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_din *)
| id_din : symb
(* id_u32 *)
| id_u32 : symb
(* id_dout *)
| id_dout : symb
(* id_plus *)
| id_plus : symb
(* id_u42 *)
| id_u42 : symb
(* id_times *)
| id_times : symb
(* id_der *)
| id_der : symb
(* id_u41 *)
| id_u41 : symb
(* id_u22 *)
| id_u22 : symb
(* id_u21 *)
| id_u21 : symb
(* id_u31 *)
| id_u31 : symb
.
Definition symb_eq_bool (f1 f2:symb) : bool :=
match f1,f2 with
| id_din,id_din => true
| id_u32,id_u32 => true
| id_dout,id_dout => true
| id_plus,id_plus => true
| id_u42,id_u42 => true
| id_times,id_times => true
| id_der,id_der => true
| id_u41,id_u41 => true
| id_u22,id_u22 => true
| id_u21,id_u21 => true
| id_u31,id_u31 => 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_din,id_din => refl_equal _
| id_u32,id_u32 => refl_equal _
| id_dout,id_dout => refl_equal _
| id_plus,id_plus => refl_equal _
| id_u42,id_u42 => refl_equal _
| id_times,id_times => refl_equal _
| id_der,id_der => refl_equal _
| id_u41,id_u41 => refl_equal _
| id_u22,id_u22 => refl_equal _
| id_u21,id_u21 => refl_equal _
| id_u31,id_u31 => refl_equal _
| _,_ => _
end;intros abs;discriminate.
Defined.
Definition arity (f:symb) :=
match f with
| id_din => term.Free 1
| id_u32 => term.Free 4
| id_dout => term.Free 1
| id_plus => term.Free 2
| id_u42 => term.Free 3
| id_times => term.Free 2
| id_der => term.Free 1
| id_u41 => term.Free 2
| id_u22 => term.Free 4
| id_u21 => term.Free 3
| id_u31 => term.Free 3
end.
Definition symb_order (f1 f2:symb) : bool :=
match f1,f2 with
| id_din,id_din => true
| id_din,id_u32 => false
| id_din,id_dout => false
| id_din,id_plus => false
| id_din,id_u42 => false
| id_din,id_times => false
| id_din,id_der => false
| id_din,id_u41 => false
| id_din,id_u22 => false
| id_din,id_u21 => false
| id_din,id_u31 => false
| id_u32,id_din => true
| id_u32,id_u32 => true
| id_u32,id_dout => false
| id_u32,id_plus => false
| id_u32,id_u42 => false
| id_u32,id_times => false
| id_u32,id_der => false
| id_u32,id_u41 => false
| id_u32,id_u22 => false
| id_u32,id_u21 => false
| id_u32,id_u31 => false
| id_dout,id_din => true
| id_dout,id_u32 => true
| id_dout,id_dout => true
| id_dout,id_plus => false
| id_dout,id_u42 => false
| id_dout,id_times => false
| id_dout,id_der => false
| id_dout,id_u41 => false
| id_dout,id_u22 => false
| id_dout,id_u21 => false
| id_dout,id_u31 => false
| id_plus,id_din => true
| id_plus,id_u32 => true
| id_plus,id_dout => true
| id_plus,id_plus => true
| id_plus,id_u42 => false
| id_plus,id_times => false
| id_plus,id_der => false
| id_plus,id_u41 => false
| id_plus,id_u22 => false
| id_plus,id_u21 => false
| id_plus,id_u31 => false
| id_u42,id_din => true
| id_u42,id_u32 => true
| id_u42,id_dout => true
| id_u42,id_plus => true
| id_u42,id_u42 => true
| id_u42,id_times => false
| id_u42,id_der => false
| id_u42,id_u41 => false
| id_u42,id_u22 => false
| id_u42,id_u21 => false
| id_u42,id_u31 => false
| id_times,id_din => true
| id_times,id_u32 => true
| id_times,id_dout => true
| id_times,id_plus => true
| id_times,id_u42 => true
| id_times,id_times => true
| id_times,id_der => false
| id_times,id_u41 => false
| id_times,id_u22 => false
| id_times,id_u21 => false
| id_times,id_u31 => false
| id_der,id_din => true
| id_der,id_u32 => true
| id_der,id_dout => true
| id_der,id_plus => true
| id_der,id_u42 => true
| id_der,id_times => true
| id_der,id_der => true
| id_der,id_u41 => false
| id_der,id_u22 => false
| id_der,id_u21 => false
| id_der,id_u31 => false
| id_u41,id_din => true
| id_u41,id_u32 => true
| id_u41,id_dout => true
| id_u41,id_plus => true
| id_u41,id_u42 => true
| id_u41,id_times => true
| id_u41,id_der => true
| id_u41,id_u41 => true
| id_u41,id_u22 => false
| id_u41,id_u21 => false
| id_u41,id_u31 => false
| id_u22,id_din => true
| id_u22,id_u32 => true
| id_u22,id_dout => true
| id_u22,id_plus => true
| id_u22,id_u42 => true
| id_u22,id_times => true
| id_u22,id_der => true
| id_u22,id_u41 => true
| id_u22,id_u22 => true
| id_u22,id_u21 => false
| id_u22,id_u31 => false
| id_u21,id_din => true
| id_u21,id_u32 => true
| id_u21,id_dout => true
| id_u21,id_plus => true
| id_u21,id_u42 => true
| id_u21,id_times => true
| id_u21,id_der => true
| id_u21,id_u41 => true
| id_u21,id_u22 => true
| id_u21,id_u21 => true
| id_u21,id_u31 => false
| id_u31,id_din => true
| id_u31,id_u32 => true
| id_u31,id_dout => true
| id_u31,id_plus => true
| id_u31,id_u42 => true
| id_u31,id_times => true
| id_u31,id_der => true
| id_u31,id_u41 => true
| id_u31,id_u22 => true
| id_u31,id_u21 => true
| id_u31,id_u31 => 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 :=
(* din(der(plus(X_,Y_))) -> u21(din(der(X_)),X_,Y_) *)
| R_xml_0_rule_0 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u21 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 1)::nil))::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))::nil))::nil))
(* u21(dout(DX_),X_,Y_) -> u22(din(der(Y_)),X_,Y_,DX_) *)
| R_xml_0_rule_1 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u22 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 2)::nil))::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::(algebra.Alg.Var 3)::nil))
(algebra.Alg.Term algebra.F.id_u21 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))
(* u22(dout(DY_),X_,Y_,DX_) -> dout(plus(DX_,DY_)) *)
| R_xml_0_rule_2 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_dout ((algebra.Alg.Term
algebra.F.id_plus ((algebra.Alg.Var 3)::
(algebra.Alg.Var 4)::nil))::nil))
(algebra.Alg.Term algebra.F.id_u22 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
(algebra.Alg.Var 3)::nil))
(* din(der(times(X_,Y_))) -> u31(din(der(X_)),X_,Y_) *)
| R_xml_0_rule_3 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u31 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 1)::nil))::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_times ((algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))::nil))::nil))
(* u31(dout(DX_),X_,Y_) -> u32(din(der(Y_)),X_,Y_,DX_) *)
| R_xml_0_rule_4 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u32 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 2)::nil))::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::(algebra.Alg.Var 3)::nil))
(algebra.Alg.Term algebra.F.id_u31 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))
(* u32(dout(DY_),X_,Y_,DX_) -> dout(plus(times(X_,DY_),times(Y_,DX_))) *)
| R_xml_0_rule_5 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_dout ((algebra.Alg.Term
algebra.F.id_plus ((algebra.Alg.Term algebra.F.id_times
((algebra.Alg.Var 1)::(algebra.Alg.Var 4)::nil))::
(algebra.Alg.Term algebra.F.id_times
((algebra.Alg.Var 2)::
(algebra.Alg.Var 3)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_u32 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
(algebra.Alg.Var 3)::nil))
(* din(der(der(X_))) -> u41(din(der(X_)),X_) *)
| R_xml_0_rule_6 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 1)::nil))::nil))::
(algebra.Alg.Var 1)::nil))
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 1)::nil))::nil))::nil))
(* u41(dout(DX_),X_) -> u42(din(der(DX_)),X_,DX_) *)
| R_xml_0_rule_7 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_u42 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 3)::nil))::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 3)::nil))
(algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::nil))
(* u42(dout(DDX_),X_,DX_) -> dout(DDX_) *)
| R_xml_0_rule_8 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 5)::nil))
(algebra.Alg.Term algebra.F.id_u42 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 3)::nil))
.
