
      | id_wb,id_size => false
      | id_wb,id_if => false
      | id_wb,id_not => false
      | id_wb,id_n => false
      | id_wb,id_1 => false
      | id_wb,id_val => false
      | id_wb,id_true => false
      | id_wb,id_max => false
      | id_ge,id_0 => true
      | id_ge,id_bs => true
      | id_ge,id_and => true
      | id_ge,id__minus_ => true
      | id_ge,id_l => true
      | id_ge,id__plus_ => true
      | id_ge,id_wb => true
      | id_ge,id_ge => true
      | id_ge,id_false => false
      | id_ge,id_min => false
      | id_ge,id__sharp_ => false
      | id_ge,id_size => false
      | id_ge,id_if => false
      | id_ge,id_not => false
      | id_ge,id_n => false
      | id_ge,id_1 => false
      | id_ge,id_val => false
      | id_ge,id_true => false
      | id_ge,id_max => false
      | id_false,id_0 => true
      | id_false,id_bs => true
      | id_false,id_and => true
      | id_false,id__minus_ => true
      | id_false,id_l => true
      | id_false,id__plus_ => true
      | id_false,id_wb => true
      | id_false,id_ge => true
      | id_false,id_false => true
      | id_false,id_min => false
      | id_false,id__sharp_ => false
      | id_false,id_size => false
      | id_false,id_if => false
      | id_false,id_not => false
      | id_false,id_n => false
      | id_false,id_1 => false
      | id_false,id_val => false
      | id_false,id_true => false
      | id_false,id_max => false
      | id_min,id_0 => true
      | id_min,id_bs => true
      | id_min,id_and => true
      | id_min,id__minus_ => true
      | id_min,id_l => true
      | id_min,id__plus_ => true
      | id_min,id_wb => true
      | id_min,id_ge => true
      | id_min,id_false => true
      | id_min,id_min => true
      | id_min,id__sharp_ => false
      | id_min,id_size => false
      | id_min,id_if => false
      | id_min,id_not => false
      | id_min,id_n => false
      | id_min,id_1 => false
      | id_min,id_val => false
      | id_min,id_true => false
      | id_min,id_max => false
      | id__sharp_,id_0 => true
      | id__sharp_,id_bs => true
      | id__sharp_,id_and => true
      | id__sharp_,id__minus_ => true
      | id__sharp_,id_l => true
      | id__sharp_,id__plus_ => true
      | id__sharp_,id_wb => true
      | id__sharp_,id_ge => true
      | id__sharp_,id_false => true
      | id__sharp_,id_min => true
      | id__sharp_,id__sharp_ => true
      | id__sharp_,id_size => false
      | id__sharp_,id_if => false
      | id__sharp_,id_not => false
      | id__sharp_,id_n => false
      | id__sharp_,id_1 => false
      | id__sharp_,id_val => false
      | id__sharp_,id_true => false
      | id__sharp_,id_max => false
      | id_size,id_0 => true
      | id_size,id_bs => true
      | id_size,id_and => true
      | id_size,id__minus_ => true
      | id_size,id_l => true
      | id_size,id__plus_ => true
      | id_size,id_wb => true
      | id_size,id_ge => true
      | id_size,id_false => true
      | id_size,id_min => true
      | id_size,id__sharp_ => true
      | id_size,id_size => true
      | id_size,id_if => false
      | id_size,id_not => false
      | id_size,id_n => false
      | id_size,id_1 => false
      | id_size,id_val => false
      | id_size,id_true => false
      | id_size,id_max => false
      | id_if,id_0 => true
      | id_if,id_bs => true
      | id_if,id_and => true
      | id_if,id__minus_ => true
      | id_if,id_l => true
      | id_if,id__plus_ => true
      | id_if,id_wb => true
      | id_if,id_ge => true
      | id_if,id_false => true
      | id_if,id_min => true
      | id_if,id__sharp_ => true
      | id_if,id_size => true
      | id_if,id_if => true
      | id_if,id_not => false
      | id_if,id_n => false
      | id_if,id_1 => false
      | id_if,id_val => false
      | id_if,id_true => false
      | id_if,id_max => false
      | id_not,id_0 => true
      | id_not,id_bs => true
      | id_not,id_and => true
      | id_not,id__minus_ => true
      | id_not,id_l => true
      | id_not,id__plus_ => true
      | id_not,id_wb => true
      | id_not,id_ge => true
      | id_not,id_false => true
      | id_not,id_min => true
      | id_not,id__sharp_ => true
      | id_not,id_size => true
      | id_not,id_if => true
      | id_not,id_not => true
      | id_not,id_n => false
      | id_not,id_1 => false
      | id_not,id_val => false
      | id_not,id_true => false
      | id_not,id_max => false
      | id_n,id_0 => true
      | id_n,id_bs => true
      | id_n,id_and => true
      | id_n,id__minus_ => true
      | id_n,id_l => true
      | id_n,id__plus_ => true
      | id_n,id_wb => true
      | id_n,id_ge => true
      | id_n,id_false => true
      | id_n,id_min => true
      | id_n,id__sharp_ => true
      | id_n,id_size => true
      | id_n,id_if => true
      | id_n,id_not => true
      | id_n,id_n => true
      | id_n,id_1 => false
      | id_n,id_val => false
      | id_n,id_true => false
      | id_n,id_max => false
      | id_1,id_0 => true
      | id_1,id_bs => true
      | id_1,id_and => true
      | id_1,id__minus_ => true
      | id_1,id_l => true
      | id_1,id__plus_ => true
      | id_1,id_wb => true
      | id_1,id_ge => true
      | id_1,id_false => true
      | id_1,id_min => true
      | id_1,id__sharp_ => true
      | id_1,id_size => true
      | id_1,id_if => true
      | id_1,id_not => true
      | id_1,id_n => true
      | id_1,id_1 => true
      | id_1,id_val => false
      | id_1,id_true => false
      | id_1,id_max => false
      | id_val,id_0 => true
      | id_val,id_bs => true
      | id_val,id_and => true
      | id_val,id__minus_ => true
      | id_val,id_l => true
      | id_val,id__plus_ => true
      | id_val,id_wb => true
      | id_val,id_ge => true
      | id_val,id_false => true
      | id_val,id_min => true
      | id_val,id__sharp_ => true
      | id_val,id_size => true
      | id_val,id_if => true
      | id_val,id_not => true
      | id_val,id_n => true
      | id_val,id_1 => true
      | id_val,id_val => true
      | id_val,id_true => false
      | id_val,id_max => false
      | id_true,id_0 => true
      | id_true,id_bs => true
      | id_true,id_and => true
      | id_true,id__minus_ => true
      | id_true,id_l => true
      | id_true,id__plus_ => true
      | id_true,id_wb => true
      | id_true,id_ge => true
      | id_true,id_false => true
      | id_true,id_min => true
      | id_true,id__sharp_ => true
      | id_true,id_size => true
      | id_true,id_if => true
      | id_true,id_not => true
      | id_true,id_n => true
      | id_true,id_1 => true
      | id_true,id_val => true
      | id_true,id_true => true
      | id_true,id_max => false
      | id_max,id_0 => true
      | id_max,id_bs => true
      | id_max,id_and => true
      | id_max,id__minus_ => true
      | id_max,id_l => true
      | id_max,id__plus_ => true
      | id_max,id_wb => true
      | id_max,id_ge => true
      | id_max,id_false => true
      | id_max,id_min => true
      | id_max,id__sharp_ => true
      | id_max,id_size => true
      | id_max,id_if => true
      | id_max,id_not => true
      | id_max,id_n => true
      | id_max,id_1 => true
      | id_max,id_val => true
      | id_max,id_true => true
      | id_max,id_max => 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 := 
    (* 0(#) -> # *)
   | R_xml_0_rule_0 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id__sharp_ nil) 
     (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
      algebra.F.id__sharp_ nil)::nil))
    (* +(x_,#) -> x_ *)
   | R_xml_0_rule_1 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))
    (* +(#,x_) -> x_ *)
   | R_xml_0_rule_2 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
      algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil))
    (* +(0(x_),0(y_)) -> 0(+(x_,y_)) *)
   | R_xml_0_rule_3 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* +(0(x_),1(y_)) -> 1(+(x_,y_)) *)
   | R_xml_0_rule_4 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* +(1(x_),0(y_)) -> 1(+(x_,y_)) *)
   | R_xml_0_rule_5 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* +(1(x_),1(y_)) -> 0(+(+(x_,y_),1(#))) *)
   | R_xml_0_rule_6 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* +(x_,+(y_,z_)) -> +(+(x_,y_),z_) *)
   | R_xml_0_rule_7 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* -(x_,#) -> x_ *)
   | R_xml_0_rule_8 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))
    (* -(#,x_) -> # *)
   | R_xml_0_rule_9 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id__sharp_ nil) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
      algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil))
    (* -(0(x_),0(y_)) -> 0(-(x_,y_)) *)
   | R_xml_0_rule_10 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__minus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
      algebra.F.id_0 ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
      algebra.F.id_0 ((algebra.Alg.Var 2)::nil))::nil))
    (* -(0(x_),1(y_)) -> 1(-(-(x_,y_),1(#))) *)
   | R_xml_0_rule_11 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__minus_ ((algebra.Alg.Term 
                   algebra.F.id__minus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
      algebra.F.id_0 ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
      algebra.F.id_1 ((algebra.Alg.Var 2)::nil))::nil))
    (* -(1(x_),0(y_)) -> 1(-(x_,y_)) *)
   | R_xml_0_rule_12 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__minus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
      algebra.F.id_1 ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
      algebra.F.id_0 ((algebra.Alg.Var 2)::nil))::nil))
    (* -(1(x_),1(y_)) -> 0(-(x_,y_)) *)
   | R_xml_0_rule_13 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__minus_ ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
      algebra.F.id_1 ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
      algebra.F.id_1 ((algebra.Alg.Var 2)::nil))::nil))
    (* not(false) -> true *)
   | R_xml_0_rule_14 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_true nil) 
     (algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term 
      algebra.F.id_false nil)::nil))
    (* not(true) -> false *)
   | R_xml_0_rule_15 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_false nil) 
     (algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term algebra.F.id_true 
      nil)::nil))
    (* and(x_,true) -> x_ *)
   | R_xml_0_rule_16 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id_true nil)::nil))
    (* and(x_,false) -> false *)
   | R_xml_0_rule_17 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_false nil) 
     (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id_false nil)::nil))
    (* if(true,x_,y_) -> x_ *)
   | R_xml_0_rule_18 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Term algebra.F.id_true 
      nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))
    (* if(false,x_,y_) -> y_ *)
   | R_xml_0_rule_19 :
    R_xml_0_rules (algebra.Alg.Var 2) 
     (algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Term algebra.F.id_false 
      nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))
    (* ge(0(x_),0(y_)) -> ge(x_,y_) *)
   | R_xml_0_rule_20 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* ge(0(x_),1(y_)) -> not(ge(y_,x_)) *)
   | R_xml_0_rule_21 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term 
                   algebra.F.id_ge ((algebra.Alg.Var 2)::
                   (algebra.Alg.Var 1)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* ge(1(x_),0(y_)) -> ge(x_,y_) *)
   | R_xml_0_rule_22 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* ge(1(x_),1(y_)) -> ge(x_,y_) *)
   | R_xml_0_rule_23 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
                   (algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 2)::nil))::nil))
    (* ge(x_,#) -> true *)
   | R_xml_0_rule_24 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_true nil) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
      (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))
    (* ge(#,1(x_)) -> false *)
   | R_xml_0_rule_25 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_false nil) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
      algebra.F.id__sharp_ nil)::(algebra.Alg.Term algebra.F.id_1 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* ge(#,0(x_)) -> ge(#,x_) *)
   | R_xml_0_rule_26 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                   algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil)) 
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
      algebra.F.id__sharp_ nil)::(algebra.Alg.Term algebra.F.id_0 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* val(l(x_)) -> x_ *)
   | R_xml_0_rule_27 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_val ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* val(n(x_,y_,z_)) -> x_ *)
   | R_xml_0_rule_28 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_val ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* min(l(x_)) -> x_ *)
   | R_xml_0_rule_29 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_min ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* min(n(x_,y_,z_)) -> min(y_) *)
   | R_xml_0_rule_30 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_min 
                   ((algebra.Alg.Var 2)::nil)) 
     (algebra.Alg.Term algebra.F.id_min ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* max(l(x_)) -> x_ *)
   | R_xml_0_rule_31 :
    R_xml_0_rules (algebra.Alg.Var 1) 
     (algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
    (* max(n(x_,y_,z_)) -> max(z_) *)
   | R_xml_0_rule_32 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_max 
                   ((algebra.Alg.Var 3)::nil)) 
     (algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* bs(l(x_)) -> true *)
   | R_xml_0_rule_33 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_true nil) 
     (algebra.Alg.Term algebra.F.id_bs ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
   
    (* bs(n(x_,y_,z_)) -> and(and(ge(x_,max(y_)),ge(min(z_),x_)),and(bs(y_),bs(z_))) *)
   | R_xml_0_rule_34 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                   algebra.F.id_and ((algebra.Alg.Term algebra.F.id_ge 
                   ((algebra.Alg.Var 1)::(algebra.Alg.Term algebra.F.id_max 
                   ((algebra.Alg.Var 2)::nil))::nil))::(algebra.Alg.Term 
                   algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_min 
                   ((algebra.Alg.Var 3)::nil))::
                   (algebra.Alg.Var 1)::nil))::nil))::(algebra.Alg.Term 
                   algebra.F.id_and ((algebra.Alg.Term algebra.F.id_bs 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_bs ((algebra.Alg.Var 3)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_bs ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* size(l(x_)) -> 1(#) *)
   | R_xml_0_rule_35 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__sharp_ nil)::nil)) 
     (algebra.Alg.Term algebra.F.id_size ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
   
    (* size(n(x_,y_,z_)) -> +(+(size(x_),size(y_)),1(#)) *)
   | R_xml_0_rule_36 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_size 
                   ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term 
                   algebra.F.id_size ((algebra.Alg.Var 2)::nil))::nil))::
                   (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                   algebra.F.id__sharp_ nil)::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_size ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
    (* wb(l(x_)) -> true *)
   | R_xml_0_rule_37 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_true nil) 
     (algebra.Alg.Term algebra.F.id_wb ((algebra.Alg.Term algebra.F.id_l 
      ((algebra.Alg.Var 1)::nil))::nil))
   
    (* wb(n(x_,y_,z_)) -> and(if(ge(size(y_),size(z_)),ge(1(#),-(size(y_),size(z_))),
     ge(1(#),-(size(z_),size(y_)))),and(wb(y_),wb(z_))) *)
   | R_xml_0_rule_38 :
    R_xml_0_rules (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                   algebra.F.id_if ((algebra.Alg.Term algebra.F.id_ge 
                   ((algebra.Alg.Term algebra.F.id_size 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_size ((algebra.Alg.Var 3)::nil))::nil))::
                   (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::(algebra.Alg.Term algebra.F.id__minus_ 
                   ((algebra.Alg.Term algebra.F.id_size 
                   ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term 
                   algebra.F.id_size 
                   ((algebra.Alg.Var 3)::nil))::nil))::nil))::
                   (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::(algebra.Alg.Term algebra.F.id__minus_ 
                   ((algebra.Alg.Term algebra.F.id_size 
                   ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Term 
                   algebra.F.id_size 
                   ((algebra.Alg.Var 2)::nil))::nil))::nil))::nil))::
                   (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                   algebra.F.id_wb ((algebra.Alg.Var 2)::nil))::
                   (algebra.Alg.Term algebra.F.id_wb 
                   ((algebra.Alg.Var 3)::nil))::nil))::nil)) 
     (algebra.Alg.Term algebra.F.id_wb ((algebra.Alg.Term algebra.F.id_n 
      ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
      (algebra.Alg.Var 3)::nil))::nil))
 .
 
 
 Definition R_xml_0_rule_as_list_0  := 
   ((algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term algebra.F.id__sharp_ 
     nil)::nil)),(algebra.Alg.Term algebra.F.id__sharp_ nil))::nil.
 
 
 Definition R_xml_0_rule_as_list_1  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_0.
 
 
 Definition R_xml_0_rule_as_list_2  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Var 1))::R_xml_0_rule_as_list_1.
 
 
 Definition R_xml_0_rule_as_list_3  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term algebra.F.id__plus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_2.
 
 
 Definition R_xml_0_rule_as_list_4  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__plus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_3.
 
 
 Definition R_xml_0_rule_as_list_5  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__plus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_4.
 
 
 Definition R_xml_0_rule_as_list_6  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term algebra.F.id__plus_ 
     ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_5.
 
 
 Definition R_xml_0_rule_as_list_7  := 
   ((algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
     algebra.F.id__plus_ ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
     (algebra.Alg.Var 3)::nil)))::R_xml_0_rule_as_list_6.
 
 
 Definition R_xml_0_rule_as_list_8  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_7.
 
