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_filter *)
| id_filter : symb
(* id_nats *)
| id_nats : symb
(* id_activate *)
| id_activate : symb
(* id_0 *)
| id_0 : symb
(* id_zprimes *)
| id_zprimes : symb
(* id_sieve *)
| id_sieve : symb
(* id_cons *)
| id_cons : symb
(* id_n__nats *)
| id_n__nats : symb
(* id_s *)
| id_s : symb
(* id_n__filter *)
| id_n__filter : symb
(* id_n__sieve *)
| id_n__sieve : symb
.
Definition symb_eq_bool (f1 f2:symb) : bool :=
match f1,f2 with
| id_filter,id_filter => true
| id_nats,id_nats => true
| id_activate,id_activate => true
| id_0,id_0 => true
| id_zprimes,id_zprimes => true
| id_sieve,id_sieve => true
| id_cons,id_cons => true
| id_n__nats,id_n__nats => true
| id_s,id_s => true
| id_n__filter,id_n__filter => true
| id_n__sieve,id_n__sieve => 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_filter,id_filter => refl_equal _
| id_nats,id_nats => refl_equal _
| id_activate,id_activate => refl_equal _
| id_0,id_0 => refl_equal _
| id_zprimes,id_zprimes => refl_equal _
| id_sieve,id_sieve => refl_equal _
| id_cons,id_cons => refl_equal _
| id_n__nats,id_n__nats => refl_equal _
| id_s,id_s => refl_equal _
| id_n__filter,id_n__filter => refl_equal _
| id_n__sieve,id_n__sieve => refl_equal _
| _,_ => _
end;intros abs;discriminate.
Defined.
Definition arity (f:symb) :=
match f with
| id_filter => term.Free 3
| id_nats => term.Free 1
| id_activate => term.Free 1
| id_0 => term.Free 0
| id_zprimes => term.Free 0
| id_sieve => term.Free 1
| id_cons => term.Free 2
| id_n__nats => term.Free 1
| id_s => term.Free 1
| id_n__filter => term.Free 3
| id_n__sieve => term.Free 1
end.
Definition symb_order (f1 f2:symb) : bool :=
match f1,f2 with
| id_filter,id_filter => true
| id_filter,id_nats => false
| id_filter,id_activate => false
| id_filter,id_0 => false
| id_filter,id_zprimes => false
| id_filter,id_sieve => false
| id_filter,id_cons => false
| id_filter,id_n__nats => false
| id_filter,id_s => false
| id_filter,id_n__filter => false
| id_filter,id_n__sieve => false
| id_nats,id_filter => true
| id_nats,id_nats => true
| id_nats,id_activate => false
| id_nats,id_0 => false
| id_nats,id_zprimes => false
| id_nats,id_sieve => false
| id_nats,id_cons => false
| id_nats,id_n__nats => false
| id_nats,id_s => false
| id_nats,id_n__filter => false
| id_nats,id_n__sieve => false
| id_activate,id_filter => true
| id_activate,id_nats => true
| id_activate,id_activate => true
| id_activate,id_0 => false
| id_activate,id_zprimes => false
| id_activate,id_sieve => false
| id_activate,id_cons => false
| id_activate,id_n__nats => false
| id_activate,id_s => false
| id_activate,id_n__filter => false
| id_activate,id_n__sieve => false
| id_0,id_filter => true
| id_0,id_nats => true
| id_0,id_activate => true
| id_0,id_0 => true
| id_0,id_zprimes => false
| id_0,id_sieve => false
| id_0,id_cons => false
| id_0,id_n__nats => false
| id_0,id_s => false
| id_0,id_n__filter => false
| id_0,id_n__sieve => false
| id_zprimes,id_filter => true
| id_zprimes,id_nats => true
| id_zprimes,id_activate => true
| id_zprimes,id_0 => true
| id_zprimes,id_zprimes => true
| id_zprimes,id_sieve => false
| id_zprimes,id_cons => false
| id_zprimes,id_n__nats => false
| id_zprimes,id_s => false
| id_zprimes,id_n__filter => false
| id_zprimes,id_n__sieve => false
| id_sieve,id_filter => true
| id_sieve,id_nats => true
| id_sieve,id_activate => true
| id_sieve,id_0 => true
| id_sieve,id_zprimes => true
| id_sieve,id_sieve => true
| id_sieve,id_cons => false
| id_sieve,id_n__nats => false
| id_sieve,id_s => false
| id_sieve,id_n__filter => false
| id_sieve,id_n__sieve => false
| id_cons,id_filter => true
| id_cons,id_nats => true
| id_cons,id_activate => true
| id_cons,id_0 => true
| id_cons,id_zprimes => true
| id_cons,id_sieve => true
| id_cons,id_cons => true
| id_cons,id_n__nats => false
| id_cons,id_s => false
| id_cons,id_n__filter => false
| id_cons,id_n__sieve => false
| id_n__nats,id_filter => true
| id_n__nats,id_nats => true
| id_n__nats,id_activate => true
| id_n__nats,id_0 => true
| id_n__nats,id_zprimes => true
| id_n__nats,id_sieve => true
| id_n__nats,id_cons => true
| id_n__nats,id_n__nats => true
| id_n__nats,id_s => false
| id_n__nats,id_n__filter => false
| id_n__nats,id_n__sieve => false
| id_s,id_filter => true
| id_s,id_nats => true
| id_s,id_activate => true
| id_s,id_0 => true
| id_s,id_zprimes => true
| id_s,id_sieve => true
| id_s,id_cons => true
| id_s,id_n__nats => true
| id_s,id_s => true
| id_s,id_n__filter => false
| id_s,id_n__sieve => false
| id_n__filter,id_filter => true
| id_n__filter,id_nats => true
| id_n__filter,id_activate => true
| id_n__filter,id_0 => true
| id_n__filter,id_zprimes => true
| id_n__filter,id_sieve => true
| id_n__filter,id_cons => true
| id_n__filter,id_n__nats => true
| id_n__filter,id_s => true
| id_n__filter,id_n__filter => true
| id_n__filter,id_n__sieve => false
| id_n__sieve,id_filter => true
| id_n__sieve,id_nats => true
| id_n__sieve,id_activate => true
| id_n__sieve,id_0 => true
| id_n__sieve,id_zprimes => true
| id_n__sieve,id_sieve => true
| id_n__sieve,id_cons => true
| id_n__sieve,id_n__nats => true
| id_n__sieve,id_s => true
| id_n__sieve,id_n__filter => true
| id_n__sieve,id_n__sieve => 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 :=
(* filter(cons(X_,Y_),0,M_) -> cons(0,n__filter(activate(Y_),M_,M_)) *)
| R_xml_0_rule_0 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_0 nil)::(algebra.Alg.Term
algebra.F.id_n__filter ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::
(algebra.Alg.Var 3)::(algebra.Alg.Var 3)::nil))::nil))
(algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
(algebra.Alg.Term algebra.F.id_0 nil)::(algebra.Alg.Var 3)::nil))
(* filter(cons(X_,Y_),s(N_),M_) -> cons(X_,n__filter(activate(Y_),N_,M_)) *)
| R_xml_0_rule_1 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::
(algebra.Alg.Term algebra.F.id_n__filter
((algebra.Alg.Term algebra.F.id_activate
((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 4)::
(algebra.Alg.Var 3)::nil))::nil))
(algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
(algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 4)::nil))::
(algebra.Alg.Var 3)::nil))
(* sieve(cons(0,Y_)) -> cons(0,n__sieve(activate(Y_))) *)
| R_xml_0_rule_2 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_0 nil)::(algebra.Alg.Term
algebra.F.id_n__sieve ((algebra.Alg.Term
algebra.F.id_activate
((algebra.Alg.Var 2)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0 nil)::
(algebra.Alg.Var 2)::nil))::nil))