Definition R_xml_0_rule_as_list_0 :=
((algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_plus ((algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))::nil))::nil)),
(algebra.Alg.Term algebra.F.id_u21 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 1)::nil))::nil))::
(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)))::nil.
Definition R_xml_0_rule_as_list_1 :=
((algebra.Alg.Term algebra.F.id_u21 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil)),
(algebra.Alg.Term algebra.F.id_u22 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 2)::nil))::nil))::
(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::(algebra.Alg.Var 3)::nil)))::
R_xml_0_rule_as_list_0.
Definition R_xml_0_rule_as_list_2 :=
((algebra.Alg.Term algebra.F.id_u22 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
(algebra.Alg.Var 3)::nil)),
(algebra.Alg.Term algebra.F.id_dout ((algebra.Alg.Term algebra.F.id_plus
((algebra.Alg.Var 3)::(algebra.Alg.Var 4)::nil))::nil)))::
R_xml_0_rule_as_list_1.
Definition R_xml_0_rule_as_list_3 :=
((algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_times ((algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil))::nil))::nil)),
(algebra.Alg.Term algebra.F.id_u31 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 1)::nil))::nil))::
(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)))::R_xml_0_rule_as_list_2
.
Definition R_xml_0_rule_as_list_4 :=
((algebra.Alg.Term algebra.F.id_u31 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 2)::nil)),
(algebra.Alg.Term algebra.F.id_u32 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 2)::nil))::nil))::
(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::(algebra.Alg.Var 3)::nil)))::
R_xml_0_rule_as_list_3.
Definition R_xml_0_rule_as_list_5 :=
((algebra.Alg.Term algebra.F.id_u32 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 1)::(algebra.Alg.Var 2)::
(algebra.Alg.Var 3)::nil)),
(algebra.Alg.Term algebra.F.id_dout ((algebra.Alg.Term algebra.F.id_plus
((algebra.Alg.Term algebra.F.id_times ((algebra.Alg.Var 1)::
(algebra.Alg.Var 4)::nil))::(algebra.Alg.Term algebra.F.id_times
((algebra.Alg.Var 2)::(algebra.Alg.Var 3)::nil))::nil))::nil)))::
R_xml_0_rule_as_list_4.
Definition R_xml_0_rule_as_list_6 :=
((algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Term algebra.F.id_der
((algebra.Alg.Var 1)::nil))::nil))::nil)),
(algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 1)::nil))::nil))::
(algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_5.
Definition R_xml_0_rule_as_list_7 :=
((algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 3)::nil))::(algebra.Alg.Var 1)::nil)),
(algebra.Alg.Term algebra.F.id_u42 ((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Var 3)::nil))::nil))::
(algebra.Alg.Var 1)::(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_u42 ((algebra.Alg.Term algebra.F.id_dout
((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 1)::
(algebra.Alg.Var 3)::nil)),
(algebra.Alg.Term algebra.F.id_dout ((algebra.Alg.Var 5)::nil)))::
R_xml_0_rule_as_list_7.
Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_8.
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.or10_equiv in H|-.
case H;clear H;intros H.
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_4 (x7 x8 x9 x10:Prop) :
Prop :=
| conj_4 : x7->x8->x9->x10->and_4 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_4 (forall x8,
t = (algebra.Alg.Term algebra.F.id_dout (x8::nil)) ->
exists x7,
t' = (algebra.Alg.Term algebra.F.id_dout (x7::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x7 x8))
(forall x8 x10,
t = (algebra.Alg.Term algebra.F.id_plus (x8::x10::nil)) ->
exists x7,
exists x9,
t' = (algebra.Alg.Term algebra.F.id_plus (x7::x9::nil))/\
(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 x8 x10,
t = (algebra.Alg.Term algebra.F.id_times (x8::x10::nil)) ->
exists x7,
exists x9,
t' = (algebra.Alg.Term algebra.F.id_times (x7::x9::nil))/\
(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 x8,
t = (algebra.Alg.Term algebra.F.id_der (x8::nil)) ->
exists x7,
t' = (algebra.Alg.Term algebra.F.id_der (x7::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x7 x8)).
Proof.
intros t t' H.
induction H as [|y IH z z_to_y] using
closure_extension.refl_trans_clos_ind2.
constructor 1.
intros x8 H;exists x8;intuition;constructor 1.
intros x8 x10 H;exists x8;exists x10;intuition;constructor 1.
intros x8 x10 H;exists x8;exists x10;intuition;constructor 1.
intros x8 H;exists x8;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_dout H_id_plus H_id_times H_id_der].
constructor.
clear H_id_plus H_id_times H_id_der;intros x8 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 x8 |- _ =>
destruct (H_id_dout y (refl_equal _)) as [x7];intros ;intuition;
exists x7;intuition;eapply closure_extension.refl_trans_clos_R;
eassumption
end
.
clear H_id_dout H_id_times H_id_der;intros 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 x8 |- _ =>
destruct (H_id_plus y x10 (refl_equal _)) as [x7 [x9]];intros ;
intuition;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_plus x8 y (refl_equal _)) as [x7 [x9]];intros ;
intuition;exists x7;exists x9;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
clear H_id_dout H_id_plus H_id_der;intros 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 x8 |- _ =>
destruct (H_id_times y x10 (refl_equal _)) as [x7 [x9]];intros ;
intuition;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_times x8 y (refl_equal _)) as [x7 [x9]];intros ;
intuition;exists x7;exists x9;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
clear H_id_dout H_id_plus H_id_times;intros x8 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 x8 |- _ =>
destruct (H_id_der y (refl_equal _)) as [x7];intros ;intuition;
exists x7;intuition;eapply closure_extension.refl_trans_clos_R;
eassumption
end
.
Qed.
Lemma id_dout_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x8,
t = (algebra.Alg.Term algebra.F.id_dout (x8::nil)) ->
exists x7,
t' = (algebra.Alg.Term algebra.F.id_dout (x7::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x7 x8).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_plus_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x8 x10,
t = (algebra.Alg.Term algebra.F.id_plus (x8::x10::nil)) ->
exists x7,
exists x9,
t' = (algebra.Alg.Term algebra.F.id_plus (x7::x9::nil))/\
(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_times_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x8 x10,
t = (algebra.Alg.Term algebra.F.id_times (x8::x10::nil)) ->
exists x7,
exists x9,
t' = (algebra.Alg.Term algebra.F.id_times (x7::x9::nil))/\
(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_der_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x8,
t = (algebra.Alg.Term algebra.F.id_der (x8::nil)) ->
exists x7,
t' = (algebra.Alg.Term algebra.F.id_der (x7::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x7 x8).