 
 Definition R_xml_0_rule_as_list_9  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil)),
    (algebra.Alg.Term algebra.F.id__sharp_ nil))::R_xml_0_rule_as_list_8.
 
 
 Definition R_xml_0_rule_as_list_10  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term algebra.F.id__minus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_9.
 
 
 Definition R_xml_0_rule_as_list_11  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__minus_ 
     ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::nil))::nil)))::
    R_xml_0_rule_as_list_10.
 
 
 Definition R_xml_0_rule_as_list_12  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__minus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_11.
 
 
 Definition R_xml_0_rule_as_list_13  := 
   ((algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term algebra.F.id__minus_ 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::nil)))::
    R_xml_0_rule_as_list_12.
 
 
 Definition R_xml_0_rule_as_list_14  := 
   ((algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term algebra.F.id_false 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_true nil))::
    R_xml_0_rule_as_list_13.
 
 
 Definition R_xml_0_rule_as_list_15  := 
   ((algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term algebra.F.id_true 
     nil)::nil)),(algebra.Alg.Term algebra.F.id_false nil))::
    R_xml_0_rule_as_list_14.
 
 
 Definition R_xml_0_rule_as_list_16  := 
   ((algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_true nil)::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_15.
 
 
 Definition R_xml_0_rule_as_list_17  := 
   ((algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_false nil)::nil)),
    (algebra.Alg.Term algebra.F.id_false nil))::R_xml_0_rule_as_list_16.
 
 
 Definition R_xml_0_rule_as_list_18  := 
   ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Term algebra.F.id_true 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)),
    (algebra.Alg.Var 1))::R_xml_0_rule_as_list_17.
 
 
 Definition R_xml_0_rule_as_list_19  := 
   ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Term algebra.F.id_false 
     nil)::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)),
    (algebra.Alg.Var 2))::R_xml_0_rule_as_list_18.
 
 
 Definition R_xml_0_rule_as_list_20  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_19.
 
 
 Definition R_xml_0_rule_as_list_21  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term algebra.F.id_ge 
     ((algebra.Alg.Var 2)::(algebra.Alg.Var 1)::nil))::nil)))::
    R_xml_0_rule_as_list_20.
 
 
 Definition R_xml_0_rule_as_list_22  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_21.
 
 
 Definition R_xml_0_rule_as_list_23  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 2)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
     (algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_22.
 
 
 Definition R_xml_0_rule_as_list_24  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id__sharp_ nil)::nil)),
    (algebra.Alg.Term algebra.F.id_true nil))::R_xml_0_rule_as_list_23.
 
 
 Definition R_xml_0_rule_as_list_25  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_false nil))::R_xml_0_rule_as_list_24.
 
 
 Definition R_xml_0_rule_as_list_26  := 
   ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::(algebra.Alg.Term algebra.F.id_0 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::(algebra.Alg.Var 1)::nil)))::
    R_xml_0_rule_as_list_25.
 
 
 Definition R_xml_0_rule_as_list_27  := 
   ((algebra.Alg.Term algebra.F.id_val ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_26.
 
 
 Definition R_xml_0_rule_as_list_28  := 
   ((algebra.Alg.Term algebra.F.id_val ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_27.
 
 
 Definition R_xml_0_rule_as_list_29  := 
   ((algebra.Alg.Term algebra.F.id_min ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_28.
 
 
 Definition R_xml_0_rule_as_list_30  := 
   ((algebra.Alg.Term algebra.F.id_min ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_min ((algebra.Alg.Var 2)::nil)))::
    R_xml_0_rule_as_list_29.
 
 
 Definition R_xml_0_rule_as_list_31  := 
   ((algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),(algebra.Alg.Var 1))::
    R_xml_0_rule_as_list_30.
 
 
 Definition R_xml_0_rule_as_list_32  := 
   ((algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Var 3)::nil)))::
    R_xml_0_rule_as_list_31.
 
 
 Definition R_xml_0_rule_as_list_33  := 
   ((algebra.Alg.Term algebra.F.id_bs ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_true nil))::R_xml_0_rule_as_list_32.
 
 
 Definition R_xml_0_rule_as_list_34  := 
   ((algebra.Alg.Term algebra.F.id_bs ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_and 
     ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Var 1)::
     (algebra.Alg.Term algebra.F.id_max ((algebra.Alg.Var 2)::nil))::nil))::
     (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_min 
     ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::nil))::nil))::
     (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_bs 
     ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_bs 
     ((algebra.Alg.Var 3)::nil))::nil))::nil)))::R_xml_0_rule_as_list_33.
 
 
 Definition R_xml_0_rule_as_list_35  := 
   ((algebra.Alg.Term algebra.F.id_size ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
     nil)::nil)))::R_xml_0_rule_as_list_34.
 
 
 Definition R_xml_0_rule_as_list_36  := 
   ((algebra.Alg.Term algebra.F.id_size ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
     algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 2)::nil))::nil))::(algebra.Alg.Term algebra.F.id_1 
     ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::nil)))::
    R_xml_0_rule_as_list_35.
 
 
 Definition R_xml_0_rule_as_list_37  := 
   ((algebra.Alg.Term algebra.F.id_wb ((algebra.Alg.Term algebra.F.id_l 
     ((algebra.Alg.Var 1)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_true nil))::R_xml_0_rule_as_list_36.
 
 
 Definition R_xml_0_rule_as_list_38  := 
   ((algebra.Alg.Term algebra.F.id_wb ((algebra.Alg.Term algebra.F.id_n 
     ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
     (algebra.Alg.Var 3)::nil))::nil)),
    (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term algebra.F.id_if 
     ((algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 3)::nil))::nil))::(algebra.Alg.Term algebra.F.id_ge 
     ((algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::nil))::(algebra.Alg.Term 
     algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 3)::nil))::nil))::nil))::(algebra.Alg.Term 
     algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
     algebra.F.id__sharp_ nil)::nil))::(algebra.Alg.Term 
     algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 3)::nil))::(algebra.Alg.Term algebra.F.id_size 
     ((algebra.Alg.Var 2)::nil))::nil))::nil))::nil))::(algebra.Alg.Term 
     algebra.F.id_and ((algebra.Alg.Term algebra.F.id_wb 
     ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_wb 
     ((algebra.Alg.Var 3)::nil))::nil))::nil)))::R_xml_0_rule_as_list_37.
 
 Definition R_xml_0_rule_as_list  := R_xml_0_rule_as_list_38.
 
 
 Lemma R_xml_0_rules_included :
  forall l r, R_xml_0_rules r l <-> In (l,r) R_xml_0_rule_as_list.
 Proof.
   intros l r.
   constructor.
   intros H.
   
   case H;clear H;
    (apply (more_list.mem_impl_in (@eq (algebra.Alg.term*algebra.Alg.term)));
     [tauto|idtac]);
    match goal with
      |  |- _ _ _ ?t ?l =>
       let u := fresh "u" in 
        (generalize (more_list.mem_bool_ok _ _ 
                      algebra.Alg_ext.eq_term_term_bool_ok t l);
          set (u:=more_list.mem_bool algebra.Alg_ext.eq_term_term_bool t l) in *;
          vm_compute in u|-;unfold u in *;clear u;intros H;refine H)
      end
    .
   intros H.
   vm_compute in H|-.
   rewrite  <- or_ext_generated.or25_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 39.
   injection H;intros ;subst;constructor 38.
   injection H;intros ;subst;constructor 37.
   injection H;intros ;subst;constructor 36.
   injection H;intros ;subst;constructor 35.
   injection H;intros ;subst;constructor 34.
   injection H;intros ;subst;constructor 33.
   injection H;intros ;subst;constructor 32.
   injection H;intros ;subst;constructor 31.
   injection H;intros ;subst;constructor 30.
   injection H;intros ;subst;constructor 29.
   injection H;intros ;subst;constructor 28.
   injection H;intros ;subst;constructor 27.
   injection H;intros ;subst;constructor 26.
   injection H;intros ;subst;constructor 25.
   injection H;intros ;subst;constructor 24.
   injection H;intros ;subst;constructor 23.
   injection H;intros ;subst;constructor 22.
   injection H;intros ;subst;constructor 21.
   injection H;intros ;subst;constructor 20.
   injection H;intros ;subst;constructor 19.
   injection H;intros ;subst;constructor 18.
   injection H;intros ;subst;constructor 17.
   injection H;intros ;subst;constructor 16.
   rewrite  <- or_ext_generated.or16_equiv in H|-.
   case H;clear H;intros H.
   injection H;intros ;subst;constructor 15.
   injection H;intros ;subst;constructor 14.
   injection H;intros ;subst;constructor 13.
   injection H;intros ;subst;constructor 12.
   injection H;intros ;subst;constructor 11.
   injection H;intros ;subst;constructor 10.
   injection H;intros ;subst;constructor 9.
   injection H;intros ;subst;constructor 8.
   injection H;intros ;subst;constructor 7.
   injection H;intros ;subst;constructor 6.
   injection H;intros ;subst;constructor 5.
   injection H;intros ;subst;constructor 4.
   injection H;intros ;subst;constructor 3.
   injection H;intros ;subst;constructor 2.
   injection H;intros ;subst;constructor 1.
   elim H.
 Qed.
 
 
 Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x).
 Proof.
   intros x t H.
   inversion H.
 Qed.
 
 
 Lemma R_xml_0_reg :
  forall s t, 
   (R_xml_0_rules s t) ->
    forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t).
 Proof.
   intros s t H.
   
   inversion H;intros x Hx;
    (apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]);
    apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx;
    vm_compute in Hx|-*;tauto.
 Qed.
 
 
 Inductive and_6 (x5 x6 x7 x8 x9 x10:Prop) :
  Prop := 
   | conj_6 : x5->x6->x7->x8->x9->x10->and_6 x5 x6 x7 x8 x9 x10
 .
 
 
 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_6 (forall x6, 
           t = (algebra.Alg.Term algebra.F.id_l (x6::nil)) ->
            exists x5,
              t' = (algebra.Alg.Term algebra.F.id_l (x5::nil))/\ 
              (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
                x5 x6)) 
     (t = (algebra.Alg.Term algebra.F.id_false nil) ->
      t' = (algebra.Alg.Term algebra.F.id_false nil)) 
     (t = (algebra.Alg.Term algebra.F.id__sharp_ nil) ->
      t' = (algebra.Alg.Term algebra.F.id__sharp_ nil)) 
     (forall x6 x8 x10, 
      t = (algebra.Alg.Term algebra.F.id_n (x6::x8::x10::nil)) ->
       exists x5,
         exists x7,
           exists x9,
             t' = (algebra.Alg.Term algebra.F.id_n (x5::x7::x9::nil))/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x5 x6)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x7 x8)/\ 
             (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
               x9 x10)) 
     (forall x6, 
      t = (algebra.Alg.Term algebra.F.id_1 (x6::nil)) ->
       exists x5,
         t' = (algebra.Alg.Term algebra.F.id_1 (x5::nil))/\ 
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x5 x6)) 
     
     (t = (algebra.Alg.Term algebra.F.id_true nil) ->
      t' = (algebra.Alg.Term algebra.F.id_true 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 x6 H;exists x6;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros H;intuition;constructor 1.
   intros x6 x8 x10 H;exists x6;exists x8;exists x10;intuition;constructor 1.
   intros x6 H;exists x6;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_l H_id_false H_id__sharp_ H_id_n H_id_1 H_id_true].
   constructor.
   
   clear H_id_false H_id__sharp_ H_id_n H_id_1 H_id_true;intros x6 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x6 |- _ =>
      destruct (H_id_l y (refl_equal _)) as [x5];intros ;intuition;exists x5;
       intuition;eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_l H_id__sharp_ H_id_n H_id_1 H_id_true;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_l H_id_false H_id_n H_id_1 H_id_true;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_l H_id_false H_id__sharp_ H_id_1 H_id_true;intros x6 x8 x10 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x6 |- _ =>
      destruct (H_id_n y x8 x10 (refl_equal _)) as [x5 [x7 [x9]]];intros ;
       intuition;exists x5;exists x7;exists x9;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x8 |- _ =>
      destruct (H_id_n x6 y x10 (refl_equal _)) as [x5 [x7 [x9]]];intros ;
       intuition;exists x5;exists x7;exists x9;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x10 |- _ =>
      destruct (H_id_n x6 x8 y (refl_equal _)) as [x5 [x7 [x9]]];intros ;
       intuition;exists x5;exists x7;exists x9;intuition;
       eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_l H_id_false H_id__sharp_ H_id_n H_id_true;intros x6 H;
    injection H;clear H;intros ;subst;
    repeat (
    match goal with
      | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
       inversion H;clear H;subst
      end
    ).
   
   match goal with
     | H:algebra.EQT.one_step _ ?y x6 |- _ =>
      destruct (H_id_1 y (refl_equal _)) as [x5];intros ;intuition;exists x5;
       intuition;eapply closure_extension.refl_trans_clos_R;eassumption
     end
   .
   
   clear H_id_l H_id_false H_id__sharp_ H_id_n H_id_1;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_l_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x6, 
     t = (algebra.Alg.Term algebra.F.id_l (x6::nil)) ->
      exists x5,
        t' = (algebra.Alg.Term algebra.F.id_l (x5::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x5 x6).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_false_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_false nil) ->
     t' = (algebra.Alg.Term algebra.F.id_false nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id__sharp__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__sharp_ nil) ->
     t' = (algebra.Alg.Term algebra.F.id__sharp_ nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_n_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x6 x8 x10, 
     t = (algebra.Alg.Term algebra.F.id_n (x6::x8::x10::nil)) ->
      exists x5,
        exists x7,
          exists x9,
            t' = (algebra.Alg.Term algebra.F.id_n (x5::x7::x9::nil))/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x5 x6)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x7 x8)/\ 
            (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
              x9 x10).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_1_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    forall x6, 
     t = (algebra.Alg.Term algebra.F.id_1 (x6::nil)) ->
      exists x5,
        t' = (algebra.Alg.Term algebra.F.id_1 (x5::nil))/\ 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x5 x6).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 Lemma id_true_is_R_xml_0_constructor :
  forall t t', 
   (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
    t = (algebra.Alg.Term algebra.F.id_true nil) ->
     t' = (algebra.Alg.Term algebra.F.id_true nil).
 Proof.
   intros t t' H.
   destruct (are_constuctors_of_R_xml_0 H).
   assumption.
 Qed.
 
 
 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_l (?x5::nil)) |- _ =>
     let x5 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_l_is_R_xml_0_constructor H (refl_equal _)) as 
           [x5 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_false nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_false_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id__sharp_ nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id__sharp__is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n (?x7::?x6::?x5::nil)) |- 
    _ =>
     let x7 := fresh "x" in 
      (let x6 := fresh "x" in 
        (let x5 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_n_is_R_xml_0_constructor H (refl_equal _))
                    as [x7 [x6 [x5 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      impossible_star_reduction_R_xml_0 ))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_1 (?x5::nil)) |- _ =>
     let x5 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_1_is_R_xml_0_constructor H (refl_equal _)) as 
           [x5 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              impossible_star_reduction_R_xml_0 ))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_true nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_true_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          impossible_star_reduction_R_xml_0 ))
    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_l (?x5::nil)) |- _ =>
     let x5 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_l_is_R_xml_0_constructor H (refl_equal _)) as 
           [x5 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_false nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_false_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id__sharp_ nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id__sharp__is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_n (?x7::?x6::?x5::nil)) |- 
    _ =>
     let x7 := fresh "x" in 
      (let x6 := fresh "x" in 
        (let x5 := fresh "x" in 
          (let Heq := fresh "Heq" in 
            (let Hred1 := fresh "Hred" in 
              (let Hred2 := fresh "Hred" in 
                (let Hred3 := fresh "Hred" in 
                  (destruct (id_n_is_R_xml_0_constructor H (refl_equal _))
                    as [x7 [x6 [x5 [Heq [Hred3 [Hred2 Hred1]]]]]];
                    (discriminate Heq)||
                    (injection Heq;intros ;subst;clear Heq;clear H;
                      try (simplify_star_reduction_R_xml_0 )))))))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_1 (?x5::nil)) |- _ =>
     let x5 := fresh "x" in 
      (let Heq := fresh "Heq" in 
        (let Hred1 := fresh "Hred" in 
          (destruct (id_1_is_R_xml_0_constructor H (refl_equal _)) as 
           [x5 [Heq Hred1]];
            (discriminate Heq)||
            (injection Heq;intros ;subst;clear Heq;clear H;
              try (simplify_star_reduction_R_xml_0 )))))
    | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) 
         _ (algebra.Alg.Term algebra.F.id_true nil) |- _ =>
     let Heq := fresh "Heq" in 
      (set (Heq:=id_true_is_R_xml_0_constructor H (refl_equal _)) in *;
        (discriminate Heq)||
        (clearbody Heq;try (subst);try (clear Heq);clear H;
          try (simplify_star_reduction_R_xml_0 )))
    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_0 : A ->A.
   