(* sieve(cons(s(N_),Y_)) -> cons(s(N_),n__sieve(filter(activate(Y_),N_,N_))) *)
| R_xml_0_rule_3 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s ((algebra.Alg.Var 4)::nil))::
(algebra.Alg.Term algebra.F.id_n__sieve
((algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::
(algebra.Alg.Var 4)::
(algebra.Alg.Var 4)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 2)::nil))::nil))
(* nats(N_) -> cons(N_,n__nats(s(N_))) *)
| R_xml_0_rule_4 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 4)::
(algebra.Alg.Term algebra.F.id_n__nats ((algebra.Alg.Term
algebra.F.id_s ((algebra.Alg.Var 4)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Var 4)::nil))
(* zprimes -> sieve(nats(s(s(0)))) *)
| R_xml_0_rule_5 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_nats ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term
algebra.F.id_0 nil)::nil))::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_zprimes nil)
(* filter(X1_,X2_,X3_) -> n__filter(X1_,X2_,X3_) *)
| R_xml_0_rule_6 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__filter
((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::
(algebra.Alg.Var 7)::nil))
(algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Var 5)::
(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))
(* sieve(X_) -> n__sieve(X_) *)
| R_xml_0_rule_7 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__sieve
((algebra.Alg.Var 1)::nil))
(algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Var 1)::nil))
(* nats(X_) -> n__nats(X_) *)
| R_xml_0_rule_8 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_n__nats
((algebra.Alg.Var 1)::nil))
(algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Var 1)::nil))
(* activate(n__filter(X1_,X2_,X3_)) -> filter(X1_,X2_,X3_) *)
| R_xml_0_rule_9 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_filter
((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::
(algebra.Alg.Var 7)::nil))
(algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__filter ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::
(algebra.Alg.Var 7)::nil))::nil))
(* activate(n__sieve(X_)) -> sieve(X_) *)
| R_xml_0_rule_10 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_sieve
((algebra.Alg.Var 1)::nil))
(algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__sieve ((algebra.Alg.Var 1)::nil))::nil))
(* activate(n__nats(X_)) -> nats(X_) *)
| R_xml_0_rule_11 :
R_xml_0_rules (algebra.Alg.Term algebra.F.id_nats
((algebra.Alg.Var 1)::nil))
(algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__nats ((algebra.Alg.Var 1)::nil))::nil))
(* activate(X_) -> X_ *)
| R_xml_0_rule_12 :
R_xml_0_rules (algebra.Alg.Var 1)
(algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 1)::nil))
.
Definition R_xml_0_rule_as_list_0 :=
((algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
(algebra.Alg.Term algebra.F.id_0 nil)::(algebra.Alg.Var 3)::nil)),
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0
nil)::(algebra.Alg.Term algebra.F.id_n__filter ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 3)::
(algebra.Alg.Var 3)::nil))::nil)))::nil.
Definition R_xml_0_rule_as_list_1 :=
((algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil))::
(algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Var 4)::nil))::
(algebra.Alg.Var 3)::nil)),
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::
(algebra.Alg.Term algebra.F.id_n__filter ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 4)::
(algebra.Alg.Var 3)::nil))::nil)))::R_xml_0_rule_as_list_0.
Definition R_xml_0_rule_as_list_2 :=
((algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0 nil)::
(algebra.Alg.Var 2)::nil))::nil)),
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0
nil)::(algebra.Alg.Term algebra.F.id_n__sieve ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::nil))::nil)))::
R_xml_0_rule_as_list_1.
Definition R_xml_0_rule_as_list_3 :=
((algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 2)::nil))::nil)),
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Var 4)::nil))::(algebra.Alg.Term algebra.F.id_n__sieve
((algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_activate ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Var 4)::
(algebra.Alg.Var 4)::nil))::nil))::nil)))::R_xml_0_rule_as_list_2.
Definition R_xml_0_rule_as_list_4 :=
((algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Var 4)::nil)),
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 4)::
(algebra.Alg.Term algebra.F.id_n__nats ((algebra.Alg.Term
algebra.F.id_s ((algebra.Alg.Var 4)::nil))::nil))::nil)))::
R_xml_0_rule_as_list_3.
Definition R_xml_0_rule_as_list_5 :=
((algebra.Alg.Term algebra.F.id_zprimes nil),
(algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_nats ((algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term
algebra.F.id_s ((algebra.Alg.Term algebra.F.id_0
nil)::nil))::nil))::nil))::nil)))::R_xml_0_rule_as_list_4.
Definition R_xml_0_rule_as_list_6 :=
((algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Var 5)::
(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)),
(algebra.Alg.Term algebra.F.id_n__filter ((algebra.Alg.Var 5)::
(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_5
.
Definition R_xml_0_rule_as_list_7 :=
((algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Var 1)::nil)),
(algebra.Alg.Term algebra.F.id_n__sieve ((algebra.Alg.Var 1)::nil)))::
R_xml_0_rule_as_list_6.
Definition R_xml_0_rule_as_list_8 :=
((algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Var 1)::nil)),
(algebra.Alg.Term algebra.F.id_n__nats ((algebra.Alg.Var 1)::nil)))::
R_xml_0_rule_as_list_7.
Definition R_xml_0_rule_as_list_9 :=
((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__filter ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::
(algebra.Alg.Var 7)::nil))::nil)),
(algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Var 5)::
(algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)))::R_xml_0_rule_as_list_8
.
Definition R_xml_0_rule_as_list_10 :=
((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__sieve ((algebra.Alg.Var 1)::nil))::nil)),
(algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Var 1)::nil)))::
R_xml_0_rule_as_list_9.
Definition R_xml_0_rule_as_list_11 :=
((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Term
algebra.F.id_n__nats ((algebra.Alg.Var 1)::nil))::nil)),
(algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Var 1)::nil)))::
R_xml_0_rule_as_list_10.
Definition R_xml_0_rule_as_list_12 :=
((algebra.Alg.Term algebra.F.id_activate ((algebra.Alg.Var 1)::nil)),
(algebra.Alg.Var 1))::R_xml_0_rule_as_list_11.
Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_12.