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_dout (?x7::nil)) |- _ =>
let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_dout_is_R_xml_0_constructor H (refl_equal _)) as
[x7 [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_plus (?x8::?x7::nil)) |- _ =>
let x8 := fresh "x" in
(let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_plus_is_R_xml_0_constructor H (refl_equal _)) as
[x8 [x7 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_times (?x8::?x7::nil)) |- _ =>
let x8 := fresh "x" in
(let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_times_is_R_xml_0_constructor H (refl_equal _))
as [x8 [x7 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_der (?x7::nil)) |- _ =>
let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_der_is_R_xml_0_constructor H (refl_equal _)) as
[x7 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))
end
.
Ltac simplify_star_reduction_R_xml_0 :=
match goal with
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_dout (?x7::nil)) |- _ =>
let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_dout_is_R_xml_0_constructor H (refl_equal _)) as
[x7 [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_plus (?x8::?x7::nil)) |- _ =>
let x8 := fresh "x" in
(let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_plus_is_R_xml_0_constructor H (refl_equal _)) as
[x8 [x7 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_times (?x8::?x7::nil)) |- _ =>
let x8 := fresh "x" in
(let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_times_is_R_xml_0_constructor H (refl_equal _))
as [x8 [x7 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_der (?x7::nil)) |- _ =>
let x7 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_der_is_R_xml_0_constructor H (refl_equal _)) as
[x7 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))
end
.
End R_xml_0_deep_rew.
Module InterpGen := interp.Interp(algebra.EQT).
Module ddp := dp.MakeDP(algebra.EQT).
Module SymbType. Definition A := algebra.Alg.F.Symb.A. End SymbType.
Module Symb_more_list := more_list_extention.Make(SymbType)(algebra.Alg.F.Symb).
Module SymbSet := list_set.Make(algebra.F.Symb).
Module Interp.
Section S.
Require Import interp.
Hypothesis A : Type.
Hypothesis Ale Alt Aeq : A -> A -> Prop.
Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
Hypothesis A0 : A.
Notation Local "a <= b" := (Ale a b).
Hypothesis P_id_din : A ->A.
Hypothesis P_id_u32 : A ->A ->A ->A ->A.
Hypothesis P_id_dout : A ->A.
Hypothesis P_id_plus : A ->A ->A.
Hypothesis P_id_u42 : A ->A ->A ->A.
Hypothesis P_id_times : A ->A ->A.
Hypothesis P_id_der : A ->A.
Hypothesis P_id_u41 : A ->A ->A.
Hypothesis P_id_u22 : A ->A ->A ->A ->A.
Hypothesis P_id_u21 : A ->A ->A ->A.
Hypothesis P_id_u31 : A ->A ->A ->A.
Hypothesis P_id_din_monotonic :
forall x8 x7, (A0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Hypothesis P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Hypothesis P_id_dout_monotonic :
forall x8 x7, (A0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Hypothesis P_id_plus_monotonic :
forall x8 x10 x9 x7,
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Hypothesis P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Hypothesis P_id_times_monotonic :
forall x8 x10 x9 x7,
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Hypothesis P_id_der_monotonic :
forall x8 x7, (A0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Hypothesis P_id_u41_monotonic :
forall x8 x10 x9 x7,
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Hypothesis P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Hypothesis P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Hypothesis P_id_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Hypothesis P_id_din_bounded : forall x7, (A0 <= x7) ->A0 <= P_id_din x7.
Hypothesis P_id_u32_bounded :
forall x8 x10 x9 x7,
(A0 <= x7) ->
(A0 <= x8) ->(A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_u32 x10 x9 x8 x7.
Hypothesis P_id_dout_bounded : forall x7, (A0 <= x7) ->A0 <= P_id_dout x7.
Hypothesis P_id_plus_bounded :
forall x8 x7, (A0 <= x7) ->(A0 <= x8) ->A0 <= P_id_plus x8 x7.
Hypothesis P_id_u42_bounded :
forall x8 x9 x7,
(A0 <= x7) ->(A0 <= x8) ->(A0 <= x9) ->A0 <= P_id_u42 x9 x8 x7.
Hypothesis P_id_times_bounded :
forall x8 x7, (A0 <= x7) ->(A0 <= x8) ->A0 <= P_id_times x8 x7.
Hypothesis P_id_der_bounded : forall x7, (A0 <= x7) ->A0 <= P_id_der x7.
Hypothesis P_id_u41_bounded :
forall x8 x7, (A0 <= x7) ->(A0 <= x8) ->A0 <= P_id_u41 x8 x7.
Hypothesis P_id_u22_bounded :
forall x8 x10 x9 x7,
(A0 <= x7) ->
(A0 <= x8) ->(A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_u22 x10 x9 x8 x7.
Hypothesis P_id_u21_bounded :
forall x8 x9 x7,
(A0 <= x7) ->(A0 <= x8) ->(A0 <= x9) ->A0 <= P_id_u21 x9 x8 x7.
Hypothesis P_id_u31_bounded :
forall x8 x9 x7,
(A0 <= x7) ->(A0 <= x8) ->(A0 <= x9) ->A0 <= P_id_u31 x9 x8 x7.
Fixpoint measure t { struct t } :=
match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => A0
end.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din => P_id_din
| algebra.F.id_u32 => P_id_u32
| algebra.F.id_dout => P_id_dout
| algebra.F.id_plus => P_id_plus
| algebra.F.id_u42 => P_id_u42
| algebra.F.id_times => P_id_times
| algebra.F.id_der => P_id_der
| algebra.F.id_u41 => P_id_u41
| algebra.F.id_u22 => P_id_u22
| algebra.F.id_u21 => P_id_u21
| algebra.F.id_u31 => P_id_u31
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_din =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_u32 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::nil => _
| _::_::_::_::_::_ => _
end
| algebra.F.id_dout =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_plus =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_u42 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_times =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_der =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_u41 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_u22 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::nil => _
| _::_::_::_::_::_ => _
end
| algebra.F.id_u21 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_u31 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
try (reflexivity );f_equal ;auto.
Qed.
Lemma measure_bounded : forall t, A0 <= measure t.
Proof.
intros t.
rewrite same_measure in |-*.
apply (InterpGen.measure_bounded Aop).
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_din_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u32_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_dout_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u42_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_times_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_der_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u41_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u22_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u21_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u31_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_din_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u32_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_dout_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_plus_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u42_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_times_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_der_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u41_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u22_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u21_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u31_monotonic;assumption.
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_din_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u32_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_dout_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u42_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_times_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_der_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u41_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u22_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u21_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u31_bounded;assumption.
intros .
do 2 (rewrite <- same_measure in |-*).
apply rules_monotonic;assumption.
assumption.
Qed.
Hypothesis P_id_U41 : A ->A ->A.
Hypothesis P_id_U21 : A ->A ->A ->A.
Hypothesis P_id_U31 : A ->A ->A ->A.
Hypothesis P_id_U42 : A ->A ->A ->A.
Hypothesis P_id_U22 : A ->A ->A ->A ->A.
Hypothesis P_id_DIN : A ->A.
Hypothesis P_id_U32 : A ->A ->A ->A ->A.