   Hypothesis P_id_bs : A ->A.
   
   Hypothesis P_id_and : A ->A ->A.
   
   Hypothesis P_id__minus_ : A ->A ->A.
   
   Hypothesis P_id_l : A ->A.
   
   Hypothesis P_id__plus_ : A ->A ->A.
   
   Hypothesis P_id_wb : A ->A.
   
   Hypothesis P_id_ge : A ->A ->A.
   
   Hypothesis P_id_false : A.
   
   Hypothesis P_id_min : A ->A.
   
   Hypothesis P_id__sharp_ : A.
   
   Hypothesis P_id_size : A ->A.
   
   Hypothesis P_id_if : A ->A ->A ->A.
   
   Hypothesis P_id_not : A ->A.
   
   Hypothesis P_id_n : A ->A ->A ->A.
   
   Hypothesis P_id_1 : A ->A.
   
   Hypothesis P_id_val : A ->A.
   
   Hypothesis P_id_true : A.
   
   Hypothesis P_id_max : A ->A.
   
   Hypothesis P_id_0_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   
   Hypothesis P_id_bs_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   
   Hypothesis P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   
   Hypothesis P_id__minus__monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   
   Hypothesis P_id_l_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   
   Hypothesis P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   
   Hypothesis P_id_wb_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   
   Hypothesis P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   
   Hypothesis P_id_min_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   
   Hypothesis P_id_size_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   
   Hypothesis P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (A0 <= x10)/\ (x10 <= x9) ->
      (A0 <= x8)/\ (x8 <= x7) ->
       (A0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   
   Hypothesis P_id_not_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   
   Hypothesis P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (A0 <= x10)/\ (x10 <= x9) ->
      (A0 <= x8)/\ (x8 <= x7) ->
       (A0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   
   Hypothesis P_id_1_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   
   Hypothesis P_id_val_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   
   Hypothesis P_id_max_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   
   Hypothesis P_id_0_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_0 x5.
   
   Hypothesis P_id_bs_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_bs x5.
   
   Hypothesis P_id_and_bounded :
    forall x6 x5, (A0 <= x5) ->(A0 <= x6) ->A0 <= P_id_and x6 x5.
   
   Hypothesis P_id__minus__bounded :
    forall x6 x5, (A0 <= x5) ->(A0 <= x6) ->A0 <= P_id__minus_ x6 x5.
   
   Hypothesis P_id_l_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_l x5.
   
   Hypothesis P_id__plus__bounded :
    forall x6 x5, (A0 <= x5) ->(A0 <= x6) ->A0 <= P_id__plus_ x6 x5.
   
   Hypothesis P_id_wb_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_wb x5.
   
   Hypothesis P_id_ge_bounded :
    forall x6 x5, (A0 <= x5) ->(A0 <= x6) ->A0 <= P_id_ge x6 x5.
   
   Hypothesis P_id_false_bounded : A0 <= P_id_false .
   
   Hypothesis P_id_min_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_min x5.
   
   Hypothesis P_id__sharp__bounded : A0 <= P_id__sharp_ .
   
   Hypothesis P_id_size_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_size x5.
   
   Hypothesis P_id_if_bounded :
    forall x6 x5 x7, 
     (A0 <= x5) ->(A0 <= x6) ->(A0 <= x7) ->A0 <= P_id_if x7 x6 x5.
   
   Hypothesis P_id_not_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_not x5.
   
   Hypothesis P_id_n_bounded :
    forall x6 x5 x7, 
     (A0 <= x5) ->(A0 <= x6) ->(A0 <= x7) ->A0 <= P_id_n x7 x6 x5.
   
   Hypothesis P_id_1_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_1 x5.
   
   Hypothesis P_id_val_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_val x5.
   
   Hypothesis P_id_true_bounded : A0 <= P_id_true .
   
   Hypothesis P_id_max_bounded : forall x5, (A0 <= x5) ->A0 <= P_id_max x5.
   
   Fixpoint measure t { struct t }  := 
     match t with
       | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) => P_id_0 (measure x5)
       | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) => P_id_bs (measure x5)
       | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
        P_id_and (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
        P_id__minus_ (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_l (x5::nil)) => P_id_l (measure x5)
       | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
        P_id__plus_ (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) => P_id_wb (measure x5)
       | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
        P_id_ge (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
       | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
        P_id_min (measure x5)
       | (algebra.Alg.Term algebra.F.id__sharp_ nil) => P_id__sharp_ 
       | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
        P_id_size (measure x5)
       | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
        P_id_if (measure x7) (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
        P_id_not (measure x5)
       | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
        P_id_n (measure x7) (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) => P_id_1 (measure x5)
       | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
        P_id_val (measure x5)
       | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
       | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
        P_id_max (measure x5)
       | _ => A0
       end.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0 => P_id_0
       | algebra.F.id_bs => P_id_bs
       | algebra.F.id_and => P_id_and
       | algebra.F.id__minus_ => P_id__minus_
       | algebra.F.id_l => P_id_l
       | algebra.F.id__plus_ => P_id__plus_
       | algebra.F.id_wb => P_id_wb
       | algebra.F.id_ge => P_id_ge
       | algebra.F.id_false => P_id_false
       | algebra.F.id_min => P_id_min
       | algebra.F.id__sharp_ => P_id__sharp_
       | algebra.F.id_size => P_id_size
       | algebra.F.id_if => P_id_if
       | algebra.F.id_not => P_id_not
       | algebra.F.id_n => P_id_n
       | algebra.F.id_1 => P_id_1
       | algebra.F.id_val => P_id_val
       | algebra.F.id_true => P_id_true
       | algebra.F.id_max => P_id_max
       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_0 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_bs =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id__minus_ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_l =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id__plus_ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_wb =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_ge =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_false => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_min =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id__sharp_ => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_size =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_if =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_not =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_n =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_val =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_true => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_max =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
      try (reflexivity );f_equal ;auto.
   Qed.
   
   Lemma measure_bounded : forall t, A0 <= measure t.
   Proof.
     intros t.
     rewrite same_measure in |-*.
     apply (InterpGen.measure_bounded Aop).
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_bs_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__minus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_l_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__plus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_wb_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ge_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_false_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_min_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__sharp__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_size_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_not_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_val_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_true_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_max_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_0_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_bs_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id__minus__monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_l_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id__plus__monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_wb_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_ge_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_min_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_size_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_not_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_val_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_max_monotonic;assumption.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_bs_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__minus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_l_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__plus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_wb_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ge_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_false_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_min_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__sharp__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_size_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_not_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_val_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_true_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_max_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     assumption.
   Qed.
   
   Hypothesis P_id_MIN : A ->A.
   
   Hypothesis P_id_0_hat_1 : A ->A.
   
   Hypothesis P_id_SIZE : A ->A.
   
   Hypothesis P_id_BS : A ->A.
   
   Hypothesis P_id_GE : A ->A ->A.
   
   Hypothesis P_id_IF : A ->A ->A ->A.
   
   Hypothesis P_id_MAX : A ->A.
   
   Hypothesis P_id__minus__hat_1 : A ->A ->A.
   
   Hypothesis P_id_WB : A ->A.
   
   Hypothesis P_id__plus__hat_1 : A ->A ->A.
   
   Hypothesis P_id_AND : A ->A ->A.
   
   Hypothesis P_id_NOT : A ->A.
   
   Hypothesis P_id_Marked_val : A ->A.
   
   Hypothesis P_id_MIN_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   
   Hypothesis P_id_0_hat_1_monotonic :
    forall x6 x5, 
     (A0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   
   Hypothesis P_id_SIZE_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   
   Hypothesis P_id_BS_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   
   Hypothesis P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   
   Hypothesis P_id_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (A0 <= x10)/\ (x10 <= x9) ->
      (A0 <= x8)/\ (x8 <= x7) ->
       (A0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   
   Hypothesis P_id_MAX_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   
   Hypothesis P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   
   Hypothesis P_id_WB_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   
   Hypothesis P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   
   Hypothesis P_id_AND_monotonic :
    forall x8 x6 x5 x7, 
     (A0 <= x8)/\ (x8 <= x7) ->
      (A0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   
   Hypothesis P_id_NOT_monotonic :
    forall x6 x5, (A0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   
   Hypothesis P_id_Marked_val_monotonic :
    forall x6 x5, 
     (A0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   
   Definition marked_measure t := 
     match t with
       | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
        P_id_MIN (measure x5)
       | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
        P_id_0_hat_1 (measure x5)
       | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
        P_id_SIZE (measure x5)
       | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) => P_id_BS (measure x5)
       | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
        P_id_GE (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
        P_id_IF (measure x7) (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
        P_id_MAX (measure x5)
       | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
        P_id__minus__hat_1 (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) => P_id_WB (measure x5)
       | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
        P_id__plus__hat_1 (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
        P_id_AND (measure x6) (measure x5)
       | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
        P_id_NOT (measure x5)
       | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
        P_id_Marked_val (measure x5)
       | _ => measure t
       end.
   
   Definition  Marked_pols :
    forall f, 
     (algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) ->
      InterpGen.Pol_type A (InterpGen.get_arity f).
   Proof.
     intros f H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_MAX x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_Marked_val x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_NOT x5).
     
     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 x7 x6 x5;apply (P_id_IF x7 x6 x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_SIZE x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_MIN x5).
     
     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 x6 x5;apply (P_id_GE x6 x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_WB x5).
     
     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 x6 x5;apply (P_id__plus__hat_1 x6 x5).
     
     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 x6 x5;apply (P_id__minus__hat_1 x6 x5).
     
     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 x6 x5;apply (P_id_AND x6 x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_BS x5).
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros x5;apply (P_id_0_hat_1 x5).
     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_0 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_bs =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_and =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id__minus_ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_l =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id__plus_ =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_wb =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_ge =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::_ => _
                 end
              | algebra.F.id_false => match l with
                                        | nil => _
                                        | _::_ => _
                                        end
              | algebra.F.id_min =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id__sharp_ => match l with
                                          | nil => _
                                          | _::_ => _
                                          end
              | algebra.F.id_size =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_if =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_not =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_n =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::nil => _
                 | _::_::_::nil => _
                 | _::_::_::_::_ => _
                 end
              | algebra.F.id_1 =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_val =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              | algebra.F.id_true => match l with
                                       | nil => _
                                       | _::_ => _
                                       end
              | algebra.F.id_max =>
               match l with
                 | nil => _
                 | _::nil => _
                 | _::_::_ => _
                 end
              end.
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     vm_compute in |-*;reflexivity .
     
     lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
      apply same_measure.
     vm_compute in |-*;reflexivity .
   Qed.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_bs_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_and_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id__minus__monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_l_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id__plus__monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_wb_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_ge_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_min_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_size_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_not_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_n_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_1_monotonic;assumption.
     vm_compute in |-*;intros ;apply P_id_val_monotonic;assumption.
     vm_compute in |-*;apply (Aop.(le_refl)).
     vm_compute in |-*;intros ;apply P_id_max_monotonic;assumption.
     clear f.
     intros f.
     case f.
     vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_bs_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_and_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__minus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_l_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__plus__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_wb_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_ge_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_false_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_min_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id__sharp__bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_size_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_if_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_not_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_n_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_1_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_val_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_true_bounded;assumption.
     vm_compute in |-*;intros ;apply P_id_max_bounded;assumption.
     intros .
     do 2 (rewrite  <- same_measure in |-*).
     apply rules_monotonic;assumption.
     clear f.
     intros f.
     clear H.
     intros H.
     generalize H.
     
     apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
     apply (Symb_more_list.change_in algebra.F.symb_order) in H .
     
     set (u := (Symb_more_list.qs algebra.F.symb_order
           (Symb_more_list.XSet.remove_red
              (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
     vm_compute in u .
     unfold u in * .
     clear u .
     unfold more_list.mem_bool in H .
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_MAX_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_Marked_val_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_NOT_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_IF_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_SIZE_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_MIN_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_GE_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_WB_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id__plus__hat_1_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id__minus__hat_1_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_AND_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_BS_monotonic;assumption.
     
     match type of H with 
       | orb ?a ?b = true => 
         assert (H':{a = true}+{b = true});[
           revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
             clear H;destruct H' as [H|H]]
     end .
     
     match type of H with 
       | _ _ ?t = true =>
         generalize (algebra.F.symb_eq_bool_ok f t);
           unfold algebra.Alg.eq_symb_bool in H;
           rewrite H;clear H;intros ;subst
     end .
     vm_compute in |-*;intros ;apply P_id_0_hat_1_monotonic;assumption.
     discriminate H.
     assumption.
   Qed.
   
   End S.
End Interp.

Module InterpZ.
 Section S.
   Open Scope Z_scope.
   
   Hypothesis min_value : Z.
   
   Import ring_extention.
   
   Notation Local "'Alt'" := (Zwf.Zwf min_value).
   
   Notation Local "'Ale'" := Zle.
   
   Notation Local "'Aeq'" := (@eq Z).
   
   Notation Local "a <= b" := (Ale a b).
   
   Notation Local "a < b" := (Alt a b).
   
   Hypothesis P_id_0 : Z ->Z.
   
   Hypothesis P_id_bs : Z ->Z.
   
   Hypothesis P_id_and : Z ->Z ->Z.
   
   Hypothesis P_id__minus_ : Z ->Z ->Z.
   
   Hypothesis P_id_l : Z ->Z.
   
   Hypothesis P_id__plus_ : Z ->Z ->Z.
   
   Hypothesis P_id_wb : Z ->Z.
   
   Hypothesis P_id_ge : Z ->Z ->Z.
   
   Hypothesis P_id_false : Z.
   
   Hypothesis P_id_min : Z ->Z.
   
   Hypothesis P_id__sharp_ : Z.
   
   Hypothesis P_id_size : Z ->Z.
   
   Hypothesis P_id_if : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_not : Z ->Z.
   
   Hypothesis P_id_n : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_1 : Z ->Z.
   
   Hypothesis P_id_val : Z ->Z.
   
   Hypothesis P_id_true : Z.
   
   Hypothesis P_id_max : Z ->Z.
   
   Hypothesis P_id_0_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   
   Hypothesis P_id_bs_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   
   Hypothesis P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   
   Hypothesis P_id__minus__monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->
       P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   
   Hypothesis P_id_l_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   
   Hypothesis P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   
   Hypothesis P_id_wb_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   
   Hypothesis P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   
   Hypothesis P_id_min_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   
   Hypothesis P_id_size_monotonic :
    forall x6 x5, 
     (min_value <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   
   Hypothesis P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (min_value <= x10)/\ (x10 <= x9) ->
      (min_value <= x8)/\ (x8 <= x7) ->
       (min_value <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   
   Hypothesis P_id_not_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   
   Hypothesis P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (min_value <= x10)/\ (x10 <= x9) ->
      (min_value <= x8)/\ (x8 <= x7) ->
       (min_value <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   
   Hypothesis P_id_1_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   
   Hypothesis P_id_val_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   
   Hypothesis P_id_max_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   
   Hypothesis P_id_0_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_0 x5.
   
   Hypothesis P_id_bs_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_bs x5.
   
   Hypothesis P_id_and_bounded :
    forall x6 x5, 
     (min_value <= x5) ->(min_value <= x6) ->min_value <= P_id_and x6 x5.
   
   Hypothesis P_id__minus__bounded :
    forall x6 x5, 
     (min_value <= x5) ->(min_value <= x6) ->min_value <= P_id__minus_ x6 x5.
   
   Hypothesis P_id_l_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_l x5.
   
   Hypothesis P_id__plus__bounded :
    forall x6 x5, 
     (min_value <= x5) ->(min_value <= x6) ->min_value <= P_id__plus_ x6 x5.
   
   Hypothesis P_id_wb_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_wb x5.
   
   Hypothesis P_id_ge_bounded :
    forall x6 x5, 
     (min_value <= x5) ->(min_value <= x6) ->min_value <= P_id_ge x6 x5.
   
   Hypothesis P_id_false_bounded : min_value <= P_id_false .
   
   Hypothesis P_id_min_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_min x5.
   
   Hypothesis P_id__sharp__bounded : min_value <= P_id__sharp_ .
   
   Hypothesis P_id_size_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_size x5.
   
   Hypothesis P_id_if_bounded :
    forall x6 x5 x7, 
     (min_value <= x5) ->
      (min_value <= x6) ->(min_value <= x7) ->min_value <= P_id_if x7 x6 x5.
   
   Hypothesis P_id_not_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_not x5.
   
   Hypothesis P_id_n_bounded :
    forall x6 x5 x7, 
     (min_value <= x5) ->
      (min_value <= x6) ->(min_value <= x7) ->min_value <= P_id_n x7 x6 x5.
   
   Hypothesis P_id_1_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_1 x5.
   
   Hypothesis P_id_val_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_val x5.
   
   Hypothesis P_id_true_bounded : min_value <= P_id_true .
   
   Hypothesis P_id_max_bounded :
    forall x5, (min_value <= x5) ->min_value <= P_id_max x5.
   