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.or14_equiv in H|-.
case H;clear H;intros H.
injection H;intros ;subst;constructor 13.
injection H;intros ;subst;constructor 12.
injection H;intros ;subst;constructor 11.
injection H;intros ;subst;constructor 10.
injection H;intros ;subst;constructor 9.
injection H;intros ;subst;constructor 8.
injection H;intros ;subst;constructor 7.
injection H;intros ;subst;constructor 6.
injection H;intros ;subst;constructor 5.
injection H;intros ;subst;constructor 4.
injection H;intros ;subst;constructor 3.
injection H;intros ;subst;constructor 2.
injection H;intros ;subst;constructor 1.
elim H.
Qed.
Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x).
Proof.
intros x t H.
inversion H.
Qed.
Lemma R_xml_0_reg :
forall s t,
(R_xml_0_rules s t) ->
forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t).
Proof.
intros s t H.
inversion H;intros x Hx;
(apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]);
apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx;
vm_compute in Hx|-*;tauto.
Qed.
Inductive and_6 (x9 x10 x11 x12 x13 x14:Prop) :
Prop :=
| conj_6 : x9->x10->x11->x12->x13->x14->and_6 x9 x10 x11 x12 x13 x14
.
Lemma are_constuctors_of_R_xml_0 :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
and_6 (t = (algebra.Alg.Term algebra.F.id_0 nil) ->
t' = (algebra.Alg.Term algebra.F.id_0 nil))
(forall x10 x12,
t = (algebra.Alg.Term algebra.F.id_cons (x10::x12::nil)) ->
exists x9,
exists x11,
t' = (algebra.Alg.Term algebra.F.id_cons (x9::x11::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x9 x10)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x11 x12))
(forall x10,
t = (algebra.Alg.Term algebra.F.id_n__nats (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_n__nats (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10))
(forall x10,
t = (algebra.Alg.Term algebra.F.id_s (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_s (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10))
(forall x10 x12 x14,
t = (algebra.Alg.Term algebra.F.id_n__filter (x10::x12::x14::nil)) ->
exists x9,
exists x11,
exists x13,
t' = (algebra.Alg.Term algebra.F.id_n__filter (x9::x11::
x13::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x9 x10)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x11 x12)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x13 x14))
(forall x10,
t = (algebra.Alg.Term algebra.F.id_n__sieve (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10))
.
Proof.
intros t t' H.
induction H as [|y IH z z_to_y] using
closure_extension.refl_trans_clos_ind2.
constructor 1.
intros H;intuition;constructor 1.
intros x10 x12 H;exists x10;exists x12;intuition;constructor 1.
intros x10 H;exists x10;intuition;constructor 1.
intros x10 H;exists x10;intuition;constructor 1.
intros x10 x12 x14 H;exists x10;exists x12;exists x14;intuition;
constructor 1.
intros x10 H;exists x10;intuition;constructor 1.
inversion z_to_y as [t1 t2 H H0 H1|f l1 l2 H0 H H2];clear z_to_y;subst.
inversion H as [t1 t2 sigma H2 H1 H0];clear H IH;subst;inversion H2;
clear ;constructor;try (intros until 0 );clear ;intros abs;
discriminate abs.
destruct IH as
[H_id_0 H_id_cons H_id_n__nats H_id_s H_id_n__filter H_id_n__sieve].
constructor.
clear H_id_cons H_id_n__nats H_id_s H_id_n__filter H_id_n__sieve;intros H;
injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
clear H_id_0 H_id_n__nats H_id_s H_id_n__filter H_id_n__sieve;
intros x10 x12 H;injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
match goal with
| H:algebra.EQT.one_step _ ?y x10 |- _ =>
destruct (H_id_cons y x12 (refl_equal _)) as [x9 [x11]];intros ;
intuition;exists x9;exists x11;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
match goal with
| H:algebra.EQT.one_step _ ?y x12 |- _ =>
destruct (H_id_cons x10 y (refl_equal _)) as [x9 [x11]];intros ;
intuition;exists x9;exists x11;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
clear H_id_0 H_id_cons H_id_s H_id_n__filter H_id_n__sieve;intros x10 H;
injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
match goal with
| H:algebra.EQT.one_step _ ?y x10 |- _ =>
destruct (H_id_n__nats y (refl_equal _)) as [x9];intros ;intuition;
exists x9;intuition;eapply closure_extension.refl_trans_clos_R;
eassumption
end
.
clear H_id_0 H_id_cons H_id_n__nats H_id_n__filter H_id_n__sieve;
intros x10 H;injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
match goal with
| H:algebra.EQT.one_step _ ?y x10 |- _ =>
destruct (H_id_s y (refl_equal _)) as [x9];intros ;intuition;exists x9;
intuition;eapply closure_extension.refl_trans_clos_R;eassumption
end
.
clear H_id_0 H_id_cons H_id_n__nats H_id_s H_id_n__sieve;
intros x10 x12 x14 H;injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
match goal with
| H:algebra.EQT.one_step _ ?y x10 |- _ =>
destruct (H_id_n__filter y x12 x14 (refl_equal _)) as [x9 [x11 [x13]]];
intros ;intuition;exists x9;exists x11;exists x13;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
match goal with
| H:algebra.EQT.one_step _ ?y x12 |- _ =>
destruct (H_id_n__filter x10 y x14 (refl_equal _)) as [x9 [x11 [x13]]];
intros ;intuition;exists x9;exists x11;exists x13;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
match goal with
| H:algebra.EQT.one_step _ ?y x14 |- _ =>
destruct (H_id_n__filter x10 x12 y (refl_equal _)) as [x9 [x11 [x13]]];
intros ;intuition;exists x9;exists x11;exists x13;intuition;
eapply closure_extension.refl_trans_clos_R;eassumption
end
.
clear H_id_0 H_id_cons H_id_n__nats H_id_s H_id_n__filter;intros x10 H;
injection H;clear H;intros ;subst;
repeat (
match goal with
| H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ =>
inversion H;clear H;subst
end
).
match goal with
| H:algebra.EQT.one_step _ ?y x10 |- _ =>
destruct (H_id_n__sieve y (refl_equal _)) as [x9];intros ;intuition;
exists x9;intuition;eapply closure_extension.refl_trans_clos_R;
eassumption
end
.
Qed.
Lemma id_0_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
t = (algebra.Alg.Term algebra.F.id_0 nil) ->
t' = (algebra.Alg.Term algebra.F.id_0 nil).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_cons_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x10 x12,
t = (algebra.Alg.Term algebra.F.id_cons (x10::x12::nil)) ->
exists x9,
exists x11,
t' = (algebra.Alg.Term algebra.F.id_cons (x9::x11::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x9 x10)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x11 x12).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_n__nats_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x10,
t = (algebra.Alg.Term algebra.F.id_n__nats (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_n__nats (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_s_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x10,
t = (algebra.Alg.Term algebra.F.id_s (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_s (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_n__filter_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x10 x12 x14,
t = (algebra.Alg.Term algebra.F.id_n__filter (x10::x12::x14::nil)) ->
exists x9,
exists x11,
exists x13,
t' = (algebra.Alg.Term algebra.F.id_n__filter (x9::x11::
x13::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x9 x10)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x11 x12)/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
x13 x14).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Lemma id_n__sieve_is_R_xml_0_constructor :
forall t t',
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) ->
forall x10,
t = (algebra.Alg.Term algebra.F.id_n__sieve (x10::nil)) ->
exists x9,
t' = (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil))/\
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x9 x10).