Hypothesis P_id_U41_monotonic :
forall x8 x10 x9 x7,
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Hypothesis P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Hypothesis P_id_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Hypothesis P_id_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Hypothesis P_id_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Hypothesis P_id_DIN_monotonic :
forall x8 x7, (A0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Hypothesis P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
(A0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Definition marked_measure t :=
match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_U21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_U31 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_U42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::x7::nil)) =>
P_id_U22 (measure x10) (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::x7::nil)) =>
P_id_U32 (measure x10) (measure x9) (measure x8) (measure x7)
| _ => measure t
end.
Definition Marked_pols :
forall f,
(algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) ->
InterpGen.Pol_type A (InterpGen.get_arity f).
Proof.
intros f H.
apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
apply (Symb_more_list.change_in algebra.F.symb_order) in H .
set (u := (Symb_more_list.qs algebra.F.symb_order
(Symb_more_list.XSet.remove_red
(ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
vm_compute in u .
unfold u in * .
clear u .
unfold more_list.mem_bool in H .
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9 x8 x7;apply (P_id_U31 x9 x8 x7).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9 x8 x7;apply (P_id_U21 x9 x8 x7).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x10 x9 x8 x7;apply (P_id_U22 x10 x9 x8 x7).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x8 x7;apply (P_id_U41 x8 x7).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9 x8 x7;apply (P_id_U42 x9 x8 x7).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x10 x9 x8 x7;apply (P_id_U32 x10 x9 x8 x7).
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;apply (P_id_DIN x7).
discriminate H.
Defined.
Lemma same_marked_measure :
forall t,
marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols
(ddp.defined_dec _ _
R_xml_0_deep_rew.R_xml_0_rules_included) t.
Proof.
intros [a| f l].
simpl in |-*.
unfold eq_rect_r, eq_rect, sym_eq in |-*.
reflexivity .
refine match f with
| algebra.F.id_din =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_u32 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::nil => _
| _::_::_::_::_::_ => _
end
| algebra.F.id_dout =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_plus =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_u42 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_times =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_der =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_u41 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_u22 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::nil => _
| _::_::_::_::_::_ => _
end
| algebra.F.id_u21 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_u31 =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
end.
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
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 .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
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_u41 (x8::x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_U21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_U31 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_U42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_U22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_U32 (measure x10) (measure x9) (measure x8)
(measure x7)
| _ => 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_din_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u32_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_dout_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_plus_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u42_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_times_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_der_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u41_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u22_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u21_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_u31_monotonic;assumption.
clear f.
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_din_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u32_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_dout_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_plus_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u42_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_times_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_der_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u41_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u22_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u21_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_u31_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_U31_monotonic;assumption.
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros ;apply P_id_U21_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_U22_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_U41_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_U42_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_U32_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_DIN_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_din : Z ->Z.
Hypothesis P_id_u32 : Z ->Z ->Z ->Z ->Z.
Hypothesis P_id_dout : Z ->Z.
Hypothesis P_id_plus : Z ->Z ->Z.
Hypothesis P_id_u42 : Z ->Z ->Z ->Z.
Hypothesis P_id_times : Z ->Z ->Z.
Hypothesis P_id_der : Z ->Z.
Hypothesis P_id_u41 : Z ->Z ->Z.
Hypothesis P_id_u22 : Z ->Z ->Z ->Z ->Z.
Hypothesis P_id_u21 : Z ->Z ->Z ->Z.
Hypothesis P_id_u31 : Z ->Z ->Z ->Z.
Hypothesis P_id_din_monotonic :
forall x8 x7, (min_value <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Hypothesis P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Hypothesis P_id_dout_monotonic :
forall x8 x7,
(min_value <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Hypothesis P_id_plus_monotonic :
forall x8 x10 x9 x7,
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Hypothesis P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Hypothesis P_id_times_monotonic :
forall x8 x10 x9 x7,
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Hypothesis P_id_der_monotonic :
forall x8 x7, (min_value <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Hypothesis P_id_u41_monotonic :
forall x8 x10 x9 x7,
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Hypothesis P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Hypothesis P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Hypothesis P_id_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Hypothesis P_id_din_bounded :
forall x7, (min_value <= x7) ->min_value <= P_id_din x7.
Hypothesis P_id_u32_bounded :
forall x8 x10 x9 x7,
(min_value <= x7) ->
(min_value <= x8) ->
(min_value <= x9) ->
(min_value <= x10) ->min_value <= P_id_u32 x10 x9 x8 x7.
Hypothesis P_id_dout_bounded :
forall x7, (min_value <= x7) ->min_value <= P_id_dout x7.
Hypothesis P_id_plus_bounded :
forall x8 x7,
(min_value <= x7) ->(min_value <= x8) ->min_value <= P_id_plus x8 x7.
Hypothesis P_id_u42_bounded :
forall x8 x9 x7,
(min_value <= x7) ->
(min_value <= x8) ->(min_value <= x9) ->min_value <= P_id_u42 x9 x8 x7.
Hypothesis P_id_times_bounded :
forall x8 x7,
(min_value <= x7) ->(min_value <= x8) ->min_value <= P_id_times x8 x7.
Hypothesis P_id_der_bounded :
forall x7, (min_value <= x7) ->min_value <= P_id_der x7.
Hypothesis P_id_u41_bounded :
forall x8 x7,
(min_value <= x7) ->(min_value <= x8) ->min_value <= P_id_u41 x8 x7.
Hypothesis P_id_u22_bounded :
forall x8 x10 x9 x7,
(min_value <= x7) ->
(min_value <= x8) ->
(min_value <= x9) ->
(min_value <= x10) ->min_value <= P_id_u22 x10 x9 x8 x7.
Hypothesis P_id_u21_bounded :
forall x8 x9 x7,
(min_value <= x7) ->
(min_value <= x8) ->(min_value <= x9) ->min_value <= P_id_u21 x9 x8 x7.
Hypothesis P_id_u31_bounded :
forall x8 x9 x7,
(min_value <= x7) ->
(min_value <= x8) ->(min_value <= x9) ->min_value <= P_id_u31 x9 x8 x7.
Definition measure :=
Interp.measure min_value P_id_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Hypothesis P_id_U41 : Z ->Z ->Z.
Hypothesis P_id_U21 : Z ->Z ->Z ->Z.
Hypothesis P_id_U31 : Z ->Z ->Z ->Z.
Hypothesis P_id_U42 : Z ->Z ->Z ->Z.
Hypothesis P_id_U22 : Z ->Z ->Z ->Z ->Z.
Hypothesis P_id_DIN : Z ->Z.
Hypothesis P_id_U32 : Z ->Z ->Z ->Z ->Z.
Hypothesis P_id_U41_monotonic :
forall x8 x10 x9 x7,
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Hypothesis P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Hypothesis P_id_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Hypothesis P_id_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Hypothesis P_id_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Hypothesis P_id_DIN_monotonic :
forall x8 x7, (min_value <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Hypothesis P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
(min_value <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Definition marked_measure :=
Interp.marked_measure min_value P_id_din P_id_u32 P_id_dout P_id_plus
P_id_u42 P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31
P_id_U41 P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_U21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_U31 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_U42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_U22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_U32 (measure x10) (measure x9) (measure x8)
(measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_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 :=
(* <din(der(plus(X_,Y_))),u21(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u21 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u21(dout(DX_),X_,Y_),u22(din(der(Y_)),X_,Y_,DX_)> *)
| DP_R_xml_0_2 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u22 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))::x1::x2::x3::nil))
(algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil))
(* <u21(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_3 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil))
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_4 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u31 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_5 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u31(dout(DX_),X_,Y_),u32(din(der(Y_)),X_,Y_,DX_)> *)
| DP_R_xml_0_6 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u32 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))::x1::x2::x3::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
(* <u31(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_7 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_8 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_9 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),u42(din(der(DX_)),X_,DX_)> *)
| DP_R_xml_0_10 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u42 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x3::nil))::nil))::x1::x3::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_11 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::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 :=
(* <u41(dout(DX_),X_),u42(din(der(DX_)),X_,DX_)> *)
| DP_R_xml_0_non_scc_1_0 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_u42
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x3::nil))::nil))::x1::x3::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
.