   Definition measure  := 
     Interp.measure min_value P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Hypothesis P_id_MIN : Z ->Z.
   
   Hypothesis P_id_0_hat_1 : Z ->Z.
   
   Hypothesis P_id_SIZE : Z ->Z.
   
   Hypothesis P_id_BS : Z ->Z.
   
   Hypothesis P_id_GE : Z ->Z ->Z.
   
   Hypothesis P_id_IF : Z ->Z ->Z ->Z.
   
   Hypothesis P_id_MAX : Z ->Z.
   
   Hypothesis P_id__minus__hat_1 : Z ->Z ->Z.
   
   Hypothesis P_id_WB : Z ->Z.
   
   Hypothesis P_id__plus__hat_1 : Z ->Z ->Z.
   
   Hypothesis P_id_AND : Z ->Z ->Z.
   
   Hypothesis P_id_NOT : Z ->Z.
   
   Hypothesis P_id_Marked_val : Z ->Z.
   
   Hypothesis P_id_MIN_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   
   Hypothesis P_id_0_hat_1_monotonic :
    forall x6 x5, 
     (min_value <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   
   Hypothesis P_id_SIZE_monotonic :
    forall x6 x5, 
     (min_value <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   
   Hypothesis P_id_BS_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   
   Hypothesis P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   
   Hypothesis P_id_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (min_value <= x10)/\ (x10 <= x9) ->
      (min_value <= x8)/\ (x8 <= x7) ->
       (min_value <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   
   Hypothesis P_id_MAX_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   
   Hypothesis P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   
   Hypothesis P_id_WB_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   
   Hypothesis P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   
   Hypothesis P_id_AND_monotonic :
    forall x8 x6 x5 x7, 
     (min_value <= x8)/\ (x8 <= x7) ->
      (min_value <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   
   Hypothesis P_id_NOT_monotonic :
    forall x6 x5, (min_value <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   
   Hypothesis P_id_Marked_val_monotonic :
    forall x6 x5, 
     (min_value <= x6)/\ (x6 <= x5) ->
      P_id_Marked_val x6 <= P_id_Marked_val x5.
   
   Definition marked_measure  := 
     Interp.marked_measure min_value P_id_0 P_id_bs P_id_and P_id__minus_ 
      P_id_l P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ 
      P_id_size P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max 
      P_id_MIN P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_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 := 
    (* <+(0(x_),0(y_)),0(+(x_,y_))> *)
   | DP_R_xml_0_0 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__plus_ (x1::x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(0(x_),0(y_)),+(x_,y_)> *)
   | DP_R_xml_0_1 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(0(x_),1(y_)),+(x_,y_)> *)
   | DP_R_xml_0_2 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(1(x_),0(y_)),+(x_,y_)> *)
   | DP_R_xml_0_3 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(1(x_),1(y_)),0(+(+(x_,y_),1(#)))> *)
   | DP_R_xml_0_4 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ (x1::x2::nil))::(algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(1(x_),1(y_)),+(+(x_,y_),1(#))> *)
   | DP_R_xml_0_5 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ (x1::x2::nil))::(algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(1(x_),1(y_)),+(x_,y_)> *)
   | DP_R_xml_0_6 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(x_,+(y_,z_)),+(+(x_,y_),z_)> *)
   | DP_R_xml_0_7 :
    forall x2 x6 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                   algebra.F.id__plus_ (x1::x2::nil))::x3::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <+(x_,+(y_,z_)),+(x_,y_)> *)
   | DP_R_xml_0_8 :
    forall x2 x6 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                x1 x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    (* <-(0(x_),0(y_)),0(-(x_,y_))> *)
   | DP_R_xml_0_9 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__minus_ (x1::x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(0(x_),0(y_)),-(x_,y_)> *)
   | DP_R_xml_0_10 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(0(x_),1(y_)),-(-(x_,y_),1(#))> *)
   | DP_R_xml_0_11 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
                   algebra.F.id__minus_ (x1::x2::nil))::(algebra.Alg.Term 
                   algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                   nil)::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(0(x_),1(y_)),-(x_,y_)> *)
   | DP_R_xml_0_12 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(1(x_),0(y_)),-(x_,y_)> *)
   | DP_R_xml_0_13 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(1(x_),1(y_)),0(-(x_,y_))> *)
   | DP_R_xml_0_14 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_0 ((algebra.Alg.Term 
                   algebra.F.id__minus_ (x1::x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <-(1(x_),1(y_)),-(x_,y_)> *)
   | DP_R_xml_0_15 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
    (* <ge(0(x_),0(y_)),ge(x_,y_)> *)
   | DP_R_xml_0_16 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <ge(0(x_),1(y_)),not(ge(y_,x_))> *)
   | DP_R_xml_0_17 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_not ((algebra.Alg.Term 
                   algebra.F.id_ge (x2::x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <ge(0(x_),1(y_)),ge(y_,x_)> *)
   | DP_R_xml_0_18 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge (x2::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <ge(1(x_),0(y_)),ge(x_,y_)> *)
   | DP_R_xml_0_19 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <ge(1(x_),1(y_)),ge(x_,y_)> *)
   | DP_R_xml_0_20 :
    forall x2 x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x2::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <ge(#,0(x_)),ge(#,x_)> *)
   | DP_R_xml_0_21 :
    forall x6 x1 x5, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                (algebra.Alg.Term algebra.F.id__sharp_ nil) 
       x6) ->
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x5) ->
       DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                   algebra.F.id__sharp_ nil)::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
    (* <min(n(x_,y_,z_)),min(y_)> *)
   | DP_R_xml_0_22 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_min (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_min (x5::nil))
    (* <max(n(x_,y_,z_)),max(z_)> *)
   | DP_R_xml_0_23 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_max (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_max (x5::nil))
   
    (* <bs(n(x_,y_,z_)),and(and(ge(x_,max(y_)),ge(min(z_),x_)),and(bs(y_),bs(z_)))> *)
   | DP_R_xml_0_24 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_and ((algebra.Alg.Term algebra.F.id_ge (x1::
                  (algebra.Alg.Term algebra.F.id_max (x2::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_min (x3::nil))::x1::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_bs (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_bs (x3::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
   
    (* <bs(n(x_,y_,z_)),and(ge(x_,max(y_)),ge(min(z_),x_))> *)
   | DP_R_xml_0_25 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_ge (x1::(algebra.Alg.Term algebra.F.id_max 
                  (x2::nil))::nil))::(algebra.Alg.Term algebra.F.id_ge 
                  ((algebra.Alg.Term algebra.F.id_min (x3::nil))::
                  x1::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),ge(x_,max(y_))> *)
   | DP_R_xml_0_26 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge (x1::(algebra.Alg.Term 
                  algebra.F.id_max (x2::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),max(y_)> *)
   | DP_R_xml_0_27 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_max (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),ge(min(z_),x_)> *)
   | DP_R_xml_0_28 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_min (x3::nil))::x1::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),min(z_)> *)
   | DP_R_xml_0_29 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_min (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),and(bs(y_),bs(z_))> *)
   | DP_R_xml_0_30 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_bs (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_bs (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),bs(y_)> *)
   | DP_R_xml_0_31 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_bs (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <bs(n(x_,y_,z_)),bs(z_)> *)
   | DP_R_xml_0_32 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_bs (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_bs (x5::nil))
    (* <size(n(x_,y_,z_)),+(+(size(x_),size(y_)),1(#))> *)
   | DP_R_xml_0_33 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                  algebra.F.id__plus_ ((algebra.Alg.Term algebra.F.id_size 
                  (x1::nil))::(algebra.Alg.Term algebra.F.id_size 
                  (x2::nil))::nil))::(algebra.Alg.Term algebra.F.id_1 
                  ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_size (x5::nil))
    (* <size(n(x_,y_,z_)),+(size(x_),size(y_))> *)
   | DP_R_xml_0_34 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id__plus_ ((algebra.Alg.Term 
                  algebra.F.id_size (x1::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_size (x5::nil))
    (* <size(n(x_,y_,z_)),size(x_)> *)
   | DP_R_xml_0_35 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_size (x1::nil)) 
       (algebra.Alg.Term algebra.F.id_size (x5::nil))
    (* <size(n(x_,y_,z_)),size(y_)> *)
   | DP_R_xml_0_36 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_size (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_size (x5::nil))
   
    (* <wb(n(x_,y_,z_)),and(if(ge(size(y_),size(z_)),ge(1(#),-(size(y_),size(z_))),
     ge(1(#),-(size(z_),size(y_)))),and(wb(y_),wb(z_)))> *)
   | DP_R_xml_0_37 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_if ((algebra.Alg.Term algebra.F.id_ge 
                  ((algebra.Alg.Term algebra.F.id_size (x2::nil))::
                  (algebra.Alg.Term algebra.F.id_size (x3::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                  nil)::nil))::(algebra.Alg.Term algebra.F.id__minus_ 
                  ((algebra.Alg.Term algebra.F.id_size (x2::nil))::
                  (algebra.Alg.Term algebra.F.id_size 
                  (x3::nil))::nil))::nil))::(algebra.Alg.Term 
                  algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
                  ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::
                  (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
                  algebra.F.id_size (x3::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::nil))::nil))::nil))::
                  (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_wb (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_wb (x3::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
   
    (* <wb(n(x_,y_,z_)),if(ge(size(y_),size(z_)),ge(1(#),-(size(y_),size(z_))),
 ge(1(#),-(size(z_),size(y_))))> *)
   | DP_R_xml_0_38 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Term 
                  algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_size 
                  (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                  (x3::nil))::nil))::(algebra.Alg.Term algebra.F.id_ge 
                  ((algebra.Alg.Term algebra.F.id_1 ((algebra.Alg.Term 
                  algebra.F.id__sharp_ nil)::nil))::(algebra.Alg.Term 
                  algebra.F.id__minus_ ((algebra.Alg.Term algebra.F.id_size 
                  (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                  (x3::nil))::nil))::nil))::(algebra.Alg.Term 
                  algebra.F.id_ge ((algebra.Alg.Term algebra.F.id_1 
                  ((algebra.Alg.Term algebra.F.id__sharp_ nil)::nil))::
                  (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
                  algebra.F.id_size (x3::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),ge(size(y_),size(z_))> *)
   | DP_R_xml_0_39 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),size(y_)> *)
   | DP_R_xml_0_40 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_size (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),size(z_)> *)
   | DP_R_xml_0_41 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_size (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),ge(1(#),-(size(y_),size(z_)))> *)
   | DP_R_xml_0_42 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                  nil)::nil))::(algebra.Alg.Term algebra.F.id__minus_ 
                  ((algebra.Alg.Term algebra.F.id_size (x2::nil))::
                  (algebra.Alg.Term algebra.F.id_size 
                  (x3::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),-(size(y_),size(z_))> *)
   | DP_R_xml_0_43 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),ge(1(#),-(size(z_),size(y_)))> *)
   | DP_R_xml_0_44 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_ge ((algebra.Alg.Term 
                  algebra.F.id_1 ((algebra.Alg.Term algebra.F.id__sharp_ 
                  nil)::nil))::(algebra.Alg.Term algebra.F.id__minus_ 
                  ((algebra.Alg.Term algebra.F.id_size (x3::nil))::
                  (algebra.Alg.Term algebra.F.id_size 
                  (x2::nil))::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),-(size(z_),size(y_))> *)
   | DP_R_xml_0_45 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id__minus_ ((algebra.Alg.Term 
                  algebra.F.id_size (x3::nil))::(algebra.Alg.Term 
                  algebra.F.id_size (x2::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),and(wb(y_),wb(z_))> *)
   | DP_R_xml_0_46 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_and ((algebra.Alg.Term 
                  algebra.F.id_wb (x2::nil))::(algebra.Alg.Term 
                  algebra.F.id_wb (x3::nil))::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),wb(y_)> *)
   | DP_R_xml_0_47 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_wb (x2::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::nil))
    (* <wb(n(x_,y_,z_)),wb(z_)> *)
   | DP_R_xml_0_48 :
    forall x2 x1 x5 x3, 
     (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                
       (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
      DP_R_xml_0 (algebra.Alg.Term algebra.F.id_wb (x3::nil)) 
       (algebra.Alg.Term algebra.F.id_wb (x5::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 := 
     (* <wb(n(x_,y_,z_)),and(wb(y_),wb(z_))> *)
    | DP_R_xml_0_non_scc_1_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_wb (x2::nil))::
                             (algebra.Alg.Term algebra.F.id_wb 
                             (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::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 := 
     (* <wb(n(x_,y_,z_)),if(ge(size(y_),size(z_)),ge(1(#),-(size(y_),size(z_))),
 ge(1(#),-(size(z_),size(y_))))> *)
    | DP_R_xml_0_non_scc_2_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_if 
                             ((algebra.Alg.Term algebra.F.id_ge 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_1 ((algebra.Alg.Term 
                             algebra.F.id__sharp_ nil)::nil))::
                             (algebra.Alg.Term algebra.F.id__minus_ 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_1 ((algebra.Alg.Term 
                             algebra.F.id__sharp_ nil)::nil))::
                             (algebra.Alg.Term algebra.F.id__minus_ 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::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 := 
     (* <wb(n(x_,y_,z_)),and(if(ge(size(y_),size(z_)),ge(1(#),-(size(y_),size(z_))),
     ge(1(#),-(size(z_),size(y_)))),and(wb(y_),wb(z_)))> *)
    | DP_R_xml_0_non_scc_3_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_if 
                             ((algebra.Alg.Term algebra.F.id_ge 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_1 ((algebra.Alg.Term 
                             algebra.F.id__sharp_ nil)::nil))::
                             (algebra.Alg.Term algebra.F.id__minus_ 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_1 ((algebra.Alg.Term 
                             algebra.F.id__sharp_ nil)::nil))::
                             (algebra.Alg.Term algebra.F.id__minus_ 
                             ((algebra.Alg.Term algebra.F.id_size 
                             (x3::nil))::(algebra.Alg.Term algebra.F.id_size 
                             (x2::nil))::nil))::nil))::nil))::
                             (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_wb (x2::nil))::
                             (algebra.Alg.Term algebra.F.id_wb 
                             (x3::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_3 :
   forall x y, 
    (DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_4  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),and(bs(y_),bs(z_))> *)
    | DP_R_xml_0_non_scc_4_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_bs (x2::nil))::
                             (algebra.Alg.Term algebra.F.id_bs 
                             (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_4 :
   forall x y, 
    (DP_R_xml_0_non_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
      (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_5  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),and(ge(x_,max(y_)),ge(min(z_),x_))> *)
    | DP_R_xml_0_non_scc_5_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_ge (x1::
                             (algebra.Alg.Term algebra.F.id_max 
                             (x2::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_min (x3::nil))::x1::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::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;
      (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 := 
     (* <bs(n(x_,y_,z_)),and(and(ge(x_,max(y_)),ge(min(z_),x_)),and(bs(y_),bs(z_)))> *)
    | DP_R_xml_0_non_scc_6_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_ge (x1::
                             (algebra.Alg.Term algebra.F.id_max 
                             (x2::nil))::nil))::(algebra.Alg.Term 
                             algebra.F.id_ge ((algebra.Alg.Term 
                             algebra.F.id_min (x3::nil))::x1::nil))::nil))::
                             (algebra.Alg.Term algebra.F.id_and 
                             ((algebra.Alg.Term algebra.F.id_bs (x2::nil))::
                             (algebra.Alg.Term algebra.F.id_bs 
                             (x3::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::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;
      (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_7  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <max(n(x_,y_,z_)),max(z_)> *)
    | DP_R_xml_0_scc_7_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_7 (algebra.Alg.Term algebra.F.id_max (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_max (x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_7.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 0.
   
   Definition P_id_bs (x5:Z) := 0.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id_l (x5:Z) := 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
   
   Definition P_id_wb (x5:Z) := 0.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 1* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 0.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 1 + 1* x5 + 1* x6 + 3* x7.
   
   Definition P_id_1 (x5:Z) := 0.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 3* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 2* x5.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_7.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_7.
  
  Definition wf_DP_R_xml_0_scc_7  := WF_DP_R_xml_0_scc_7.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_7 :
   forall x y, (DP_R_xml_0_scc_7 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_7).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    apply wf_DP_R_xml_0_scc_7.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_8  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),max(y_)> *)
    | DP_R_xml_0_non_scc_8_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_max (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::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_7;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_9  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <min(n(x_,y_,z_)),min(y_)> *)
    | DP_R_xml_0_scc_9_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_9 (algebra.Alg.Term algebra.F.id_min (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_min (x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_9.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_bs (x5:Z) := 2 + 2* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 1 + 1* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
   
   Definition P_id_l (x5:Z) := 3 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 2 + 2* x5.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 1* x6.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 2* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 1* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 1* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 2* x5 + 2* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 1 + 3* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 3* x5.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_9.
   Proof.
     intros x.
     
     apply well_founded_ind with
       (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
     apply Inverse_Image.wf_inverse_image with  (B:=Z).
     apply Zwf.Zwf_well_founded.
     clear x.
     intros x IHx.
     
     repeat (
     constructor;inversion 1;subst;
      full_prove_ineq algebra.Alg.Term 
      ltac:(algebra.Alg_ext.find_replacement ) 
      algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure 
      marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) 
      ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) 
      ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) 
      ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
                      try (constructor))
       IHx
     ).
   Qed.
  End WF_DP_R_xml_0_scc_9.
  