Proof.
intros t t' H.
destruct (are_constuctors_of_R_xml_0 H).
assumption.
Qed.
Ltac impossible_star_reduction_R_xml_0 :=
match goal with
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_0 nil) |- _ =>
let Heq := fresh "Heq" in
(set (Heq:=id_0_is_R_xml_0_constructor H (refl_equal _)) in *;
(discriminate Heq)||
(clearbody Heq;try (subst);try (clear Heq);clear H;
impossible_star_reduction_R_xml_0 ))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_cons (?x10::?x9::nil)) |- _ =>
let x10 := fresh "x" in
(let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as
[x10 [x9 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__nats (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_n__nats_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_s (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_s_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__filter (?x11::?x10::?x9::nil)) |-
_ =>
let x11 := fresh "x" in
(let x10 := fresh "x" in
(let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(let Hred3 := fresh "Hred" in
(destruct (id_n__filter_is_R_xml_0_constructor H
(refl_equal _))
as [x11 [x10 [x9 [Heq [Hred3 [Hred2 Hred1]]]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__sieve (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_n__sieve_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
impossible_star_reduction_R_xml_0 ))))
end
.
Ltac simplify_star_reduction_R_xml_0 :=
match goal with
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_0 nil) |- _ =>
let Heq := fresh "Heq" in
(set (Heq:=id_0_is_R_xml_0_constructor H (refl_equal _)) in *;
(discriminate Heq)||
(clearbody Heq;try (subst);try (clear Heq);clear H;
try (simplify_star_reduction_R_xml_0 )))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_cons (?x10::?x9::nil)) |- _ =>
let x10 := fresh "x" in
(let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as
[x10 [x9 [Heq [Hred2 Hred1]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__nats (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_n__nats_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_s (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_s_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__filter (?x11::?x10::?x9::nil)) |-
_ =>
let x11 := fresh "x" in
(let x10 := fresh "x" in
(let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(let Hred2 := fresh "Hred" in
(let Hred3 := fresh "Hred" in
(destruct (id_n__filter_is_R_xml_0_constructor H
(refl_equal _))
as [x11 [x10 [x9 [Heq [Hred3 [Hred2 Hred1]]]]]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))))))
| H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules)
_ (algebra.Alg.Term algebra.F.id_n__sieve (?x9::nil)) |- _ =>
let x9 := fresh "x" in
(let Heq := fresh "Heq" in
(let Hred1 := fresh "Hred" in
(destruct (id_n__sieve_is_R_xml_0_constructor H (refl_equal _)) as
[x9 [Heq Hred1]];
(discriminate Heq)||
(injection Heq;intros ;subst;clear Heq;clear H;
try (simplify_star_reduction_R_xml_0 )))))
end
.
End R_xml_0_deep_rew.
Module InterpGen := interp.Interp(algebra.EQT).
Module ddp := dp.MakeDP(algebra.EQT).
Module SymbType. Definition A := algebra.Alg.F.Symb.A. End SymbType.
Module Symb_more_list := more_list_extention.Make(SymbType)(algebra.Alg.F.Symb).
Module SymbSet := list_set.Make(algebra.F.Symb).
Module Interp.
Section S.
Require Import interp.
Hypothesis A : Type.
Hypothesis Ale Alt Aeq : A -> A -> Prop.
Hypothesis Aop : interp.ordering_pair Aeq Alt Ale.
Hypothesis A0 : A.
Notation Local "a <= b" := (Ale a b).
Hypothesis P_id_filter : A ->A ->A ->A.
Hypothesis P_id_nats : A ->A.
Hypothesis P_id_activate : A ->A.
Hypothesis P_id_0 : A.
Hypothesis P_id_zprimes : A.
Hypothesis P_id_sieve : A ->A.
Hypothesis P_id_cons : A ->A ->A.
Hypothesis P_id_n__nats : A ->A.
Hypothesis P_id_s : A ->A.
Hypothesis P_id_n__filter : A ->A ->A ->A.
Hypothesis P_id_n__sieve : A ->A.
Hypothesis P_id_filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
P_id_filter x10 x12 x14 <= P_id_filter x9 x11 x13.
Hypothesis P_id_nats_monotonic :
forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_nats x10 <= P_id_nats x9.
Hypothesis P_id_activate_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
Hypothesis P_id_sieve_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_sieve x10 <= P_id_sieve x9.
Hypothesis P_id_cons_monotonic :
forall x12 x10 x9 x11,
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11.
Hypothesis P_id_n__nats_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_n__nats x10 <= P_id_n__nats x9.
Hypothesis P_id_s_monotonic :
forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
Hypothesis P_id_n__filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
P_id_n__filter x10 x12 x14 <= P_id_n__filter x9 x11 x13.
Hypothesis P_id_n__sieve_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_n__sieve x10 <= P_id_n__sieve x9.
Hypothesis P_id_filter_bounded :
forall x10 x9 x11,
(A0 <= x9) ->(A0 <= x10) ->(A0 <= x11) ->A0 <= P_id_filter x11 x10 x9.
Hypothesis P_id_nats_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_nats x9.
Hypothesis P_id_activate_bounded :
forall x9, (A0 <= x9) ->A0 <= P_id_activate x9.
Hypothesis P_id_0_bounded : A0 <= P_id_0 .
Hypothesis P_id_zprimes_bounded : A0 <= P_id_zprimes .
Hypothesis P_id_sieve_bounded :
forall x9, (A0 <= x9) ->A0 <= P_id_sieve x9.
Hypothesis P_id_cons_bounded :
forall x10 x9, (A0 <= x9) ->(A0 <= x10) ->A0 <= P_id_cons x10 x9.
Hypothesis P_id_n__nats_bounded :
forall x9, (A0 <= x9) ->A0 <= P_id_n__nats x9.
Hypothesis P_id_s_bounded : forall x9, (A0 <= x9) ->A0 <= P_id_s x9.
Hypothesis P_id_n__filter_bounded :
forall x10 x9 x11,
(A0 <= x9) ->(A0 <= x10) ->(A0 <= x11) ->A0 <= P_id_n__filter x11 x10 x9.
Hypothesis P_id_n__sieve_bounded :
forall x9, (A0 <= x9) ->A0 <= P_id_n__sieve x9.