Lemma acc_DP_R_xml_0_non_scc_1 :
forall x y,
(DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_non_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u31(dout(DX_),X_,Y_),u32(din(der(Y_)),X_,Y_,DX_)> *)
| DP_R_xml_0_non_scc_2_0 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_u32
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))::x1::x2::x3::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
.
Lemma acc_DP_R_xml_0_non_scc_2 :
forall x y,
(DP_R_xml_0_non_scc_2 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_non_scc_3 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u21(dout(DX_),X_,Y_),u22(din(der(Y_)),X_,Y_,DX_)> *)
| DP_R_xml_0_non_scc_3_0 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_u22
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))::x1::x2::x3::nil))
(algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil))
.
Lemma acc_DP_R_xml_0_non_scc_3 :
forall x y,
(DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_scc_4 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u21(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_0 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil))
(* <din(der(plus(X_,Y_))),u21(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_u21
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_2 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_3 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_u31
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u31(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_4 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_5 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_6 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_u41
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_7 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_8 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Module WF_DP_R_xml_0_scc_4.
Inductive DP_R_xml_0_scc_4_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(plus(X_,Y_))),u21(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_u21
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_2 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_u31
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u31(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_large_3 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_4 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_5 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_u41
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_large_6 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_7 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Inductive DP_R_xml_0_scc_4_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u21(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_strict_0 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4_strict (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large.
Inductive DP_R_xml_0_scc_4_large_non_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(plus(X_,Y_))),u21(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_non_scc_1_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_non_scc_1 (algebra.Alg.Term algebra.F.id_u21
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Lemma acc_DP_R_xml_0_scc_4_large_non_scc_1 :
forall x y,
(DP_R_xml_0_scc_4_large_non_scc_1 x y) ->
Acc WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_scc_4_large_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_scc_2_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_u31
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u31(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_large_scc_2_1 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_2 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_3 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_scc_2_4 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_u41
((algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_large_scc_2_5 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_6 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2 (algebra.Alg.Term algebra.F.id_din
((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2.
Inductive DP_R_xml_0_scc_4_large_scc_2_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_u31
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::x1::
x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_2 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_3 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_u41
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_4 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_5 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Inductive DP_R_xml_0_scc_4_large_scc_2_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u31(dout(DX_),X_,Y_),din(der(Y_))> *)
| DP_R_xml_0_scc_4_large_scc_2_strict_0 :
forall x8 x2 x1 x9 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x9) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x2 x7) ->
DP_R_xml_0_scc_4_large_scc_2_strict (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x2::nil))::nil))
(algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2_large.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(times(X_,Y_))),u31(din(der(X_)),X_,Y_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1 (algebra.Alg.Term
algebra.F.id_u31
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::
x1::x2::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Lemma acc_DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1 :
forall x y,
(DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1 x y) ->
Acc WF_DP_R_xml_0_scc_4_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large
x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_2 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_u41
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::
x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_3 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_4 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_2 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large (algebra.Alg.Term
algebra.F.id_u41
((algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))::
x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_3 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <u41(dout(DX_),X_),din(der(DX_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict_0 :
forall x8 x1 x3 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_dout (x3::nil)) x8) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x1 x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict (algebra.Alg.Term
algebra.F.id_din
((algebra.Alg.Term
algebra.F.id_der
(x3::nil))::nil))
(algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(der(X_))),u41(din(der(X_)),X_)> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1_0 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1
(algebra.Alg.Term algebra.F.id_u41 ((algebra.Alg.Term
algebra.F.id_din ((algebra.Alg.Term algebra.F.id_der
(x1::nil))::nil))::x1::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Lemma acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1 :
forall x y,
(DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1 x y) ->
Acc WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large
x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((
algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((
algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_2 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 (algebra.Alg.Term
algebra.F.id_din
((
algebra.Alg.Term
algebra.F.id_der
(x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(der(X_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large_0 :
forall x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Inductive DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <din(der(plus(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict_0 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_plus (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
(* <din(der(times(X_,Y_))),din(der(X_))> *)
| DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict_1 :
forall x2 x1 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_der ((algebra.Alg.Term
algebra.F.id_times (x1::x2::nil))::nil)) x7) ->
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict
(algebra.Alg.Term algebra.F.id_din ((algebra.Alg.Term
algebra.F.id_der (x1::nil))::nil))
(algebra.Alg.Term algebra.F.id_din (x7::nil))
.
Module WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_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_din (x7:Z) := 2.
Definition P_id_u32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_dout (x7:Z) := 1* x7.
Definition P_id_plus (x7:Z) (x8:Z) := 0.
Definition P_id_u42 (x7:Z) (x8:Z) (x9:Z) := 1* x7.
Definition P_id_times (x7:Z) (x8:Z) := 0.
Definition P_id_der (x7:Z) := 1 + 1* x7.
Definition P_id_u41 (x7:Z) (x8:Z) := 2.
Definition P_id_u22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_u21 (x7:Z) (x8:Z) (x9:Z) := 2.
Definition P_id_u31 (x7:Z) (x8:Z) (x9:Z) := 1* x7.
Lemma P_id_din_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_monotonic :
forall x8 x7,
(0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Proof.
intros x8 x7.
intros [H_1 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 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_times_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_din_bounded : forall x7, (0 <= x7) ->0 <= P_id_din x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u32 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_bounded : forall x7, (0 <= x7) ->0 <= P_id_dout x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_plus_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_plus x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u42 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_times_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_times x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_bounded : forall x7, (0 <= x7) ->0 <= P_id_der x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_u41 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u22 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u21 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u31_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u31 x9 x8 x7.
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_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::
x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::
x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_U41 (x7:Z) (x8:Z) := 0.
Definition P_id_U21 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U31 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_DIN (x7:Z) := 1* x7.
Definition P_id_U32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Lemma P_id_U41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_DIN_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_din P_id_u32 P_id_dout P_id_plus
P_id_u42 P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21
P_id_u31 P_id_U41 P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN
P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::
x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::
x8::x7::nil)) =>
P_id_U21 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::
x8::x7::nil)) =>
P_id_U31 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::
x8::x7::nil)) =>
P_id_U42 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::
x9::x8::x7::nil)) =>
P_id_U22 (measure x10) (measure x9)
(measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_din
(x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::
x9::x8::x7::nil)) =>
P_id_U32 (measure x10) (measure x9)
(measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_monotonic;assumption.
Qed.
Ltac rewrite_and_unfold :=
do 2 (rewrite marked_measure_equation);
repeat (
match goal with
| |- context [measure (algebra.Alg.Term ?f ?t)] =>
rewrite (measure_equation (algebra.Alg.Term f t))
| H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
end
).
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large
.