  Definition wf_DP_R_xml_0_scc_9  := WF_DP_R_xml_0_scc_9.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_9 :
   forall x y, (DP_R_xml_0_scc_9 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x.
    pattern x.
    apply (@Acc_ind _ DP_R_xml_0_scc_9).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
    apply wf_DP_R_xml_0_scc_9.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_10  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),min(z_)> *)
    | DP_R_xml_0_non_scc_10_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_min (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::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_9;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
       (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_11  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <ge(0(x_),1(y_)),not(ge(y_,x_))> *)
    | DP_R_xml_0_non_scc_11_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_not 
                               ((algebra.Alg.Term algebra.F.id_ge (x2::
                               x1::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_11 :
   forall x y, 
    (DP_R_xml_0_non_scc_11 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (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_12  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <ge(#,0(x_)),ge(#,x_)> *)
    | DP_R_xml_0_scc_12_0 :
     forall x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id__sharp_ nil) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x1::nil)) x5) ->
        DP_R_xml_0_scc_12 (algebra.Alg.Term algebra.F.id_ge 
                           ((algebra.Alg.Term algebra.F.id__sharp_ nil)::
                           x1::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_12.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_bs (x5:Z) := 1* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5 + 1* x6.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
   
   Definition P_id_l (x5:Z) := 1 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
   
   Definition P_id_wb (x5:Z) := 0.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 2* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 2* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 2 + 3* x5 + 2* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 3* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 2* x6.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_12.
   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_12.
  
  Definition wf_DP_R_xml_0_scc_12  := WF_DP_R_xml_0_scc_12.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_12 :
   forall x y, (DP_R_xml_0_scc_12 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_12).
    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_11;
         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_12.
  Qed.
  
  
  Inductive DP_R_xml_0_scc_13  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <ge(0(x_),1(y_)),ge(y_,x_)> *)
    | DP_R_xml_0_scc_13_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_ge (x2::x1::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
     (* <ge(0(x_),0(y_)),ge(x_,y_)> *)
    | DP_R_xml_0_scc_13_1 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
     (* <ge(1(x_),0(y_)),ge(x_,y_)> *)
    | DP_R_xml_0_scc_13_2 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
     (* <ge(1(x_),1(y_)),ge(x_,y_)> *)
    | DP_R_xml_0_scc_13_3 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_ge (x1::x2::nil)) 
         (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_13.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_bs (x5:Z) := 3.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
   
   Definition P_id_l (x5:Z) := 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
   
   Definition P_id_wb (x5:Z) := 1.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 3* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 1 + 2* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 1* x5 + 2* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 1* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 3* x5 + 3* x6.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_13.
   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_13.
  
  Definition wf_DP_R_xml_0_scc_13  := WF_DP_R_xml_0_scc_13.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_13 :
   forall x y, (DP_R_xml_0_scc_13 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_13).
    intros x' _ Hrec y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply Hrec;econstructor eassumption)||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          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_13.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_14  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),ge(1(#),-(size(z_),size(y_)))> *)
    | DP_R_xml_0_non_scc_14_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_14 (algebra.Alg.Term algebra.F.id_ge 
                              ((algebra.Alg.Term algebra.F.id_1 
                              ((algebra.Alg.Term algebra.F.id__sharp_ 
                              nil)::nil))::(algebra.Alg.Term 
                              algebra.F.id__minus_ ((algebra.Alg.Term 
                              algebra.F.id_size (x3::nil))::
                              (algebra.Alg.Term algebra.F.id_size 
                              (x2::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_14 :
   forall x y, 
    (DP_R_xml_0_non_scc_14 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_13;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          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_15  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),ge(1(#),-(size(y_),size(z_)))> *)
    | DP_R_xml_0_non_scc_15_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_ge 
                              ((algebra.Alg.Term algebra.F.id_1 
                              ((algebra.Alg.Term algebra.F.id__sharp_ 
                              nil)::nil))::(algebra.Alg.Term 
                              algebra.F.id__minus_ ((algebra.Alg.Term 
                              algebra.F.id_size (x2::nil))::
                              (algebra.Alg.Term algebra.F.id_size 
                              (x3::nil))::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_15 :
   forall x y, 
    (DP_R_xml_0_non_scc_15 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
  Proof.
    intros x y h.
    
    inversion h;clear h;subst;
     constructor;intros _y _h;inversion _h;clear _h;subst;
      (eapply acc_DP_R_xml_0_scc_13;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          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_16  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),ge(size(y_),size(z_))> *)
    | DP_R_xml_0_non_scc_16_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_16 (algebra.Alg.Term algebra.F.id_ge 
                              ((algebra.Alg.Term algebra.F.id_size 
                              (x2::nil))::(algebra.Alg.Term 
                              algebra.F.id_size (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_16 :
   forall x y, 
    (DP_R_xml_0_non_scc_16 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_13;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          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_17  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),ge(min(z_),x_)> *)
    | DP_R_xml_0_non_scc_17_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_17 (algebra.Alg.Term algebra.F.id_ge 
                              ((algebra.Alg.Term algebra.F.id_min 
                              (x3::nil))::x1::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_17 :
   forall x y, 
    (DP_R_xml_0_non_scc_17 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_13;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          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_18  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),ge(x_,max(y_))> *)
    | DP_R_xml_0_non_scc_18_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_18 (algebra.Alg.Term algebra.F.id_ge (x1::
                              (algebra.Alg.Term algebra.F.id_max 
                              (x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_18 :
   forall x y, 
    (DP_R_xml_0_non_scc_18 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_13;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_scc_12;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_11;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_19  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <bs(n(x_,y_,z_)),bs(z_)> *)
    | DP_R_xml_0_scc_19_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_bs (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::nil))
     (* <bs(n(x_,y_,z_)),bs(y_)> *)
    | DP_R_xml_0_scc_19_1 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_bs (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_bs (x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_19.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 0.
   
   Definition P_id_bs (x5:Z) := 3* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
   
   Definition P_id_l (x5:Z) := 2 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 2 + 2* x5.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 1.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 3* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 1* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 1* x5 + 2* x6 + 1* x7.
   
   Definition P_id_not (x5:Z) := 1.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 3* x5 + 3* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 2.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 1.
   
   Definition P_id_max (x5:Z) := 1* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 2* x5.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_19.
   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_19.
  
  Definition wf_DP_R_xml_0_scc_19  := WF_DP_R_xml_0_scc_19.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_19 :
   forall x y, (DP_R_xml_0_scc_19 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_19).
    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_18;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_17;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_10;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_8;
            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' ))||
             ((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_19.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_20  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <-(1(x_),1(y_)),0(-(x_,y_))> *)
    | DP_R_xml_0_non_scc_20_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_non_scc_20 (algebra.Alg.Term algebra.F.id_0 
                               ((algebra.Alg.Term algebra.F.id__minus_ (x1::
                               x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_20 :
   forall x y, 
    (DP_R_xml_0_non_scc_20 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_21  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <-(0(x_),0(y_)),0(-(x_,y_))> *)
    | DP_R_xml_0_non_scc_21_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_non_scc_21 (algebra.Alg.Term algebra.F.id_0 
                               ((algebra.Alg.Term algebra.F.id__minus_ (x1::
                               x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_21 :
   forall x y, 
    (DP_R_xml_0_non_scc_21 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_22  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <-(0(x_),1(y_)),-(-(x_,y_),1(#))> *)
    | DP_R_xml_0_scc_22_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_22 (algebra.Alg.Term algebra.F.id__minus_ 
                           ((algebra.Alg.Term algebra.F.id__minus_ (x1::
                           x2::nil))::(algebra.Alg.Term algebra.F.id_1 
                           ((algebra.Alg.Term algebra.F.id__sharp_ 
                           nil)::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
     (* <-(0(x_),1(y_)),-(x_,y_)> *)
    | DP_R_xml_0_scc_22_1 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_22 (algebra.Alg.Term algebra.F.id__minus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
     (* <-(0(x_),0(y_)),-(x_,y_)> *)
    | DP_R_xml_0_scc_22_2 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_22 (algebra.Alg.Term algebra.F.id__minus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
     (* <-(1(x_),0(y_)),-(x_,y_)> *)
    | DP_R_xml_0_scc_22_3 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_22 (algebra.Alg.Term algebra.F.id__minus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
     (* <-(1(x_),1(y_)),-(x_,y_)> *)
    | DP_R_xml_0_scc_22_4 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_22 (algebra.Alg.Term algebra.F.id__minus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_22.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 2 + 1* x5.
   
   Definition P_id_bs (x5:Z) := 2* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 1 + 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
   
   Definition P_id_l (x5:Z) := 2 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 2.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 3* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 1 + 2* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 2 + 1* x5 + 2* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 2 + 1* x5.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 1* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_22.
   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_22.
  
  Definition wf_DP_R_xml_0_scc_22  := WF_DP_R_xml_0_scc_22.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_22 :
   forall x y, (DP_R_xml_0_scc_22 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_22).
    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_21;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_20;
          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_22.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_23  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),-(size(z_),size(y_))> *)
    | DP_R_xml_0_non_scc_23_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_23 (algebra.Alg.Term algebra.F.id__minus_ 
                              ((algebra.Alg.Term algebra.F.id_size 
                              (x3::nil))::(algebra.Alg.Term 
                              algebra.F.id_size (x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_23 :
   forall x y, 
    (DP_R_xml_0_non_scc_23 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_22;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_21;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_20;
          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_24  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),-(size(y_),size(z_))> *)
    | DP_R_xml_0_non_scc_24_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_24 (algebra.Alg.Term algebra.F.id__minus_ 
                              ((algebra.Alg.Term algebra.F.id_size 
                              (x2::nil))::(algebra.Alg.Term 
                              algebra.F.id_size (x3::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_24 :
   forall x y, 
    (DP_R_xml_0_non_scc_24 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_22;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_21;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_20;
          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_25  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <+(1(x_),1(y_)),0(+(+(x_,y_),1(#)))> *)
    | DP_R_xml_0_non_scc_25_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_non_scc_25 (algebra.Alg.Term algebra.F.id_0 
                               ((algebra.Alg.Term algebra.F.id__plus_ 
                               ((algebra.Alg.Term algebra.F.id__plus_ (x1::
                               x2::nil))::(algebra.Alg.Term algebra.F.id_1 
                               ((algebra.Alg.Term algebra.F.id__sharp_ 
                               nil)::nil))::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_25 :
   forall x y, 
    (DP_R_xml_0_non_scc_25 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_26  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <+(0(x_),0(y_)),0(+(x_,y_))> *)
    | DP_R_xml_0_non_scc_26_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_non_scc_26 (algebra.Alg.Term algebra.F.id_0 
                               ((algebra.Alg.Term algebra.F.id__plus_ (x1::
                               x2::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_26 :
   forall x y, 
    (DP_R_xml_0_non_scc_26 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_27  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <+(0(x_),1(y_)),+(x_,y_)> *)
    | DP_R_xml_0_scc_27_0 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(0(x_),0(y_)),+(x_,y_)> *)
    | DP_R_xml_0_scc_27_1 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_0 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(1(x_),0(y_)),+(x_,y_)> *)
    | DP_R_xml_0_scc_27_2 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(1(x_),1(y_)),+(+(x_,y_),1(#))> *)
    | DP_R_xml_0_scc_27_3 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ 
                           ((algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil))::(algebra.Alg.Term algebra.F.id_1 
                           ((algebra.Alg.Term algebra.F.id__sharp_ 
                           nil)::nil))::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(1(x_),1(y_)),+(x_,y_)> *)
    | DP_R_xml_0_scc_27_4 :
     forall x2 x6 x1 x5, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 (algebra.Alg.Term algebra.F.id_1 (x1::nil)) 
        x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(x_,+(y_,z_)),+(+(x_,y_),z_)> *)
    | DP_R_xml_0_scc_27_5 :
     forall x2 x6 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ 
                           ((algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil))::x3::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
     (* <+(x_,+(y_,z_)),+(x_,y_)> *)
    | DP_R_xml_0_scc_27_6 :
     forall x2 x6 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 x1 x6) ->
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
        DP_R_xml_0_scc_27 (algebra.Alg.Term algebra.F.id__plus_ (x1::
                           x2::nil)) 
         (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_27.
   Inductive DP_R_xml_0_scc_27_large  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <+(1(x_),1(y_)),+(+(x_,y_),1(#))> *)
     | DP_R_xml_0_scc_27_large_0 :
      forall x2 x6 x1 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x1::nil)) x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
         DP_R_xml_0_scc_27_large (algebra.Alg.Term algebra.F.id__plus_ 
                                  ((algebra.Alg.Term algebra.F.id__plus_ 
                                  (x1::x2::nil))::(algebra.Alg.Term 
                                  algebra.F.id_1 ((algebra.Alg.Term 
                                  algebra.F.id__sharp_ nil)::nil))::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      (* <+(x_,+(y_,z_)),+(+(x_,y_),z_)> *)
     | DP_R_xml_0_scc_27_large_1 :
      forall x2 x6 x1 x5 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
         DP_R_xml_0_scc_27_large (algebra.Alg.Term algebra.F.id__plus_ 
                                  ((algebra.Alg.Term algebra.F.id__plus_ 
                                  (x1::x2::nil))::x3::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      (* <+(x_,+(y_,z_)),+(x_,y_)> *)
     | DP_R_xml_0_scc_27_large_2 :
      forall x2 x6 x1 x5 x3, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  x1 x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
         DP_R_xml_0_scc_27_large (algebra.Alg.Term algebra.F.id__plus_ (x1::
                                  x2::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
   .
   
   
   Inductive DP_R_xml_0_scc_27_strict  :
    algebra.Alg.term ->algebra.Alg.term ->Prop := 
      (* <+(0(x_),1(y_)),+(x_,y_)> *)
     | DP_R_xml_0_scc_27_strict_0 :
      forall x2 x6 x1 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x1::nil)) x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
         DP_R_xml_0_scc_27_strict (algebra.Alg.Term algebra.F.id__plus_ (x1::
                                   x2::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      (* <+(0(x_),0(y_)),+(x_,y_)> *)
     | DP_R_xml_0_scc_27_strict_1 :
      forall x2 x6 x1 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_0 (x1::nil)) x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
         DP_R_xml_0_scc_27_strict (algebra.Alg.Term algebra.F.id__plus_ (x1::
                                   x2::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      (* <+(1(x_),0(y_)),+(x_,y_)> *)
     | DP_R_xml_0_scc_27_strict_2 :
      forall x2 x6 x1 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x1::nil)) x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_0 (x2::nil)) x5) ->
         DP_R_xml_0_scc_27_strict (algebra.Alg.Term algebra.F.id__plus_ (x1::
                                   x2::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      (* <+(1(x_),1(y_)),+(x_,y_)> *)
     | DP_R_xml_0_scc_27_strict_3 :
      forall x2 x6 x1 x5, 
       (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                  
         (algebra.Alg.Term algebra.F.id_1 (x1::nil)) x6) ->
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
         DP_R_xml_0_scc_27_strict (algebra.Alg.Term algebra.F.id__plus_ (x1::
                                   x2::nil)) 
          (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
   .
   
   
   Module WF_DP_R_xml_0_scc_27_large.
    Inductive DP_R_xml_0_scc_27_large_scc_1  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <+(1(x_),1(y_)),+(+(x_,y_),1(#))> *)
      | DP_R_xml_0_scc_27_large_scc_1_0 :
       forall x2 x6 x1 x5, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   
          (algebra.Alg.Term algebra.F.id_1 (x1::nil)) x6) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id_1 (x2::nil)) x5) ->
          DP_R_xml_0_scc_27_large_scc_1 (algebra.Alg.Term 
                                         algebra.F.id__plus_ 
                                         ((algebra.Alg.Term 
                                         algebra.F.id__plus_ (x1::x2::nil))::
                                         (algebra.Alg.Term algebra.F.id_1 
                                         ((algebra.Alg.Term 
                                         algebra.F.id__sharp_ 
                                         nil)::nil))::nil)) 
           (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_27_large_scc_1.
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_0 (x5:Z) := 2 + 1* x5.
     
     Definition P_id_bs (x5:Z) := 3.
     
     Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
     
     Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
     
     Definition P_id_l (x5:Z) := 1 + 1* x5.
     
     Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
     
     Definition P_id_wb (x5:Z) := 1.
     
     Definition P_id_ge (x5:Z) (x6:Z) := 0.
     
     Definition P_id_false  := 0.
     
     Definition P_id_min (x5:Z) := 3* x5.
     