Fixpoint measure t { struct t } :=
match t with
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil)) =>
P_id_filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_nats (measure x9)
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_activate (measure x9)
| (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0
| (algebra.Alg.Term algebra.F.id_zprimes nil) => P_id_zprimes
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_sieve (measure x9)
| (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) =>
P_id_cons (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__nats (x9::nil)) =>
P_id_n__nats (measure x9)
| (algebra.Alg.Term algebra.F.id_s (x9::nil)) => P_id_s (measure x9)
| (algebra.Alg.Term algebra.F.id_n__filter (x11::x10::x9::nil)) =>
P_id_n__filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil)) =>
P_id_n__sieve (measure x9)
| _ => A0
end.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_nats (measure x9)
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_activate (measure x9)
| (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_zprimes
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_sieve (measure x9)
| (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) =>
P_id_cons (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__nats (x9::nil)) =>
P_id_n__nats (measure x9)
| (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
P_id_s (measure x9)
| (algebra.Alg.Term algebra.F.id_n__filter (x11::x10::
x9::nil)) =>
P_id_n__filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil)) =>
P_id_n__sieve (measure x9)
| _ => A0
end.
Proof.
intros t;case t;intros ;apply refl_equal.
Qed.
Definition Pols f : InterpGen.Pol_type A (InterpGen.get_arity f) :=
match f with
| algebra.F.id_filter => P_id_filter
| algebra.F.id_nats => P_id_nats
| algebra.F.id_activate => P_id_activate
| algebra.F.id_0 => P_id_0
| algebra.F.id_zprimes => P_id_zprimes
| algebra.F.id_sieve => P_id_sieve
| algebra.F.id_cons => P_id_cons
| algebra.F.id_n__nats => P_id_n__nats
| algebra.F.id_s => P_id_s
| algebra.F.id_n__filter => P_id_n__filter
| algebra.F.id_n__sieve => P_id_n__sieve
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_filter =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_nats =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_activate =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_0 => match l with
| nil => _
| _::_ => _
end
| algebra.F.id_zprimes => match l with
| nil => _
| _::_ => _
end
| algebra.F.id_sieve =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_cons =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_n__nats =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_s =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_n__filter =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_n__sieve =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*;
try (reflexivity );f_equal ;auto.
Qed.
Lemma measure_bounded : forall t, A0 <= measure t.
Proof.
intros t.
rewrite same_measure in |-*.
apply (InterpGen.measure_bounded Aop).
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_zprimes_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_sieve_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__sieve_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_filter_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_nats_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_activate_monotonic;assumption.
vm_compute in |-*;apply (Aop.(le_refl)).
vm_compute in |-*;apply (Aop.(le_refl)).
vm_compute in |-*;intros ;apply P_id_sieve_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__nats_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__filter_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__sieve_monotonic;assumption.
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_zprimes_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_sieve_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__sieve_bounded;assumption.
intros .
do 2 (rewrite <- same_measure in |-*).
apply rules_monotonic;assumption.
assumption.
Qed.
Hypothesis P_id_ACTIVATE : A ->A.
Hypothesis P_id_ZPRIMES : A.
Hypothesis P_id_SIEVE : A ->A.
Hypothesis P_id_FILTER : A ->A ->A ->A.
Hypothesis P_id_NATS : A ->A.
Hypothesis P_id_ACTIVATE_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
Hypothesis P_id_SIEVE_monotonic :
forall x10 x9,
(A0 <= x10)/\ (x10 <= x9) ->P_id_SIEVE x10 <= P_id_SIEVE x9.
Hypothesis P_id_FILTER_monotonic :
forall x12 x10 x14 x9 x13 x11,
(A0 <= x14)/\ (x14 <= x13) ->
(A0 <= x12)/\ (x12 <= x11) ->
(A0 <= x10)/\ (x10 <= x9) ->
P_id_FILTER x10 x12 x14 <= P_id_FILTER x9 x11 x13.
Hypothesis P_id_NATS_monotonic :
forall x10 x9, (A0 <= x10)/\ (x10 <= x9) ->P_id_NATS x10 <= P_id_NATS x9.
Definition marked_measure t :=
match t with
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_ACTIVATE (measure x9)
| (algebra.Alg.Term algebra.F.id_zprimes nil) => P_id_ZPRIMES
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_SIEVE (measure x9)
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil)) =>
P_id_FILTER (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_NATS (measure x9)
| _ => measure t
end.
Definition Marked_pols :
forall f,
(algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) ->
InterpGen.Pol_type A (InterpGen.get_arity f).
Proof.
intros f H.
apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H .
apply (Symb_more_list.change_in algebra.F.symb_order) in H .
set (u := (Symb_more_list.qs algebra.F.symb_order
(Symb_more_list.XSet.remove_red
(ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * .
vm_compute in u .
unfold u in * .
clear u .
unfold more_list.mem_bool in H .
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9;apply (P_id_SIEVE x9).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros ;apply (P_id_ZPRIMES ).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9;apply (P_id_ACTIVATE x9).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x9;apply (P_id_NATS x9).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros x11 x10 x9;apply (P_id_FILTER x11 x10 x9).
discriminate H.
Defined.
Lemma same_marked_measure :
forall t,
marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols
(ddp.defined_dec _ _
R_xml_0_deep_rew.R_xml_0_rules_included) t.
Proof.
intros [a| f l].
simpl in |-*.
unfold eq_rect_r, eq_rect, sym_eq in |-*.
reflexivity .
refine match f with
| algebra.F.id_filter =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_nats =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_activate =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_0 => match l with
| nil => _
| _::_ => _
end
| algebra.F.id_zprimes => match l with
| nil => _
| _::_ => _
end
| algebra.F.id_sieve =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_cons =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::_ => _
end
| algebra.F.id_n__nats =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_s =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
| algebra.F.id_n__filter =>
match l with
| nil => _
| _::nil => _
| _::_::nil => _
| _::_::_::nil => _
| _::_::_::_::_ => _
end
| algebra.F.id_n__sieve =>
match l with
| nil => _
| _::nil => _
| _::_::_ => _
end
end.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
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 .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
vm_compute in |-*;reflexivity .
lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ;
apply same_measure.
vm_compute in |-*;reflexivity .
Qed.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_activate
(x9::nil)) =>
P_id_ACTIVATE (measure x9)
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_ZPRIMES
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_SIEVE (measure x9)
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_FILTER (measure x11) (measure x10)
(measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_NATS (measure x9)
| _ => measure t
end.
Proof.
reflexivity .
Qed.
Lemma marked_measure_star_monotonic :
forall f l1 l2,
(closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step
R_xml_0_deep_rew.R_xml_0_rules)
) l1 l2) ->
marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term
f l2).
Proof.
intros .
do 2 (rewrite same_marked_measure in |-*).
apply InterpGen.marked_measure_star_monotonic with (1:=Aop) (Pols:=
Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules).
clear f.