Proof.
intros x.
apply well_founded_ind with
(R:=fun x
y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)).
apply Inverse_Image.wf_inverse_image with (B:=Z).
apply Zwf.Zwf_well_founded.
clear x.
intros x IHx.
repeat (
constructor;inversion 1;subst;
full_prove_ineq algebra.Alg.Term
ltac:(algebra.Alg_ext.find_replacement )
algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure
marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0)
ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )
ltac:(fun _ => rewrite_and_unfold )
ltac:(fun _ => generate_pos_hyp )
ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ;
try (constructor))
IHx
).
Qed.
End WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_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_din (x7:Z) := 2.
Definition P_id_u32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_dout (x7:Z) := 0.
Definition P_id_plus (x7:Z) (x8:Z) := 3 + 1* x7.
Definition P_id_u42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_times (x7:Z) (x8:Z) := 3 + 1* x7.
Definition P_id_der (x7:Z) := 2* x7.
Definition P_id_u41 (x7:Z) (x8:Z) := 0.
Definition P_id_u22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_u21 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_u31 (x7:Z) (x8:Z) (x9:Z) := 1.
Lemma P_id_din_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Proof.
intros x8 x7.
intros [H_1 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 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_times_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_din_bounded : forall x7, (0 <= x7) ->0 <= P_id_din x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u32 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_bounded : forall x7, (0 <= x7) ->0 <= P_id_dout x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_plus_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_plus x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u42 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_times_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_times x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_bounded : forall x7, (0 <= x7) ->0 <= P_id_der x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_u41 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u22 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u21 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u31_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u31 x9 x8 x7.
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_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::
x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_U41 (x7:Z) (x8:Z) := 0.
Definition P_id_U21 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U31 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_DIN (x7:Z) := 2* x7.
Definition P_id_U32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Lemma P_id_U41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_DIN_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_din P_id_u32 P_id_dout P_id_plus
P_id_u42 P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31
P_id_U41 P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::
x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::
x8::x7::nil)) =>
P_id_U21 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::
x8::x7::nil)) =>
P_id_U31 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::
x8::x7::nil)) =>
P_id_U42 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::
x9::x8::x7::nil)) =>
P_id_U22 (measure x10) (measure x9)
(measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_din
(x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::
x9::x8::x7::nil)) =>
P_id_U32 (measure x10) (measure x9)
(measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_monotonic;assumption.
Qed.
Ltac rewrite_and_unfold :=
do 2 (rewrite marked_measure_equation);
repeat (
match goal with
| |- context [measure (algebra.Alg.Term ?f ?t)] =>
rewrite (measure_equation (algebra.Alg.Term f t))
| H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
end
).
Definition lt a b :=
(Zwf.Zwf 0) (marked_measure a) (marked_measure b).
Definition le a b := marked_measure a <= marked_measure b.
Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
Proof.
unfold lt, le in *.
intros a b c.
apply (interp.le_lt_compat_right (interp.o_Z 0)).
Qed.
Lemma wf_lt : well_founded lt.
Proof.
unfold lt in *.
apply Inverse_Image.wf_inverse_image with (B:=Z).
apply Zwf.Zwf_well_founded.
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict_in_lt :
Relation_Definitions.inclusion _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict
lt.
Proof.
unfold Relation_Definitions.inclusion, lt in *.
intros a b H;destruct H;
match goal with
| |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (
apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large_in_le :
Relation_Definitions.inclusion _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large
le.
Proof.
unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
intros a b H;destruct H;
match goal with
| |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_trans (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (
apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large :=
WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large.wf.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large.DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2
.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';
apply DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_strict_in_lt;
econstructor eassumption)||
((apply IHx;[econstructor eassumption|
intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
apply DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large_in_le;
econstructor eassumption])).
apply wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2_large.
Qed.
End WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2.
Definition wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 :=
WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2.wf.
Lemma acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 :
forall x y,
(DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2 x y) ->
Acc WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large
x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_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_4_large_scc_2_large_scc_2_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
apply wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2.
Qed.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large
.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_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_4_large_scc_2_large_scc_2_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_din (x7:Z) := 0.
Definition P_id_u32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 3* x7.
Definition P_id_dout (x7:Z) := 3.
Definition P_id_plus (x7:Z) (x8:Z) := 0.
Definition P_id_u42 (x7:Z) (x8:Z) (x9:Z) := 1* x7.
Definition P_id_times (x7:Z) (x8:Z) := 0.
Definition P_id_der (x7:Z) := 0.
Definition P_id_u41 (x7:Z) (x8:Z) := 0.
Definition P_id_u22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 2 + 2* x7.
Definition P_id_u21 (x7:Z) (x8:Z) (x9:Z) := 3* x7.
Definition P_id_u31 (x7:Z) (x8:Z) (x9:Z) := 0.
Lemma P_id_din_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Proof.
intros x8 x7.
intros [H_1 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 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_times_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_din_bounded : forall x7, (0 <= x7) ->0 <= P_id_din x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u32 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_bounded : forall x7, (0 <= x7) ->0 <= P_id_dout x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_plus_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_plus x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u42 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_times_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_times x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_bounded : forall x7, (0 <= x7) ->0 <= P_id_der x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_u41 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u22 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u21 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u31_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u31 x9 x8 x7.
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_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_U41 (x7:Z) (x8:Z) := 2* x7.
Definition P_id_U21 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U31 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_DIN (x7:Z) := 0.
Definition P_id_U32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Lemma P_id_U41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_DIN_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_din P_id_u32 P_id_dout P_id_plus
P_id_u42 P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31
P_id_U41 P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::
x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_U21 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_U31 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_U42 (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::
x8::x7::nil)) =>
P_id_U22 (measure x10) (measure x9)
(measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::
x8::x7::nil)) =>
P_id_U32 (measure x10) (measure x9)
(measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_monotonic;assumption.
Qed.
Ltac rewrite_and_unfold :=
do 2 (rewrite marked_measure_equation);
repeat (
match goal with
| |- context [measure (algebra.Alg.Term ?f ?t)] =>
rewrite (measure_equation (algebra.Alg.Term f t))
| H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
end
).
Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b)
.
Definition le a b := marked_measure a <= marked_measure b.
Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
Proof.
unfold lt, le in *.
intros a b c.
apply (interp.le_lt_compat_right (interp.o_Z 0)).
Qed.
Lemma wf_lt : well_founded lt.
Proof.
unfold lt in *.
apply Inverse_Image.wf_inverse_image with (B:=Z).
apply Zwf.Zwf_well_founded.
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict_in_lt :
Relation_Definitions.inclusion _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict lt.
Proof.
unfold Relation_Definitions.inclusion, lt in *.
intros a b H;destruct H;
match goal with
| |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_in_le :
Relation_Definitions.inclusion _
DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large le.
Proof.
unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
intros a b H;destruct H;
match goal with
| |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_trans (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large :=
WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large.wf.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large_scc_2_large.DP_R_xml_0_scc_4_large_scc_2_large_scc_2
.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';
apply DP_R_xml_0_scc_4_large_scc_2_large_scc_2_strict_in_lt;
econstructor eassumption)||
((apply IHx;[econstructor eassumption|
intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
apply DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large_in_le;
econstructor eassumption])).
apply wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2_large.
Qed.
End WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.
Definition wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2 :=
WF_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.wf.
Lemma acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2 :
forall x y,
(DP_R_xml_0_scc_4_large_scc_2_large_scc_2 x y) ->
Acc WF_DP_R_xml_0_scc_4_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large
x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4_large_scc_2_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_4_large_scc_2_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
apply wf_DP_R_xml_0_scc_4_large_scc_2_large_scc_2.