     Definition P_id__sharp_  := 0.
     
     Definition P_id_size (x5:Z) := 2* x5.
     
     Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
     
     Definition P_id_not (x5:Z) := 0.
     
     Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 2* x5 + 2* x6 + 2* x7.
     
     Definition P_id_1 (x5:Z) := 2 + 1* x5.
     
     Definition P_id_val (x5:Z) := 1* x5.
     
     Definition P_id_true  := 0.
     
     Definition P_id_max (x5:Z) := 1* x5.
     
     Lemma P_id_0_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_bs_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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__minus__monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_l_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_wb_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ge_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_min_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_size_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_if_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_not_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_n_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_1_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_val_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_max_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__minus__bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ge_bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_false_bounded : 0 <= P_id_false .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_if_bounded :
      forall x6 x5 x7, 
       (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_n_bounded :
      forall x6 x5 x7, 
       (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_true_bounded : 0 <= P_id_true .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
     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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
        P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ 
        P_id_size P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                      P_id_0 (measure x5)
                     | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                      P_id_bs (measure x5)
                     | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                      P_id_and (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                      P_id__minus_ (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                      P_id_l (measure x5)
                     | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                      P_id__plus_ (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                      P_id_wb (measure x5)
                     | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                      P_id_ge (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_false nil) =>
                      P_id_false 
                     | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                      P_id_min (measure x5)
                     | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                      P_id__sharp_ 
                     | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                      P_id_size (measure x5)
                     | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                      P_id_if (measure x7) (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                      P_id_not (measure x5)
                     | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                      P_id_n (measure x7) (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                      P_id_1 (measure x5)
                     | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                      P_id_val (measure x5)
                     | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                     | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                      P_id_max (measure x5)
                     | _ => 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_0_monotonic;assumption.
       intros ;apply P_id_bs_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id__minus__monotonic;assumption.
       intros ;apply P_id_l_monotonic;assumption.
       intros ;apply P_id__plus__monotonic;assumption.
       intros ;apply P_id_wb_monotonic;assumption.
       intros ;apply P_id_ge_monotonic;assumption.
       intros ;apply P_id_min_monotonic;assumption.
       intros ;apply P_id_size_monotonic;assumption.
       intros ;apply P_id_if_monotonic;assumption.
       intros ;apply P_id_not_monotonic;assumption.
       intros ;apply P_id_n_monotonic;assumption.
       intros ;apply P_id_1_monotonic;assumption.
       intros ;apply P_id_val_monotonic;assumption.
       intros ;apply P_id_max_monotonic;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_bs_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id__minus__bounded;assumption.
       intros ;apply P_id_l_bounded;assumption.
       intros ;apply P_id__plus__bounded;assumption.
       intros ;apply P_id_wb_bounded;assumption.
       intros ;apply P_id_ge_bounded;assumption.
       intros ;apply P_id_false_bounded;assumption.
       intros ;apply P_id_min_bounded;assumption.
       intros ;apply P_id__sharp__bounded;assumption.
       intros ;apply P_id_size_bounded;assumption.
       intros ;apply P_id_if_bounded;assumption.
       intros ;apply P_id_not_bounded;assumption.
       intros ;apply P_id_n_bounded;assumption.
       intros ;apply P_id_1_bounded;assumption.
       intros ;apply P_id_val_bounded;assumption.
       intros ;apply P_id_true_bounded;assumption.
       intros ;apply P_id_max_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_MIN (x5:Z) := 0.
     
     Definition P_id_0_hat_1 (x5:Z) := 0.
     
     Definition P_id_SIZE (x5:Z) := 0.
     
     Definition P_id_BS (x5:Z) := 0.
     
     Definition P_id_GE (x5:Z) (x6:Z) := 0.
     
     Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
     
     Definition P_id_MAX (x5:Z) := 0.
     
     Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
     
     Definition P_id_WB (x5:Z) := 0.
     
     Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 1* x5 + 2* x6.
     
     Definition P_id_AND (x5:Z) (x6:Z) := 0.
     
     Definition P_id_NOT (x5:Z) := 0.
     
     Definition P_id_Marked_val (x5:Z) := 0.
     
     Lemma P_id_MIN_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_hat_1_monotonic :
      forall x6 x5, 
       (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_SIZE_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_BS_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_GE_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_IF_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MAX_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__minus__hat_1_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->
         P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_WB_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__hat_1_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->
         P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_AND_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_NOT_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_Marked_val_monotonic :
      forall x6 x5, 
       (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ 
        P_id_l P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ 
        P_id_size P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max 
        P_id_MIN P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
        P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
        P_id_Marked_val.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                             P_id_MIN (measure x5)
                            | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                             P_id_0_hat_1 (measure x5)
                            | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                             P_id_SIZE (measure x5)
                            | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                             P_id_BS (measure x5)
                            | (algebra.Alg.Term algebra.F.id_ge (x6::
                               x5::nil)) =>
                             P_id_GE (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                               x5::nil)) =>
                             P_id_IF (measure x7) (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                             P_id_MAX (measure x5)
                            | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                               x5::nil)) =>
                             P_id__minus__hat_1 (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                             P_id_WB (measure x5)
                            | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                               x5::nil)) =>
                             P_id__plus__hat_1 (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_and (x6::
                               x5::nil)) =>
                             P_id_AND (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                             P_id_NOT (measure x5)
                            | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                             P_id_Marked_val (measure x5)
                            | _ => 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_0_monotonic;assumption.
       intros ;apply P_id_bs_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id__minus__monotonic;assumption.
       intros ;apply P_id_l_monotonic;assumption.
       intros ;apply P_id__plus__monotonic;assumption.
       intros ;apply P_id_wb_monotonic;assumption.
       intros ;apply P_id_ge_monotonic;assumption.
       intros ;apply P_id_min_monotonic;assumption.
       intros ;apply P_id_size_monotonic;assumption.
       intros ;apply P_id_if_monotonic;assumption.
       intros ;apply P_id_not_monotonic;assumption.
       intros ;apply P_id_n_monotonic;assumption.
       intros ;apply P_id_1_monotonic;assumption.
       intros ;apply P_id_val_monotonic;assumption.
       intros ;apply P_id_max_monotonic;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_bs_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id__minus__bounded;assumption.
       intros ;apply P_id_l_bounded;assumption.
       intros ;apply P_id__plus__bounded;assumption.
       intros ;apply P_id_wb_bounded;assumption.
       intros ;apply P_id_ge_bounded;assumption.
       intros ;apply P_id_false_bounded;assumption.
       intros ;apply P_id_min_bounded;assumption.
       intros ;apply P_id__sharp__bounded;assumption.
       intros ;apply P_id_size_bounded;assumption.
       intros ;apply P_id_if_bounded;assumption.
       intros ;apply P_id_not_bounded;assumption.
       intros ;apply P_id_n_bounded;assumption.
       intros ;apply P_id_1_bounded;assumption.
       intros ;apply P_id_val_bounded;assumption.
       intros ;apply P_id_true_bounded;assumption.
       intros ;apply P_id_max_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_MIN_monotonic;assumption.
       intros ;apply P_id_0_hat_1_monotonic;assumption.
       intros ;apply P_id_SIZE_monotonic;assumption.
       intros ;apply P_id_BS_monotonic;assumption.
       intros ;apply P_id_GE_monotonic;assumption.
       intros ;apply P_id_IF_monotonic;assumption.
       intros ;apply P_id_MAX_monotonic;assumption.
       intros ;apply P_id__minus__hat_1_monotonic;assumption.
       intros ;apply P_id_WB_monotonic;assumption.
       intros ;apply P_id__plus__hat_1_monotonic;assumption.
       intros ;apply P_id_AND_monotonic;assumption.
       intros ;apply P_id_NOT_monotonic;assumption.
       intros ;apply P_id_Marked_val_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_27_large.DP_R_xml_0_scc_27_large_scc_1.
     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_27_large_scc_1.
    
    Definition wf_DP_R_xml_0_scc_27_large_scc_1  := 
      WF_DP_R_xml_0_scc_27_large_scc_1.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_27_large_scc_1 :
     forall x y, 
      (DP_R_xml_0_scc_27_large_scc_1 x y) ->
       Acc WF_DP_R_xml_0_scc_27.DP_R_xml_0_scc_27_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_27_large_scc_1).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
      apply wf_DP_R_xml_0_scc_27_large_scc_1.
    Qed.
    
    
    Inductive DP_R_xml_0_scc_27_large_scc_2  :
     algebra.Alg.term ->algebra.Alg.term ->Prop := 
       (* <+(x_,+(y_,z_)),+(x_,y_)> *)
      | DP_R_xml_0_scc_27_large_scc_2_0 :
       forall x2 x6 x1 x5 x3, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x6) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
          DP_R_xml_0_scc_27_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id__plus_ (x1::x2::nil)) 
           (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
      
       (* <+(x_,+(y_,z_)),+(+(x_,y_),z_)> *)
      | DP_R_xml_0_scc_27_large_scc_2_1 :
       forall x2 x6 x1 x5 x3, 
        (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                   x1 x6) ->
         (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                    
           (algebra.Alg.Term algebra.F.id__plus_ (x2::x3::nil)) x5) ->
          DP_R_xml_0_scc_27_large_scc_2 (algebra.Alg.Term 
                                         algebra.F.id__plus_ 
                                         ((algebra.Alg.Term 
                                         algebra.F.id__plus_ (x1::x2::nil))::
                                         x3::nil)) 
           (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil))
    .
    
    
    Module WF_DP_R_xml_0_scc_27_large_scc_2.
     Open Scope Z_scope.
     
     Import ring_extention.
     
     Notation Local "a <= b" := (Zle a b).
     
     Notation Local "a < b" := (Zlt a b).
     
     Definition P_id_0 (x5:Z) := 1.
     
     Definition P_id_bs (x5:Z) := 1.
     
     Definition P_id_and (x5:Z) (x6:Z) := 1* x5.
     
     Definition P_id__minus_ (x5:Z) (x6:Z) := 1 + 1* x5.
     
     Definition P_id_l (x5:Z) := 1* x5.
     
     Definition P_id__plus_ (x5:Z) (x6:Z) := 1 + 1* x5 + 2* x6.
     
     Definition P_id_wb (x5:Z) := 1 + 2* x5.
     
     Definition P_id_ge (x5:Z) (x6:Z) := 0.
     
     Definition P_id_false  := 0.
     
     Definition P_id_min (x5:Z) := 1* x5.
     
     Definition P_id__sharp_  := 0.
     
     Definition P_id_size (x5:Z) := 3* x5.
     
     Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 1* x6 + 1* x7.
     
     Definition P_id_not (x5:Z) := 0.
     
     Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 1 + 2* x5 + 2* x6 + 2* x7.
     
     Definition P_id_1 (x5:Z) := 0.
     
     Definition P_id_val (x5:Z) := 1* x5.
     
     Definition P_id_true  := 0.
     
     Definition P_id_max (x5:Z) := 1* x5.
     
     Lemma P_id_0_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_bs_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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__minus__monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_l_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_wb_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ge_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_min_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_size_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_if_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_not_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_n_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_1_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_val_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_max_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_and_bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__minus__bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_ge_bounded :
      forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_false_bounded : 0 <= P_id_false .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_if_bounded :
      forall x6 x5 x7, 
       (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_n_bounded :
      forall x6 x5 x7, 
       (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_true_bounded : 0 <= P_id_true .
     Proof.
       intros .
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
     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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
        P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ 
        P_id_size P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
     
     Lemma measure_equation :
      forall t, 
       measure t = match t with
                     | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                      P_id_0 (measure x5)
                     | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                      P_id_bs (measure x5)
                     | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                      P_id_and (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                      P_id__minus_ (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                      P_id_l (measure x5)
                     | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                      P_id__plus_ (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                      P_id_wb (measure x5)
                     | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                      P_id_ge (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_false nil) =>
                      P_id_false 
                     | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                      P_id_min (measure x5)
                     | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                      P_id__sharp_ 
                     | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                      P_id_size (measure x5)
                     | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                      P_id_if (measure x7) (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                      P_id_not (measure x5)
                     | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                      P_id_n (measure x7) (measure x6) (measure x5)
                     | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                      P_id_1 (measure x5)
                     | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                      P_id_val (measure x5)
                     | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                     | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                      P_id_max (measure x5)
                     | _ => 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_0_monotonic;assumption.
       intros ;apply P_id_bs_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id__minus__monotonic;assumption.
       intros ;apply P_id_l_monotonic;assumption.
       intros ;apply P_id__plus__monotonic;assumption.
       intros ;apply P_id_wb_monotonic;assumption.
       intros ;apply P_id_ge_monotonic;assumption.
       intros ;apply P_id_min_monotonic;assumption.
       intros ;apply P_id_size_monotonic;assumption.
       intros ;apply P_id_if_monotonic;assumption.
       intros ;apply P_id_not_monotonic;assumption.
       intros ;apply P_id_n_monotonic;assumption.
       intros ;apply P_id_1_monotonic;assumption.
       intros ;apply P_id_val_monotonic;assumption.
       intros ;apply P_id_max_monotonic;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_bs_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id__minus__bounded;assumption.
       intros ;apply P_id_l_bounded;assumption.
       intros ;apply P_id__plus__bounded;assumption.
       intros ;apply P_id_wb_bounded;assumption.
       intros ;apply P_id_ge_bounded;assumption.
       intros ;apply P_id_false_bounded;assumption.
       intros ;apply P_id_min_bounded;assumption.
       intros ;apply P_id__sharp__bounded;assumption.
       intros ;apply P_id_size_bounded;assumption.
       intros ;apply P_id_if_bounded;assumption.
       intros ;apply P_id_not_bounded;assumption.
       intros ;apply P_id_n_bounded;assumption.
       intros ;apply P_id_1_bounded;assumption.
       intros ;apply P_id_val_bounded;assumption.
       intros ;apply P_id_true_bounded;assumption.
       intros ;apply P_id_max_bounded;assumption.
       apply rules_monotonic.
     Qed.
     
     Definition P_id_MIN (x5:Z) := 0.
     
     Definition P_id_0_hat_1 (x5:Z) := 0.
     
     Definition P_id_SIZE (x5:Z) := 0.
     
     Definition P_id_BS (x5:Z) := 0.
     
     Definition P_id_GE (x5:Z) (x6:Z) := 0.
     
     Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
     
     Definition P_id_MAX (x5:Z) := 0.
     
     Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
     
     Definition P_id_WB (x5:Z) := 0.
     
     Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 1* x6.
     
     Definition P_id_AND (x5:Z) (x6:Z) := 0.
     
     Definition P_id_NOT (x5:Z) := 0.
     
     Definition P_id_Marked_val (x5:Z) := 0.
     
     Lemma P_id_MIN_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_0_hat_1_monotonic :
      forall x6 x5, 
       (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_SIZE_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_BS_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_GE_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_IF_monotonic :
      forall x8 x10 x6 x9 x5 x7, 
       (0 <= x10)/\ (x10 <= x9) ->
        (0 <= x8)/\ (x8 <= x7) ->
         (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
     Proof.
       intros x10 x9 x8 x7 x6 x5.
       intros [H_1 H_0].
       intros [H_3 H_2].
       intros [H_5 H_4].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_MAX_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__minus__hat_1_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->
         P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_WB_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id__plus__hat_1_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->
         P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_AND_monotonic :
      forall x8 x6 x5 x7, 
       (0 <= x8)/\ (x8 <= x7) ->
        (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
     Proof.
       intros x8 x7 x6 x5.
       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_NOT_monotonic :
      forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Lemma P_id_Marked_val_monotonic :
      forall x6 x5, 
       (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
     Proof.
       intros x6 x5.
       intros [H_1 H_0].
       
       cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
        (auto with zarith)||(repeat (translate_vars );prove_ineq ).
     Qed.
     
     Definition marked_measure  := 
       InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ 
        P_id_l P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ 
        P_id_size P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max 
        P_id_MIN P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
        P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
        P_id_Marked_val.
     