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_filter_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_nats_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_activate_monotonic;assumption.
vm_compute in |-*;apply (Aop.(le_refl)).
vm_compute in |-*;apply (Aop.(le_refl)).
vm_compute in |-*;intros ;apply P_id_sieve_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__nats_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_s_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__filter_monotonic;assumption.
vm_compute in |-*;intros ;apply P_id_n__sieve_monotonic;assumption.
clear f.
intros f.
case f.
vm_compute in |-*;intros ;apply P_id_filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_activate_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_0_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_zprimes_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_sieve_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__nats_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_s_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__filter_bounded;assumption.
vm_compute in |-*;intros ;apply P_id_n__sieve_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_SIEVE_monotonic;assumption.
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros ;apply (Aop.(le_refl)).
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros ;apply P_id_ACTIVATE_monotonic;assumption.
match type of H with
| orb ?a ?b = true =>
assert (H':{a = true}+{b = true});[
revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]|
clear H;destruct H' as [H|H]]
end .
match type of H with
| _ _ ?t = true =>
generalize (algebra.F.symb_eq_bool_ok f t);
unfold algebra.Alg.eq_symb_bool in H;
rewrite H;clear H;intros ;subst
end .
vm_compute in |-*;intros ;apply P_id_NATS_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_FILTER_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_filter : Z ->Z ->Z ->Z.
Hypothesis P_id_nats : Z ->Z.
Hypothesis P_id_activate : Z ->Z.
Hypothesis P_id_0 : Z.
Hypothesis P_id_zprimes : Z.
Hypothesis P_id_sieve : Z ->Z.
Hypothesis P_id_cons : Z ->Z ->Z.
Hypothesis P_id_n__nats : Z ->Z.
Hypothesis P_id_s : Z ->Z.
Hypothesis P_id_n__filter : Z ->Z ->Z ->Z.
Hypothesis P_id_n__sieve : Z ->Z.
Hypothesis P_id_filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
P_id_filter x10 x12 x14 <= P_id_filter x9 x11 x13.
Hypothesis P_id_nats_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_nats x10 <= P_id_nats x9.
Hypothesis P_id_activate_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
Hypothesis P_id_sieve_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_sieve x10 <= P_id_sieve x9.
Hypothesis P_id_cons_monotonic :
forall x12 x10 x9 x11,
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
P_id_cons x10 x12 <= P_id_cons x9 x11.
Hypothesis P_id_n__nats_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_n__nats x10 <= P_id_n__nats x9.
Hypothesis P_id_s_monotonic :
forall x10 x9, (min_value <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
Hypothesis P_id_n__filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
P_id_n__filter x10 x12 x14 <= P_id_n__filter x9 x11 x13.
Hypothesis P_id_n__sieve_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_n__sieve x10 <= P_id_n__sieve x9.
Hypothesis P_id_filter_bounded :
forall x10 x9 x11,
(min_value <= x9) ->
(min_value <= x10) ->
(min_value <= x11) ->min_value <= P_id_filter x11 x10 x9.
Hypothesis P_id_nats_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_nats x9.
Hypothesis P_id_activate_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_activate x9.
Hypothesis P_id_0_bounded : min_value <= P_id_0 .
Hypothesis P_id_zprimes_bounded : min_value <= P_id_zprimes .
Hypothesis P_id_sieve_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_sieve x9.
Hypothesis P_id_cons_bounded :
forall x10 x9,
(min_value <= x9) ->(min_value <= x10) ->min_value <= P_id_cons x10 x9.
Hypothesis P_id_n__nats_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_n__nats x9.
Hypothesis P_id_s_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_s x9.
Hypothesis P_id_n__filter_bounded :
forall x10 x9 x11,
(min_value <= x9) ->
(min_value <= x10) ->
(min_value <= x11) ->min_value <= P_id_n__filter x11 x10 x9.
Hypothesis P_id_n__sieve_bounded :
forall x9, (min_value <= x9) ->min_value <= P_id_n__sieve x9.
Definition measure :=
Interp.measure min_value P_id_filter P_id_nats P_id_activate P_id_0
P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s P_id_n__filter
P_id_n__sieve.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_nats (measure x9)
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_activate (measure x9)
| (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_zprimes
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_sieve (measure x9)
| (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) =>
P_id_cons (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__nats (x9::nil)) =>
P_id_n__nats (measure x9)
| (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
P_id_s (measure x9)
| (algebra.Alg.Term algebra.F.id_n__filter (x11::x10::
x9::nil)) =>
P_id_n__filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil)) =>
P_id_n__sieve (measure x9)
| _ => min_value
end.
Proof.
intros t;case t;intros ;apply refl_equal.
Qed.
Lemma measure_bounded : forall t, min_value <= measure t.
Proof.
unfold measure in |-*.
apply Interp.measure_bounded with Alt Aeq;
(apply interp.o_Z)||
(cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
Qed.
Ltac generate_pos_hyp :=
match goal with
| H:context [measure ?x] |- _ =>
let v := fresh "v" in
(let H := fresh "h" in
(set (H:=measure_bounded x) in *;set (v:=measure x) in *;
clearbody H;clearbody v))
| |- context [measure ?x] =>
let v := fresh "v" in
(let H := fresh "h" in
(set (H:=measure_bounded x) in *;set (v:=measure x) in *;
clearbody H;clearbody v))
end
.
Hypothesis rules_monotonic :
forall l r,
(algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) ->
measure r <= measure l.
Lemma measure_star_monotonic :
forall l r,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
r l) ->measure r <= measure l.
Proof.
unfold measure in *.
apply Interp.measure_star_monotonic with Alt Aeq.
(apply interp.o_Z)||
(cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
intros ;apply P_id_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
Qed.
Hypothesis P_id_ACTIVATE : Z ->Z.
Hypothesis P_id_ZPRIMES : Z.
Hypothesis P_id_SIEVE : Z ->Z.
Hypothesis P_id_FILTER : Z ->Z ->Z ->Z.
Hypothesis P_id_NATS : Z ->Z.
Hypothesis P_id_ACTIVATE_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
Hypothesis P_id_SIEVE_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_SIEVE x10 <= P_id_SIEVE x9.
Hypothesis P_id_FILTER_monotonic :
forall x12 x10 x14 x9 x13 x11,
(min_value <= x14)/\ (x14 <= x13) ->
(min_value <= x12)/\ (x12 <= x11) ->
(min_value <= x10)/\ (x10 <= x9) ->
P_id_FILTER x10 x12 x14 <= P_id_FILTER x9 x11 x13.
Hypothesis P_id_NATS_monotonic :
forall x10 x9,
(min_value <= x10)/\ (x10 <= x9) ->P_id_NATS x10 <= P_id_NATS x9.
Definition marked_measure :=
Interp.marked_measure min_value P_id_filter P_id_nats P_id_activate
P_id_0 P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s
P_id_n__filter P_id_n__sieve P_id_ACTIVATE P_id_ZPRIMES P_id_SIEVE
P_id_FILTER P_id_NATS.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_activate
(x9::nil)) =>
P_id_ACTIVATE (measure x9)
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_ZPRIMES
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_SIEVE (measure x9)
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_FILTER (measure x11) (measure x10)
(measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_NATS (measure x9)
| _ => measure t
end.