Qed.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large_scc_2.DP_R_xml_0_scc_4_large_scc_2_large
.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2_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_4_large_scc_2_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_din (x7:Z) := 0.
Definition P_id_u32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 3* x7 + 3* x10.
Definition P_id_dout (x7:Z) := 2 + 1* x7.
Definition P_id_plus (x7:Z) (x8:Z) := 3.
Definition P_id_u42 (x7:Z) (x8:Z) (x9:Z) := 1 + 2* x7 + 1* x9.
Definition P_id_times (x7:Z) (x8:Z) := 0.
Definition P_id_der (x7:Z) := 0.
Definition P_id_u41 (x7:Z) (x8:Z) := 2* x7.
Definition P_id_u22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 2 + 3* x7 + 1* x10.
Definition P_id_u21 (x7:Z) (x8:Z) (x9:Z) := 2* x7.
Definition P_id_u31 (x7:Z) (x8:Z) (x9:Z) := 3* x7.
Lemma P_id_din_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Proof.
intros x8 x7.
intros [H_1 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 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_times_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_din_bounded : forall x7, (0 <= x7) ->0 <= P_id_din x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u32 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_bounded : forall x7, (0 <= x7) ->0 <= P_id_dout x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_plus_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_plus x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u42 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_times_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_times x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_bounded : forall x7, (0 <= x7) ->0 <= P_id_der x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_u41 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u22 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u21 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u31_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u31 x9 x8 x7.
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_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_U41 (x7:Z) (x8:Z) := 0.
Definition P_id_U21 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U31 (x7:Z) (x8:Z) (x9:Z) := 1* x7.
Definition P_id_U42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_DIN (x7:Z) := 0.
Definition P_id_U32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Lemma P_id_U41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_DIN_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_din P_id_u32 P_id_dout P_id_plus
P_id_u42 P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31
P_id_U41 P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::
x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_U21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_U31 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_U42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::
x8::x7::nil)) =>
P_id_U22 (measure x10) (measure x9)
(measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::
x8::x7::nil)) =>
P_id_U32 (measure x10) (measure x9)
(measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_monotonic;assumption.
Qed.
Ltac rewrite_and_unfold :=
do 2 (rewrite marked_measure_equation);
repeat (
match goal with
| |- context [measure (algebra.Alg.Term ?f ?t)] =>
rewrite (measure_equation (algebra.Alg.Term f t))
| H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
end
).
Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
Definition le a b := marked_measure a <= marked_measure b.
Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
Proof.
unfold lt, le in *.
intros a b c.
apply (interp.le_lt_compat_right (interp.o_Z 0)).
Qed.
Lemma wf_lt : well_founded lt.
Proof.
unfold lt in *.
apply Inverse_Image.wf_inverse_image with (B:=Z).
apply Zwf.Zwf_well_founded.
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_strict_in_lt :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large_scc_2_strict lt.
Proof.
unfold Relation_Definitions.inclusion, lt in *.
intros a b H;destruct H;
match goal with
| |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma DP_R_xml_0_scc_4_large_scc_2_large_in_le :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large_scc_2_large le.
Proof.
unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
intros a b H;destruct H;
match goal with
| |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_trans (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition wf_DP_R_xml_0_scc_4_large_scc_2_large :=
WF_DP_R_xml_0_scc_4_large_scc_2_large.wf.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_4_large.DP_R_xml_0_scc_4_large_scc_2.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4_large_scc_2_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';apply DP_R_xml_0_scc_4_large_scc_2_strict_in_lt;
econstructor eassumption)||
((apply IHx;[econstructor eassumption|
intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
apply DP_R_xml_0_scc_4_large_scc_2_large_in_le;
econstructor eassumption])).
apply wf_DP_R_xml_0_scc_4_large_scc_2_large.
Qed.
End WF_DP_R_xml_0_scc_4_large_scc_2.
Definition wf_DP_R_xml_0_scc_4_large_scc_2 :=
WF_DP_R_xml_0_scc_4_large_scc_2.wf.
Lemma acc_DP_R_xml_0_scc_4_large_scc_2 :
forall x y,
(DP_R_xml_0_scc_4_large_scc_2 x y) ->
Acc WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4_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_4_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))).
apply wf_DP_R_xml_0_scc_4_large_scc_2.
Qed.
Lemma wf : well_founded WF_DP_R_xml_0_scc_4.DP_R_xml_0_scc_4_large.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_4_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4_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_4_large.
Open Scope Z_scope.
Import ring_extention.
Notation Local "a <= b" := (Zle a b).
Notation Local "a < b" := (Zlt a b).
Definition P_id_din (x7:Z) := 0.
Definition P_id_u32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 1* x7.
Definition P_id_dout (x7:Z) := 2.
Definition P_id_plus (x7:Z) (x8:Z) := 0.
Definition P_id_u42 (x7:Z) (x8:Z) (x9:Z) := 3 + 3* x7.
Definition P_id_times (x7:Z) (x8:Z) := 0.
Definition P_id_der (x7:Z) := 0.
Definition P_id_u41 (x7:Z) (x8:Z) := 2* x7.
Definition P_id_u22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 3 + 3* x7.
Definition P_id_u21 (x7:Z) (x8:Z) (x9:Z) := 2* x7.
Definition P_id_u31 (x7:Z) (x8:Z) (x9:Z) := 2* x7.
Lemma P_id_din_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_din x8 <= P_id_din x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u32 x8 x10 x12 x14 <= P_id_u32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_dout x8 <= P_id_dout x7.
Proof.
intros x8 x7.
intros [H_1 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 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_plus x8 x10 <= P_id_plus x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u42 x8 x10 x12 <= P_id_u42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_times_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_times x8 x10 <= P_id_times x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_der x8 <= P_id_der x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u41 x8 x10 <= P_id_u41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_u22 x8 x10 x12 x14 <= P_id_u22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u21 x8 x10 x12 <= P_id_u21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_u31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_u31 x8 x10 x12 <= P_id_u31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_din_bounded : forall x7, (0 <= x7) ->0 <= P_id_din x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u32_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u32 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_dout_bounded : forall x7, (0 <= x7) ->0 <= P_id_dout x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_plus_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_plus x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u42_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u42 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_times_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_times x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_der_bounded : forall x7, (0 <= x7) ->0 <= P_id_der x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u41_bounded :
forall x8 x7, (0 <= x7) ->(0 <= x8) ->0 <= P_id_u41 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u22_bounded :
forall x8 x10 x9 x7,
(0 <= x7) ->
(0 <= x8) ->(0 <= x9) ->(0 <= x10) ->0 <= P_id_u22 x10 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u21_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u21 x9 x8 x7.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_u31_bounded :
forall x8 x9 x7,
(0 <= x7) ->(0 <= x8) ->(0 <= x9) ->0 <= P_id_u31 x9 x8 x7.
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_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_din (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_u32 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_dout (x7::nil)) =>
P_id_dout (measure x7)
| (algebra.Alg.Term algebra.F.id_plus (x8::x7::nil)) =>
P_id_plus (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::x7::nil)) =>
P_id_u42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_times (x8::x7::nil)) =>
P_id_times (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_der (x7::nil)) =>
P_id_der (measure x7)
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_u41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_u22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::x7::nil)) =>
P_id_u21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::x7::nil)) =>
P_id_u31 (measure x9) (measure x8) (measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_U41 (x7:Z) (x8:Z) := 0.