     Lemma marked_measure_equation :
      forall t, 
       marked_measure t = match t with
                            | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                             P_id_MIN (measure x5)
                            | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                             P_id_0_hat_1 (measure x5)
                            | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                             P_id_SIZE (measure x5)
                            | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                             P_id_BS (measure x5)
                            | (algebra.Alg.Term algebra.F.id_ge (x6::
                               x5::nil)) =>
                             P_id_GE (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                               x5::nil)) =>
                             P_id_IF (measure x7) (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                             P_id_MAX (measure x5)
                            | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                               x5::nil)) =>
                             P_id__minus__hat_1 (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                             P_id_WB (measure x5)
                            | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                               x5::nil)) =>
                             P_id__plus__hat_1 (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_and (x6::
                               x5::nil)) =>
                             P_id_AND (measure x6) (measure x5)
                            | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                             P_id_NOT (measure x5)
                            | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                             P_id_Marked_val (measure x5)
                            | _ => 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_0_monotonic;assumption.
       intros ;apply P_id_bs_monotonic;assumption.
       intros ;apply P_id_and_monotonic;assumption.
       intros ;apply P_id__minus__monotonic;assumption.
       intros ;apply P_id_l_monotonic;assumption.
       intros ;apply P_id__plus__monotonic;assumption.
       intros ;apply P_id_wb_monotonic;assumption.
       intros ;apply P_id_ge_monotonic;assumption.
       intros ;apply P_id_min_monotonic;assumption.
       intros ;apply P_id_size_monotonic;assumption.
       intros ;apply P_id_if_monotonic;assumption.
       intros ;apply P_id_not_monotonic;assumption.
       intros ;apply P_id_n_monotonic;assumption.
       intros ;apply P_id_1_monotonic;assumption.
       intros ;apply P_id_val_monotonic;assumption.
       intros ;apply P_id_max_monotonic;assumption.
       intros ;apply P_id_0_bounded;assumption.
       intros ;apply P_id_bs_bounded;assumption.
       intros ;apply P_id_and_bounded;assumption.
       intros ;apply P_id__minus__bounded;assumption.
       intros ;apply P_id_l_bounded;assumption.
       intros ;apply P_id__plus__bounded;assumption.
       intros ;apply P_id_wb_bounded;assumption.
       intros ;apply P_id_ge_bounded;assumption.
       intros ;apply P_id_false_bounded;assumption.
       intros ;apply P_id_min_bounded;assumption.
       intros ;apply P_id__sharp__bounded;assumption.
       intros ;apply P_id_size_bounded;assumption.
       intros ;apply P_id_if_bounded;assumption.
       intros ;apply P_id_not_bounded;assumption.
       intros ;apply P_id_n_bounded;assumption.
       intros ;apply P_id_1_bounded;assumption.
       intros ;apply P_id_val_bounded;assumption.
       intros ;apply P_id_true_bounded;assumption.
       intros ;apply P_id_max_bounded;assumption.
       apply rules_monotonic.
       intros ;apply P_id_MIN_monotonic;assumption.
       intros ;apply P_id_0_hat_1_monotonic;assumption.
       intros ;apply P_id_SIZE_monotonic;assumption.
       intros ;apply P_id_BS_monotonic;assumption.
       intros ;apply P_id_GE_monotonic;assumption.
       intros ;apply P_id_IF_monotonic;assumption.
       intros ;apply P_id_MAX_monotonic;assumption.
       intros ;apply P_id__minus__hat_1_monotonic;assumption.
       intros ;apply P_id_WB_monotonic;assumption.
       intros ;apply P_id__plus__hat_1_monotonic;assumption.
       intros ;apply P_id_AND_monotonic;assumption.
       intros ;apply P_id_NOT_monotonic;assumption.
       intros ;apply P_id_Marked_val_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_27_large.DP_R_xml_0_scc_27_large_scc_2.
     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_27_large_scc_2.
    
    Definition wf_DP_R_xml_0_scc_27_large_scc_2  := 
      WF_DP_R_xml_0_scc_27_large_scc_2.wf.
    
    
    Lemma acc_DP_R_xml_0_scc_27_large_scc_2 :
     forall x y, 
      (DP_R_xml_0_scc_27_large_scc_2 x y) ->
       Acc WF_DP_R_xml_0_scc_27.DP_R_xml_0_scc_27_large x.
    Proof.
      intros x.
      pattern x.
      apply (@Acc_ind _ DP_R_xml_0_scc_27_large_scc_2).
      intros x' _ Hrec y h.
      
      inversion h;clear h;subst;
       constructor;intros _y _h;inversion _h;clear _h;subst;
        (eapply Hrec;econstructor eassumption)||
        ((eapply acc_DP_R_xml_0_scc_27_large_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_27_large_scc_2.
    Qed.
    
    
    Lemma wf : well_founded WF_DP_R_xml_0_scc_27.DP_R_xml_0_scc_27_large.
    Proof.
      constructor;intros _y _h;inversion _h;clear _h;subst;
       (eapply acc_DP_R_xml_0_scc_27_large_scc_2;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_scc_27_large_scc_1;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_scc_27_large_scc_0;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))).
    Qed.
   End WF_DP_R_xml_0_scc_27_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_0 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_bs (x5:Z) := 0.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5.
   
   Definition P_id_l (x5:Z) := 3 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 0.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 2* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 2 + 2* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 2* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 1* x5 + 1* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 1 + 1* x5.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 2* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 2* x6.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_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_27_strict_in_lt :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_27_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_27_large_in_le :
    Relation_Definitions.inclusion _ DP_R_xml_0_scc_27_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_27_large  := WF_DP_R_xml_0_scc_27_large.wf.
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_27.
   Proof.
     intros x.
     apply (well_founded_ind wf_lt).
     clear x.
     intros x.
     pattern x.
     apply (@Acc_ind _ DP_R_xml_0_scc_27_large).
     clear x.
     intros x _ IHx IHx'.
     constructor.
     intros y H.
     
     destruct H;
      (apply IHx';apply DP_R_xml_0_scc_27_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_27_large_in_le;econstructor eassumption])).
     apply wf_DP_R_xml_0_scc_27_large.
   Qed.
  End WF_DP_R_xml_0_scc_27.
  
  Definition wf_DP_R_xml_0_scc_27  := WF_DP_R_xml_0_scc_27.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_27 :
   forall x y, (DP_R_xml_0_scc_27 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_27).
    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_26;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_25;
          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_27.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_28  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <size(n(x_,y_,z_)),+(size(x_),size(y_))> *)
    | DP_R_xml_0_non_scc_28_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_28 (algebra.Alg.Term algebra.F.id__plus_ 
                              ((algebra.Alg.Term algebra.F.id_size 
                              (x1::nil))::(algebra.Alg.Term 
                              algebra.F.id_size (x2::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_size (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_28 :
   forall x y, 
    (DP_R_xml_0_non_scc_28 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_27;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_26;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_25;
          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_29  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <size(n(x_,y_,z_)),+(+(size(x_),size(y_)),1(#))> *)
    | DP_R_xml_0_non_scc_29_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_29 (algebra.Alg.Term algebra.F.id__plus_ 
                              ((algebra.Alg.Term algebra.F.id__plus_ 
                              ((algebra.Alg.Term algebra.F.id_size 
                              (x1::nil))::(algebra.Alg.Term 
                              algebra.F.id_size (x2::nil))::nil))::
                              (algebra.Alg.Term algebra.F.id_1 
                              ((algebra.Alg.Term algebra.F.id__sharp_ 
                              nil)::nil))::nil)) 
        (algebra.Alg.Term algebra.F.id_size (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_29 :
   forall x y, 
    (DP_R_xml_0_non_scc_29 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_27;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_26;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_25;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_30  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <size(n(x_,y_,z_)),size(y_)> *)
    | DP_R_xml_0_scc_30_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_30 (algebra.Alg.Term algebra.F.id_size (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_size (x5::nil))
     (* <size(n(x_,y_,z_)),size(x_)> *)
    | DP_R_xml_0_scc_30_1 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_30 (algebra.Alg.Term algebra.F.id_size (x1::nil)) 
        (algebra.Alg.Term algebra.F.id_size (x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_30.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 2.
   
   Definition P_id_bs (x5:Z) := 3 + 3* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id_l (x5:Z) := 3 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 1.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 0.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 3* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 2 + 2* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 1* x6 + 2* x7.
   
   Definition P_id_not (x5:Z) := 0.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 3* x5 + 2* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 2.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 0.
   
   Definition P_id_max (x5:Z) := 1* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 2* x5.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 0.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_30.
   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_30.
  
  Definition wf_DP_R_xml_0_scc_30  := WF_DP_R_xml_0_scc_30.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_30 :
   forall x y, (DP_R_xml_0_scc_30 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_30).
    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_29;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_28;
          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_30.
  Qed.
  
  
  Inductive DP_R_xml_0_non_scc_31  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),size(z_)> *)
    | DP_R_xml_0_non_scc_31_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_31 (algebra.Alg.Term algebra.F.id_size (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_31 :
   forall x y, 
    (DP_R_xml_0_non_scc_31 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_30;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_29;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_28;
          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_32  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),size(y_)> *)
    | DP_R_xml_0_non_scc_32_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_non_scc_32 (algebra.Alg.Term algebra.F.id_size (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Lemma acc_DP_R_xml_0_non_scc_32 :
   forall x y, 
    (DP_R_xml_0_non_scc_32 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_30;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_29;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_28;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
         (eapply Hrec;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))).
  Qed.
  
  
  Inductive DP_R_xml_0_scc_33  :
   algebra.Alg.term ->algebra.Alg.term ->Prop := 
     (* <wb(n(x_,y_,z_)),wb(z_)> *)
    | DP_R_xml_0_scc_33_0 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_33 (algebra.Alg.Term algebra.F.id_wb (x3::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
     (* <wb(n(x_,y_,z_)),wb(y_)> *)
    | DP_R_xml_0_scc_33_1 :
     forall x2 x1 x5 x3, 
      (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
                                 
        (algebra.Alg.Term algebra.F.id_n (x1::x2::x3::nil)) x5) ->
       DP_R_xml_0_scc_33 (algebra.Alg.Term algebra.F.id_wb (x2::nil)) 
        (algebra.Alg.Term algebra.F.id_wb (x5::nil))
  .
  
  
  Module WF_DP_R_xml_0_scc_33.
   Open Scope Z_scope.
   
   Import ring_extention.
   
   Notation Local "a <= b" := (Zle a b).
   
   Notation Local "a < b" := (Zlt a b).
   
   Definition P_id_0 (x5:Z) := 0.
   
   Definition P_id_bs (x5:Z) := 3* x5.
   
   Definition P_id_and (x5:Z) (x6:Z) := 2* x5.
   
   Definition P_id__minus_ (x5:Z) (x6:Z) := 1* x5 + 2* x6.
   
   Definition P_id_l (x5:Z) := 2 + 1* x5.
   
   Definition P_id__plus_ (x5:Z) (x6:Z) := 1* x5 + 1* x6.
   
   Definition P_id_wb (x5:Z) := 2 + 2* x5.
   
   Definition P_id_ge (x5:Z) (x6:Z) := 1.
   
   Definition P_id_false  := 0.
   
   Definition P_id_min (x5:Z) := 3* x5.
   
   Definition P_id__sharp_  := 0.
   
   Definition P_id_size (x5:Z) := 1* x5.
   
   Definition P_id_if (x5:Z) (x6:Z) (x7:Z) := 1* x5 + 2* x6 + 1* x7.
   
   Definition P_id_not (x5:Z) := 1.
   
   Definition P_id_n (x5:Z) (x6:Z) (x7:Z) := 3 + 3* x5 + 3* x6 + 2* x7.
   
   Definition P_id_1 (x5:Z) := 2.
   
   Definition P_id_val (x5:Z) := 1* x5.
   
   Definition P_id_true  := 1.
   
   Definition P_id_max (x5:Z) := 1* x5.
   
   Lemma P_id_0_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0 x6 <= P_id_0 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_bs x6 <= P_id_bs x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_and x6 x8 <= P_id_and x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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__minus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__minus_ x6 x8 <= P_id__minus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_l_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_l x6 <= P_id_l x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id__plus_ x6 x8 <= P_id__plus_ x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_wb_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_wb x6 <= P_id_wb x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_ge x6 x8 <= P_id_ge x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_min_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_min x6 <= P_id_min x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_size x6 <= P_id_size x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_if x6 x8 x10 <= P_id_if x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_not x6 <= P_id_not x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_n x6 x8 x10 <= P_id_n x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_1 x6 <= P_id_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_val x6 <= P_id_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_max x6 <= P_id_max x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_bounded : forall x5, (0 <= x5) ->0 <= P_id_0 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_bs_bounded : forall x5, (0 <= x5) ->0 <= P_id_bs x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_and_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_and x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__minus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_l_bounded : forall x5, (0 <= x5) ->0 <= P_id_l x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id__plus_ x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_wb_bounded : forall x5, (0 <= x5) ->0 <= P_id_wb x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_ge_bounded :
    forall x6 x5, (0 <= x5) ->(0 <= x6) ->0 <= P_id_ge x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_false_bounded : 0 <= P_id_false .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_min_bounded : forall x5, (0 <= x5) ->0 <= P_id_min x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__sharp__bounded : 0 <= P_id__sharp_ .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_size_bounded : forall x5, (0 <= x5) ->0 <= P_id_size x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_if_bounded :
    forall x6 x5 x7, 
     (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_if x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_not_bounded : forall x5, (0 <= x5) ->0 <= P_id_not x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_n_bounded :
    forall x6 x5 x7, (0 <= x5) ->(0 <= x6) ->(0 <= x7) ->0 <= P_id_n x7 x6 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_1_bounded : forall x5, (0 <= x5) ->0 <= P_id_1 x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_val_bounded : forall x5, (0 <= x5) ->0 <= P_id_val x5.
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_true_bounded : 0 <= P_id_true .
   Proof.
     intros .
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_max_bounded : forall x5, (0 <= x5) ->0 <= P_id_max x5.
   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_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max.
   
   Lemma measure_equation :
    forall t, 
     measure t = match t with
                   | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                    P_id_0 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                    P_id_bs (measure x5)
                   | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                    P_id_and (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id__minus_ (x6::x5::nil)) =>
                    P_id__minus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_l (x5::nil)) =>
                    P_id_l (measure x5)
                   | (algebra.Alg.Term algebra.F.id__plus_ (x6::x5::nil)) =>
                    P_id__plus_ (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                    P_id_wb (measure x5)
                   | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                    P_id_ge (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_false nil) => P_id_false 
                   | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                    P_id_min (measure x5)
                   | (algebra.Alg.Term algebra.F.id__sharp_ nil) =>
                    P_id__sharp_ 
                   | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                    P_id_size (measure x5)
                   | (algebra.Alg.Term algebra.F.id_if (x7::x6::x5::nil)) =>
                    P_id_if (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                    P_id_not (measure x5)
                   | (algebra.Alg.Term algebra.F.id_n (x7::x6::x5::nil)) =>
                    P_id_n (measure x7) (measure x6) (measure x5)
                   | (algebra.Alg.Term algebra.F.id_1 (x5::nil)) =>
                    P_id_1 (measure x5)
                   | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                    P_id_val (measure x5)
                   | (algebra.Alg.Term algebra.F.id_true nil) => P_id_true 
                   | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                    P_id_max (measure x5)
                   | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
   Qed.
   
   Definition P_id_MIN (x5:Z) := 0.
   
   Definition P_id_0_hat_1 (x5:Z) := 0.
   
   Definition P_id_SIZE (x5:Z) := 0.
   
   Definition P_id_BS (x5:Z) := 0.
   
   Definition P_id_GE (x5:Z) (x6:Z) := 0.
   
   Definition P_id_IF (x5:Z) (x6:Z) (x7:Z) := 0.
   
   Definition P_id_MAX (x5:Z) := 0.
   
   Definition P_id__minus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_WB (x5:Z) := 2* x5.
   
   Definition P_id__plus__hat_1 (x5:Z) (x6:Z) := 0.
   
   Definition P_id_AND (x5:Z) (x6:Z) := 0.
   
   Definition P_id_NOT (x5:Z) := 0.
   
   Definition P_id_Marked_val (x5:Z) := 0.
   
   Lemma P_id_MIN_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MIN x6 <= P_id_MIN x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_0_hat_1_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_0_hat_1 x6 <= P_id_0_hat_1 x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_SIZE_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_SIZE x6 <= P_id_SIZE x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_BS_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_BS x6 <= P_id_BS x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_GE_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_GE x6 x8 <= P_id_GE x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_IF_monotonic :
    forall x8 x10 x6 x9 x5 x7, 
     (0 <= x10)/\ (x10 <= x9) ->
      (0 <= x8)/\ (x8 <= x7) ->
       (0 <= x6)/\ (x6 <= x5) ->P_id_IF x6 x8 x10 <= P_id_IF x5 x7 x9.
   Proof.
     intros x10 x9 x8 x7 x6 x5.
     intros [H_1 H_0].
     intros [H_3 H_2].
     intros [H_5 H_4].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_MAX_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_MAX x6 <= P_id_MAX x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__minus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__minus__hat_1 x6 x8 <= P_id__minus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_WB_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_WB x6 <= P_id_WB x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id__plus__hat_1_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->
       P_id__plus__hat_1 x6 x8 <= P_id__plus__hat_1 x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_AND_monotonic :
    forall x8 x6 x5 x7, 
     (0 <= x8)/\ (x8 <= x7) ->
      (0 <= x6)/\ (x6 <= x5) ->P_id_AND x6 x8 <= P_id_AND x5 x7.
   Proof.
     intros x8 x7 x6 x5.
     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_NOT_monotonic :
    forall x6 x5, (0 <= x6)/\ (x6 <= x5) ->P_id_NOT x6 <= P_id_NOT x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Lemma P_id_Marked_val_monotonic :
    forall x6 x5, 
     (0 <= x6)/\ (x6 <= x5) ->P_id_Marked_val x6 <= P_id_Marked_val x5.
   Proof.
     intros x6 x5.
     intros [H_1 H_0].
     
     cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
      (auto with zarith)||(repeat (translate_vars );prove_ineq ).
   Qed.
   