Proof.
reflexivity .
Qed.
Lemma marked_measure_star_monotonic :
forall f l1 l2,
(closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step
R_xml_0_deep_rew.R_xml_0_rules)
) l1 l2) ->
marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term
f l2).
Proof.
unfold marked_measure in *.
apply Interp.marked_measure_star_monotonic with Alt Aeq.
(apply interp.o_Z)||
(cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith).
intros ;apply P_id_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_ACTIVATE_monotonic;assumption.
intros ;apply P_id_SIEVE_monotonic;assumption.
intros ;apply P_id_FILTER_monotonic;assumption.
intros ;apply P_id_NATS_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 :=
(* <filter(cons(X_,Y_),0,M_),activate(Y_)> *)
| DP_R_xml_0_0 :
forall x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_0 nil)
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <filter(cons(X_,Y_),s(N_),M_),activate(Y_)> *)
| DP_R_xml_0_1 :
forall x4 x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_s (x4::nil))
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <sieve(cons(0,Y_)),activate(Y_)> *)
| DP_R_xml_0_2 :
forall x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_0
nil)::x2::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <sieve(cons(s(N_),Y_)),filter(activate(Y_),N_,N_)> *)
| DP_R_xml_0_3 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_s
(x4::nil))::x2::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_filter ((algebra.Alg.Term
algebra.F.id_activate (x2::nil))::x4::x4::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <sieve(cons(s(N_),Y_)),activate(Y_)> *)
| DP_R_xml_0_4 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_s
(x4::nil))::x2::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <zprimes,sieve(nats(s(s(0))))> *)
| DP_R_xml_0_5 :
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_sieve ((algebra.Alg.Term
algebra.F.id_nats ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_s ((algebra.Alg.Term
algebra.F.id_0 nil)::nil))::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_zprimes nil)
(* <zprimes,nats(s(s(0)))> *)
| DP_R_xml_0_6 :
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_nats ((algebra.Alg.Term
algebra.F.id_s ((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_0 nil)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_zprimes nil)
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_7 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_filter (x5::x6::x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <activate(n__sieve(X_)),sieve(X_)> *)
| DP_R_xml_0_8 :
forall x1 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__sieve (x1::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_sieve (x1::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <activate(n__nats(X_)),nats(X_)> *)
| DP_R_xml_0_9 :
forall x1 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__nats (x1::nil)) x9) ->
DP_R_xml_0 (algebra.Alg.Term algebra.F.id_nats (x1::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
.
Module ddp := dp.MakeDP(algebra.EQT).
Lemma R_xml_0_dp_step_spec :
forall x y,
(ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) ->
exists f,
exists l1,
exists l2,
y = algebra.Alg.Term f l2/\
(closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step
R_xml_0_deep_rew.R_xml_0_rules)
) l1 l2)/\
(ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)).
Proof.
intros x y H.
induction H.
inversion H.
subst.
destruct t0.
refine ((False_ind) _ _).
refine (R_xml_0_deep_rew.R_xml_0_non_var H0).
simpl in H|-*.
exists a.
exists ((List.map) (algebra.Alg.apply_subst sigma) l).
exists ((List.map) (algebra.Alg.apply_subst sigma) l).
repeat (constructor).
assumption.
exists f.
exists l2.
exists l1.
constructor.
constructor.
constructor.
constructor.
rewrite <- closure.rwr_list_trans_clos_one_step_list.
assumption.
assumption.
Qed.
Ltac included_dp_tac H :=
injection H;clear H;intros;subst;
repeat (match goal with
| H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=>
let x := fresh "x" in
let l := fresh "l" in
let h1 := fresh "h" in
let h2 := fresh "h" in
let h3 := fresh "h" in
destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _ H) as [x [l[h1[h2 h3]]]];clear H;subst
| H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ =>
rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H
end
);simpl;
econstructor eassumption
.
Ltac dp_concl_tac h2 h cont_tac
t :=
match t with
| False => let h' := fresh "a" in
(set (h':=t) in *;cont_tac h';
repeat (
let e := type of h in
(match e with
| ?t => unfold t in h|-;
(case h;
[abstract (clear h;intros h;injection h;
clear h;intros ;subst;
included_dp_tac h2)|
clear h;intros h;clear t])
| ?t => unfold t in h|-;elim h
end
)
))
| or ?a ?b => let cont_tac
h' := let h'' := fresh "a" in
(set (h'':=or a h') in *;cont_tac h'') in
(dp_concl_tac h2 h cont_tac b)
end
.
Module WF_DP_R_xml_0.
Inductive DP_R_xml_0_non_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__nats(X_)),nats(X_)> *)
| DP_R_xml_0_non_scc_1_0 :
forall x1 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__nats (x1::nil)) x9) ->
DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_nats (x1::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
.
Lemma acc_DP_R_xml_0_non_scc_1 :
forall x y,
(DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Inductive DP_R_xml_0_non_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <zprimes,nats(s(s(0)))> *)
| DP_R_xml_0_non_scc_2_0 :
DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_nats
((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_0
nil)::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_zprimes 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_scc_3 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_scc_3_0 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil)) x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_filter (x5::x6::
x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <filter(cons(X_,Y_),0,M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_1 :
forall x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_0 nil)
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <activate(n__sieve(X_)),sieve(X_)> *)
| DP_R_xml_0_scc_3_2 :
forall x1 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__sieve (x1::nil)) x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_sieve (x1::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <sieve(cons(0,Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_3 :
forall x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_0 nil)::x2::nil)) x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <sieve(cons(s(N_),Y_)),filter(activate(Y_),N_,N_)> *)
| DP_R_xml_0_scc_3_4 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_filter
((algebra.Alg.Term algebra.F.id_activate
(x2::nil))::x4::x4::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <filter(cons(X_,Y_),s(N_),M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_5 :
forall x4 x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_s (x4::nil)) x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <sieve(cons(s(N_),Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_6 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3 (algebra.Alg.Term algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
.
Module WF_DP_R_xml_0_scc_3.
Inductive DP_R_xml_0_scc_3_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_scc_3_large_0 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil))
x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_filter (x5::
x6::x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <filter(cons(X_,Y_),0,M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_1 :
forall x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_0 nil)
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <sieve(cons(0,Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_2 :
forall x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_0 nil)::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <sieve(cons(s(N_),Y_)),filter(activate(Y_),N_,N_)> *)
| DP_R_xml_0_scc_3_large_3 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_filter
((algebra.Alg.Term algebra.F.id_activate
(x2::nil))::x4::x4::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
(* <filter(cons(X_,Y_),s(N_),M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_4 :
forall x4 x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_s (x4::nil)) x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <sieve(cons(s(N_),Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_5 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large (algebra.Alg.Term algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
.
Inductive DP_R_xml_0_scc_3_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__sieve(X_)),sieve(X_)> *)
| DP_R_xml_0_scc_3_strict_0 :
forall x1 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__sieve (x1::nil)) x9) ->
DP_R_xml_0_scc_3_strict (algebra.Alg.Term algebra.F.id_sieve
(x1::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
.