Definition P_id_U21 (x7:Z) (x8:Z) (x9:Z) := 2* x7.
Definition P_id_U31 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U42 (x7:Z) (x8:Z) (x9:Z) := 0.
Definition P_id_U22 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Definition P_id_DIN (x7:Z) := 0.
Definition P_id_U32 (x7:Z) (x8:Z) (x9:Z) (x10:Z) := 0.
Lemma P_id_U41_monotonic :
forall x8 x10 x9 x7,
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U41 x8 x10 <= P_id_U41 x7 x9.
Proof.
intros x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U21_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U21 x8 x10 x12 <= P_id_U21 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U31_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U31 x8 x10 x12 <= P_id_U31 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U42_monotonic :
forall x8 x12 x10 x9 x11 x7,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->P_id_U42 x8 x10 x12 <= P_id_U42 x7 x9 x11.
Proof.
intros x12 x11 x10 x9 x8 x7.
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_U22_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U22 x8 x10 x12 x14 <= P_id_U22 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_DIN_monotonic :
forall x8 x7, (0 <= x8)/\ (x8 <= x7) ->P_id_DIN x8 <= P_id_DIN x7.
Proof.
intros x8 x7.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_U32_monotonic :
forall x8 x12 x10 x14 x9 x13 x11 x7,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
(0 <= x8)/\ (x8 <= x7) ->
P_id_U32 x8 x10 x12 x14 <= P_id_U32 x7 x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9 x8 x7.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
intros [H_7 H_6].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_din P_id_u32 P_id_dout P_id_plus P_id_u42
P_id_times P_id_der P_id_u41 P_id_u22 P_id_u21 P_id_u31 P_id_U41
P_id_U21 P_id_U31 P_id_U42 P_id_U22 P_id_DIN P_id_U32.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_u41 (x8::x7::nil)) =>
P_id_U41 (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u21 (x9::x8::
x7::nil)) =>
P_id_U21 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u31 (x9::x8::
x7::nil)) =>
P_id_U31 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u42 (x9::x8::
x7::nil)) =>
P_id_U42 (measure x9) (measure x8) (measure x7)
| (algebra.Alg.Term algebra.F.id_u22 (x10::x9::x8::
x7::nil)) =>
P_id_U22 (measure x10) (measure x9) (measure x8)
(measure x7)
| (algebra.Alg.Term algebra.F.id_din (x7::nil)) =>
P_id_DIN (measure x7)
| (algebra.Alg.Term algebra.F.id_u32 (x10::x9::x8::
x7::nil)) =>
P_id_U32 (measure x10) (measure x9) (measure x8)
(measure x7)
| _ => 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_din_monotonic;assumption.
intros ;apply P_id_u32_monotonic;assumption.
intros ;apply P_id_dout_monotonic;assumption.
intros ;apply P_id_plus_monotonic;assumption.
intros ;apply P_id_u42_monotonic;assumption.
intros ;apply P_id_times_monotonic;assumption.
intros ;apply P_id_der_monotonic;assumption.
intros ;apply P_id_u41_monotonic;assumption.
intros ;apply P_id_u22_monotonic;assumption.
intros ;apply P_id_u21_monotonic;assumption.
intros ;apply P_id_u31_monotonic;assumption.
intros ;apply P_id_din_bounded;assumption.
intros ;apply P_id_u32_bounded;assumption.
intros ;apply P_id_dout_bounded;assumption.
intros ;apply P_id_plus_bounded;assumption.
intros ;apply P_id_u42_bounded;assumption.
intros ;apply P_id_times_bounded;assumption.
intros ;apply P_id_der_bounded;assumption.
intros ;apply P_id_u41_bounded;assumption.
intros ;apply P_id_u22_bounded;assumption.
intros ;apply P_id_u21_bounded;assumption.
intros ;apply P_id_u31_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_U41_monotonic;assumption.
intros ;apply P_id_U21_monotonic;assumption.
intros ;apply P_id_U31_monotonic;assumption.
intros ;apply P_id_U42_monotonic;assumption.
intros ;apply P_id_U22_monotonic;assumption.
intros ;apply P_id_DIN_monotonic;assumption.
intros ;apply P_id_U32_monotonic;assumption.
Qed.
Ltac rewrite_and_unfold :=
do 2 (rewrite marked_measure_equation);
repeat (
match goal with
| |- context [measure (algebra.Alg.Term ?f ?t)] =>
rewrite (measure_equation (algebra.Alg.Term f t))
| H:context [measure (algebra.Alg.Term ?f ?t)] |- _ =>
rewrite (measure_equation (algebra.Alg.Term f t)) in H|-
end
).
Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b).
Definition le a b := marked_measure a <= marked_measure b.
Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c.
Proof.
unfold lt, le in *.
intros a b c.
apply (interp.le_lt_compat_right (interp.o_Z 0)).
Qed.
Lemma wf_lt : well_founded lt.
Proof.
unfold lt in *.
apply Inverse_Image.wf_inverse_image with (B:=Z).
apply Zwf.Zwf_well_founded.
Qed.
Lemma DP_R_xml_0_scc_4_strict_in_lt :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_strict lt.
Proof.
unfold Relation_Definitions.inclusion, lt in *.
intros a b H;destruct H;
match goal with
| |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma DP_R_xml_0_scc_4_large_in_le :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_4_large le.
Proof.
unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
intros a b H;destruct H;
match goal with
| |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_trans (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition wf_DP_R_xml_0_scc_4_large := WF_DP_R_xml_0_scc_4_large.wf.
Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_4.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';apply DP_R_xml_0_scc_4_strict_in_lt;
econstructor eassumption)||
((apply IHx;[econstructor eassumption|
intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
apply DP_R_xml_0_scc_4_large_in_le;econstructor eassumption])).
apply wf_DP_R_xml_0_scc_4_large.
Qed.
End WF_DP_R_xml_0_scc_4.
Definition wf_DP_R_xml_0_scc_4 := WF_DP_R_xml_0_scc_4.wf.
Lemma acc_DP_R_xml_0_scc_4 :
forall x y, (DP_R_xml_0_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_4).
intros x' _ Hrec y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply Hrec;econstructor eassumption)||
((eapply acc_DP_R_xml_0_non_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))).
apply wf_DP_R_xml_0_scc_4.
Qed.
Lemma wf : well_founded WF_R_xml_0_deep_rew.DP_R_xml_0.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_non_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_4;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))))).
Qed.
End WF_DP_R_xml_0.
Definition wf_H := WF_DP_R_xml_0.wf.
Lemma wf :
well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
Proof.
apply ddp.dp_criterion.
apply R_xml_0_deep_rew.R_xml_0_non_var.
apply R_xml_0_deep_rew.R_xml_0_reg.
intros ;
apply (ddp.constructor_defined_dec _ _
R_xml_0_deep_rew.R_xml_0_rules_included).
refine (Inclusion.wf_incl _ _ _ _ wf_H).
intros x y H.
destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
destruct (ddp.dp_list_complete _ _
R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
as [x' [y' [sigma [h1 [h2 h3]]]]].
clear H3.
subst.
vm_compute in h3|-.
let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
Qed.
End WF_R_xml_0_deep_rew.
(*
*** Local Variables: ***
*** coq-prog-name: "coqtop" ***
*** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") ***
*** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/ c_output/strat/tpdb-5.0___TRS___Rubio___wst99.trs/a3pat.v" ***
*** End: ***
*)