   Definition marked_measure  := 
     InterpZ.marked_measure 0 P_id_0 P_id_bs P_id_and P_id__minus_ P_id_l 
      P_id__plus_ P_id_wb P_id_ge P_id_false P_id_min P_id__sharp_ P_id_size 
      P_id_if P_id_not P_id_n P_id_1 P_id_val P_id_true P_id_max P_id_MIN 
      P_id_0_hat_1 P_id_SIZE P_id_BS P_id_GE P_id_IF P_id_MAX 
      P_id__minus__hat_1 P_id_WB P_id__plus__hat_1 P_id_AND P_id_NOT 
      P_id_Marked_val.
   
   Lemma marked_measure_equation :
    forall t, 
     marked_measure t = match t with
                          | (algebra.Alg.Term algebra.F.id_min (x5::nil)) =>
                           P_id_MIN (measure x5)
                          | (algebra.Alg.Term algebra.F.id_0 (x5::nil)) =>
                           P_id_0_hat_1 (measure x5)
                          | (algebra.Alg.Term algebra.F.id_size (x5::nil)) =>
                           P_id_SIZE (measure x5)
                          | (algebra.Alg.Term algebra.F.id_bs (x5::nil)) =>
                           P_id_BS (measure x5)
                          | (algebra.Alg.Term algebra.F.id_ge (x6::x5::nil)) =>
                           P_id_GE (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_if (x7::x6::
                             x5::nil)) =>
                           P_id_IF (measure x7) (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_max (x5::nil)) =>
                           P_id_MAX (measure x5)
                          | (algebra.Alg.Term algebra.F.id__minus_ (x6::
                             x5::nil)) =>
                           P_id__minus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_wb (x5::nil)) =>
                           P_id_WB (measure x5)
                          | (algebra.Alg.Term algebra.F.id__plus_ (x6::
                             x5::nil)) =>
                           P_id__plus__hat_1 (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_and (x6::x5::nil)) =>
                           P_id_AND (measure x6) (measure x5)
                          | (algebra.Alg.Term algebra.F.id_not (x5::nil)) =>
                           P_id_NOT (measure x5)
                          | (algebra.Alg.Term algebra.F.id_val (x5::nil)) =>
                           P_id_Marked_val (measure x5)
                          | _ => 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_0_monotonic;assumption.
     intros ;apply P_id_bs_monotonic;assumption.
     intros ;apply P_id_and_monotonic;assumption.
     intros ;apply P_id__minus__monotonic;assumption.
     intros ;apply P_id_l_monotonic;assumption.
     intros ;apply P_id__plus__monotonic;assumption.
     intros ;apply P_id_wb_monotonic;assumption.
     intros ;apply P_id_ge_monotonic;assumption.
     intros ;apply P_id_min_monotonic;assumption.
     intros ;apply P_id_size_monotonic;assumption.
     intros ;apply P_id_if_monotonic;assumption.
     intros ;apply P_id_not_monotonic;assumption.
     intros ;apply P_id_n_monotonic;assumption.
     intros ;apply P_id_1_monotonic;assumption.
     intros ;apply P_id_val_monotonic;assumption.
     intros ;apply P_id_max_monotonic;assumption.
     intros ;apply P_id_0_bounded;assumption.
     intros ;apply P_id_bs_bounded;assumption.
     intros ;apply P_id_and_bounded;assumption.
     intros ;apply P_id__minus__bounded;assumption.
     intros ;apply P_id_l_bounded;assumption.
     intros ;apply P_id__plus__bounded;assumption.
     intros ;apply P_id_wb_bounded;assumption.
     intros ;apply P_id_ge_bounded;assumption.
     intros ;apply P_id_false_bounded;assumption.
     intros ;apply P_id_min_bounded;assumption.
     intros ;apply P_id__sharp__bounded;assumption.
     intros ;apply P_id_size_bounded;assumption.
     intros ;apply P_id_if_bounded;assumption.
     intros ;apply P_id_not_bounded;assumption.
     intros ;apply P_id_n_bounded;assumption.
     intros ;apply P_id_1_bounded;assumption.
     intros ;apply P_id_val_bounded;assumption.
     intros ;apply P_id_true_bounded;assumption.
     intros ;apply P_id_max_bounded;assumption.
     apply rules_monotonic.
     intros ;apply P_id_MIN_monotonic;assumption.
     intros ;apply P_id_0_hat_1_monotonic;assumption.
     intros ;apply P_id_SIZE_monotonic;assumption.
     intros ;apply P_id_BS_monotonic;assumption.
     intros ;apply P_id_GE_monotonic;assumption.
     intros ;apply P_id_IF_monotonic;assumption.
     intros ;apply P_id_MAX_monotonic;assumption.
     intros ;apply P_id__minus__hat_1_monotonic;assumption.
     intros ;apply P_id_WB_monotonic;assumption.
     intros ;apply P_id__plus__hat_1_monotonic;assumption.
     intros ;apply P_id_AND_monotonic;assumption.
     intros ;apply P_id_NOT_monotonic;assumption.
     intros ;apply P_id_Marked_val_monotonic;assumption.
   Qed.
   
   Ltac rewrite_and_unfold  :=
    do 2 (rewrite marked_measure_equation);
     repeat (
     match goal with
       |  |- context [measure (algebra.Alg.Term ?f ?t)] =>
        rewrite (measure_equation (algebra.Alg.Term f t))
       | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
        rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
       end
     ).
   
   
   Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_33.
   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_33.
  
  Definition wf_DP_R_xml_0_scc_33  := WF_DP_R_xml_0_scc_33.wf.
  
  
  Lemma acc_DP_R_xml_0_scc_33 :
   forall x y, (DP_R_xml_0_scc_33 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_33).
    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_32;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_31;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_24;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_23;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_16;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_15;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_14;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_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_33.
  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_32;
       econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
     ((eapply acc_DP_R_xml_0_non_scc_31;
        econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
      ((eapply acc_DP_R_xml_0_non_scc_30;
         econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
       ((eapply acc_DP_R_xml_0_non_scc_29;
          econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
        ((eapply acc_DP_R_xml_0_non_scc_28;
           econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
         ((eapply acc_DP_R_xml_0_non_scc_27;
            econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
          ((eapply acc_DP_R_xml_0_non_scc_26;
             econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
           ((eapply acc_DP_R_xml_0_non_scc_25;
              econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
            ((eapply acc_DP_R_xml_0_non_scc_24;
               econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
             ((eapply acc_DP_R_xml_0_non_scc_23;
                econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
              ((eapply acc_DP_R_xml_0_non_scc_22;
                 econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
               ((eapply acc_DP_R_xml_0_non_scc_21;
                  econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                ((eapply acc_DP_R_xml_0_non_scc_20;
                   econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                 ((eapply acc_DP_R_xml_0_non_scc_19;
                    econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
                  ((eapply acc_DP_R_xml_0_non_scc_18;
                     econstructor 
                     (eassumption)||(algebra.Alg_ext.star_refl' ))||
                   ((eapply acc_DP_R_xml_0_non_scc_17;
                      econstructor 
                      (eassumption)||(algebra.Alg_ext.star_refl' ))||
                    ((eapply acc_DP_R_xml_0_non_scc_16;
                       econstructor 
                       (eassumption)||(algebra.Alg_ext.star_refl' ))||
                     ((eapply acc_DP_R_xml_0_non_scc_15;
                        econstructor 
                        (eassumption)||(algebra.Alg_ext.star_refl' ))||
                      ((eapply acc_DP_R_xml_0_non_scc_14;
                         econstructor 
                         (eassumption)||(algebra.Alg_ext.star_refl' ))||
                       ((eapply acc_DP_R_xml_0_non_scc_13;
                          econstructor 
                          (eassumption)||(algebra.Alg_ext.star_refl' ))||
                        ((eapply acc_DP_R_xml_0_non_scc_12;
                           econstructor 
                           (eassumption)||(algebra.Alg_ext.star_refl' ))||
                         ((eapply acc_DP_R_xml_0_non_scc_11;
                            econstructor 
                            (eassumption)||(algebra.Alg_ext.star_refl' ))||
                          ((eapply acc_DP_R_xml_0_non_scc_10;
                             econstructor 
                             (eassumption)||(algebra.Alg_ext.star_refl' ))||
                           ((eapply acc_DP_R_xml_0_non_scc_9;
                              econstructor 
                              (eassumption)||(algebra.Alg_ext.star_refl' ))||
                            ((eapply acc_DP_R_xml_0_non_scc_8;
                               econstructor 
                               (eassumption)||(algebra.Alg_ext.star_refl' ))||
                             ((eapply acc_DP_R_xml_0_non_scc_7;
                                econstructor 
                                (eassumption)||(algebra.Alg_ext.star_refl' ))||
                              ((eapply acc_DP_R_xml_0_non_scc_6;
                                 econstructor 
                                 (eassumption)||(algebra.Alg_ext.star_refl' ))||
                               ((eapply acc_DP_R_xml_0_non_scc_5;
                                  econstructor 
                                  (eassumption)||
                                  (algebra.Alg_ext.star_refl' ))||
                                ((eapply acc_DP_R_xml_0_non_scc_4;
                                   econstructor 
                                   (eassumption)||
                                   (algebra.Alg_ext.star_refl' ))||
                                 ((eapply acc_DP_R_xml_0_non_scc_3;
                                    econstructor 
                                    (eassumption)||
                                    (algebra.Alg_ext.star_refl' ))||
                                  ((eapply acc_DP_R_xml_0_non_scc_2;
                                     econstructor 
                                     (eassumption)||
                                     (algebra.Alg_ext.star_refl' ))||
                                   ((eapply acc_DP_R_xml_0_non_scc_1;
                                      econstructor 
                                      (eassumption)||
                                      (algebra.Alg_ext.star_refl' ))||
                                    ((eapply acc_DP_R_xml_0_non_scc_0;
                                       econstructor 
                                       (eassumption)||
                                       (algebra.Alg_ext.star_refl' ))||
                                     ((eapply acc_DP_R_xml_0_scc_33;
                                        econstructor 
                                        (eassumption)||
                                        (algebra.Alg_ext.star_refl' ))||
                                      ((eapply acc_DP_R_xml_0_scc_32;
                                         econstructor 
                                         (eassumption)||
                                         (algebra.Alg_ext.star_refl' ))||
                                       ((eapply acc_DP_R_xml_0_scc_31;
                                          econstructor 
                                          (eassumption)||
                                          (algebra.Alg_ext.star_refl' ))||
                                        ((eapply acc_DP_R_xml_0_scc_30;
                                           econstructor 
                                           (eassumption)||
                                           (algebra.Alg_ext.star_refl' ))||
                                         ((eapply acc_DP_R_xml_0_scc_29;
                                            econstructor 
                                            (eassumption)||
                                            (algebra.Alg_ext.star_refl' ))||
                                          ((eapply acc_DP_R_xml_0_scc_28;
                                             econstructor 
                                             (eassumption)||
                                             (algebra.Alg_ext.star_refl' ))||
                                           ((eapply acc_DP_R_xml_0_scc_27;
                                              econstructor 
                                              (eassumption)||
                                              (algebra.Alg_ext.star_refl' ))||
                                            ((eapply acc_DP_R_xml_0_scc_26;
                                               econstructor 
                                               (eassumption)||
                                               (algebra.Alg_ext.star_refl' ))||
                                             ((eapply acc_DP_R_xml_0_scc_25;
                                                econstructor 
                                                (eassumption)||
                                                (algebra.Alg_ext.star_refl' ))||
                                              ((eapply acc_DP_R_xml_0_scc_24;
                                                 econstructor 
                                                 (eassumption)||
                                                 (algebra.Alg_ext.star_refl' ))||
                                               ((eapply acc_DP_R_xml_0_scc_23;
                                                  econstructor 
                                                  (eassumption)||
                                                  (algebra.Alg_ext.star_refl' ))||
                                                ((eapply acc_DP_R_xml_0_scc_22;
                                                   
                                                   econstructor 
                                                   (eassumption)||
                                                   (algebra.Alg_ext.star_refl' ))||
                                                 ((eapply acc_DP_R_xml_0_scc_21;
                                                    
                                                    econstructor 
                                                    (eassumption)||
                                                    (algebra.Alg_ext.star_refl' ))||
                                                  ((eapply acc_DP_R_xml_0_scc_20;
                                                     
                                                     econstructor 
                                                     (eassumption)||
                                                     (algebra.Alg_ext.star_refl' ))||
                                                   ((eapply acc_DP_R_xml_0_scc_19;
                                                      
                                                      econstructor 
                                                      (eassumption)||
                                                      (algebra.Alg_ext.star_refl' ))||
                                                    ((eapply acc_DP_R_xml_0_scc_18;
                                                       
                                                       econstructor 
                                                       (eassumption)||
                                                       (algebra.Alg_ext.star_refl' ))||
                                                     ((eapply acc_DP_R_xml_0_scc_17;
                                                        
                                                        econstructor 
                                                        (eassumption)||
                                                        (algebra.Alg_ext.star_refl' ))||
                                                      ((eapply acc_DP_R_xml_0_scc_16;
                                                         
                                                         econstructor 
                                                         (eassumption)||
                                                         (algebra.Alg_ext.star_refl' ))||
                                                       ((eapply acc_DP_R_xml_0_scc_15;
                                                          
                                                          econstructor 
                                                          (eassumption)||
                                                          (algebra.Alg_ext.star_refl' ))||
                                                        ((eapply acc_DP_R_xml_0_scc_14;
                                                           
                                                           econstructor 
                                                           (eassumption)||
                                                           (algebra.Alg_ext.star_refl' ))||
                                                         ((eapply acc_DP_R_xml_0_scc_13;
                                                            
                                                            econstructor 
                                                            (eassumption)||
                                                            (algebra.Alg_ext.star_refl' ))||
                                                          ((eapply acc_DP_R_xml_0_scc_12;
                                                             
                                                             econstructor 
                                                             (eassumption)||
                                                             (algebra.Alg_ext.star_refl' ))||
                                                           ((eapply acc_DP_R_xml_0_scc_11;
                                                              
                                                              econstructor 
                                                              (eassumption)||
                                                              (algebra.Alg_ext.star_refl' ))||
                                                            ((eapply 
                                                              acc_DP_R_xml_0_scc_10;
                                                               
                                                               econstructor 
                                                               (eassumption)||
                                                               (algebra.Alg_ext.star_refl' ))||
                                                             ((eapply 
                                                               acc_DP_R_xml_0_scc_9;
                                                                
                                                                econstructor 
                                                                (eassumption)||
                                                                (algebra.Alg_ext.star_refl' ))||
                                                              ((eapply 
                                                                acc_DP_R_xml_0_scc_8;
                                                                 
                                                                 econstructor 
                                                                 (eassumption)||
                                                                 (algebra.Alg_ext.star_refl' ))||
                                                               ((eapply 
                                                                 acc_DP_R_xml_0_scc_7;
                                                                  
                                                                  econstructor 
                                                                  (eassumption)||
                                                                  (algebra.Alg_ext.star_refl' ))||
                                                                ((eapply 
                                                                  acc_DP_R_xml_0_scc_6;
                                                                   
                                                                   econstructor 
                                                                   (eassumption)||
                                                                   (algebra.Alg_ext.star_refl' ))||
                                                                 ((eapply 
                                                                   acc_DP_R_xml_0_scc_5;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                  ((eapply 
                                                                    acc_DP_R_xml_0_scc_4;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                   ((
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_3;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_2;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_1;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    eapply 
                                                                    acc_DP_R_xml_0_scc_0;
                                                                    
                                                                    econstructor 
                                                                    (
                                                                    eassumption)||
                                                                    (
                                                                    algebra.Alg_ext.star_refl' ))||
                                                                    (
                                                                    (
                                                                    R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
                                                                    (
                                                                    fail)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
  Qed.
 End WF_DP_R_xml_0.
 
 Definition wf_H  := WF_DP_R_xml_0.wf.
 
 Lemma wf :
  well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
 Proof.
   apply ddp.dp_criterion.
   apply R_xml_0_deep_rew.R_xml_0_non_var.
   apply R_xml_0_deep_rew.R_xml_0_reg.
   
   intros ;
    apply (ddp.constructor_defined_dec _ _ 
            R_xml_0_deep_rew.R_xml_0_rules_included).
   refine (Inclusion.wf_incl _ _ _ _ wf_H).
   intros x y H.
   destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
   
   destruct (ddp.dp_list_complete _ _ 
              R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
    as [x' [y' [sigma [h1 [h2 h3]]]]].
   clear H3.
   subst.
   vm_compute in h3|-.
   let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
 Qed.
End WF_R_xml_0_deep_rew.


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