Module WF_DP_R_xml_0_scc_3_large.
Inductive DP_R_xml_0_scc_3_large_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <filter(cons(X_,Y_),0,M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_scc_1_0 :
forall x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_0 nil)
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large_scc_1 (algebra.Alg.Term
algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_scc_3_large_scc_1_1 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil))
x9) ->
DP_R_xml_0_scc_3_large_scc_1 (algebra.Alg.Term algebra.F.id_filter
(x5::x6::x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
(* <filter(cons(X_,Y_),s(N_),M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_scc_1_2 :
forall x4 x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_s (x4::nil)) x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large_scc_1 (algebra.Alg.Term
algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
.
Module WF_DP_R_xml_0_scc_3_large_scc_1.
Inductive DP_R_xml_0_scc_3_large_scc_1_large :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_scc_3_large_scc_1_large_0 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil))
x9) ->
DP_R_xml_0_scc_3_large_scc_1_large (algebra.Alg.Term
algebra.F.id_filter (x5::x6::
x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
.
Inductive DP_R_xml_0_scc_3_large_scc_1_strict :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <filter(cons(X_,Y_),0,M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_scc_1_strict_0 :
forall x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_0 nil)
x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large_scc_1_strict (algebra.Alg.Term
algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
(* <filter(cons(X_,Y_),s(N_),M_),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_scc_1_strict_1 :
forall x4 x2 x10 x1 x9 x3 x11,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons (x1::x2::nil)) x11) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_s (x4::nil)) x10) ->
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
x3 x9) ->
DP_R_xml_0_scc_3_large_scc_1_strict (algebra.Alg.Term
algebra.F.id_activate
(x2::nil))
(algebra.Alg.Term algebra.F.id_filter (x11::x10::x9::nil))
.
Module WF_DP_R_xml_0_scc_3_large_scc_1_large.
Inductive DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <activate(n__filter(X1_,X2_,X3_)),filter(X1_,X2_,X3_)> *)
| DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1_0 :
forall x6 x9 x5 x7,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_n__filter (x5::x6::x7::nil))
x9) ->
DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1 (algebra.Alg.Term
algebra.F.id_filter
(x5::x6::x7::nil))
(algebra.Alg.Term algebra.F.id_activate (x9::nil))
.
Lemma acc_DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1 :
forall x y,
(DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1 x y) ->
Acc WF_DP_R_xml_0_scc_3_large_scc_1.DP_R_xml_0_scc_3_large_scc_1_large
x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' )).
Qed.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_3_large_scc_1.DP_R_xml_0_scc_3_large_scc_1_large
.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3_large_scc_1_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_scc_1_large_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))).
Qed.
End WF_DP_R_xml_0_scc_3_large_scc_1_large.
Open Scope Z_scope.
Import ring_extention.
Notation Local "a <= b" := (Zle a b).
Notation Local "a < b" := (Zlt a b).
Definition P_id_filter (x9:Z) (x10:Z) (x11:Z) := 2* x9 + 1* x11.
Definition P_id_nats (x9:Z) := 2 + 2* x9.
Definition P_id_activate (x9:Z) := 2 + 2* x9.
Definition P_id_0 := 0.
Definition P_id_zprimes := 2.
Definition P_id_sieve (x9:Z) := 2.
Definition P_id_cons (x9:Z) (x10:Z) := 2 + 1* x9 + 1* x10.
Definition P_id_n__nats (x9:Z) := 2* x9.
Definition P_id_s (x9:Z) := 0.
Definition P_id_n__filter (x9:Z) (x10:Z) (x11:Z) := 1* x9 + 1* x11.
Definition P_id_n__sieve (x9:Z) := 0.
Lemma P_id_filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_filter x10 x12 x14 <= P_id_filter x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_nats_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_nats x10 <= P_id_nats x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_activate_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_sieve_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_sieve x10 <= P_id_sieve x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_cons_monotonic :
forall x12 x10 x9 x11,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11.
Proof.
intros x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__nats_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_n__nats x10 <= P_id_n__nats x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_s_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_n__filter x10 x12 x14 <= P_id_n__filter x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__sieve_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_n__sieve x10 <= P_id_n__sieve x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_filter_bounded :
forall x10 x9 x11,
(0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_filter x11 x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_nats_bounded : forall x9, (0 <= x9) ->0 <= P_id_nats x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_activate_bounded :
forall x9, (0 <= x9) ->0 <= P_id_activate x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_0_bounded : 0 <= P_id_0 .
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_zprimes_bounded : 0 <= P_id_zprimes .
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_sieve_bounded : forall x9, (0 <= x9) ->0 <= P_id_sieve x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_cons_bounded :
forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_cons x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__nats_bounded :
forall x9, (0 <= x9) ->0 <= P_id_n__nats x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_s_bounded : forall x9, (0 <= x9) ->0 <= P_id_s x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__filter_bounded :
forall x10 x9 x11,
(0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_n__filter x11 x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__sieve_bounded :
forall x9, (0 <= x9) ->0 <= P_id_n__sieve x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition measure :=
InterpZ.measure 0 P_id_filter P_id_nats P_id_activate P_id_0
P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s P_id_n__filter
P_id_n__sieve.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_nats (measure x9)
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_activate (measure x9)
| (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_zprimes
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_sieve (measure x9)
| (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) =>
P_id_cons (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__nats (x9::nil)) =>
P_id_n__nats (measure x9)
| (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
P_id_s (measure x9)
| (algebra.Alg.Term algebra.F.id_n__filter (x11::x10::
x9::nil)) =>
P_id_n__filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil)) =>
P_id_n__sieve (measure x9)
| _ => 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_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_ACTIVATE (x9:Z) := 1* x9.
Definition P_id_ZPRIMES := 0.
Definition P_id_SIEVE (x9:Z) := 0.
Definition P_id_FILTER (x9:Z) (x10:Z) (x11:Z) := 1* x9.
Definition P_id_NATS (x9:Z) := 0.
Lemma P_id_ACTIVATE_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_SIEVE_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_SIEVE x10 <= P_id_SIEVE x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_FILTER_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_FILTER x10 x12 x14 <= P_id_FILTER x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_NATS_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_NATS x10 <= P_id_NATS x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_filter P_id_nats P_id_activate P_id_0
P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s P_id_n__filter
P_id_n__sieve P_id_ACTIVATE P_id_ZPRIMES P_id_SIEVE P_id_FILTER
P_id_NATS.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_activate
(x9::nil)) =>
P_id_ACTIVATE (measure x9)
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_ZPRIMES
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_SIEVE (measure x9)
| (algebra.Alg.Term algebra.F.id_filter (x11::
x10::x9::nil)) =>
P_id_FILTER (measure x11) (measure x10)
(measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_NATS (measure x9)
| _ => measure t
end.
Proof.
reflexivity .
Qed.
Lemma marked_measure_star_monotonic :
forall f l1 l2,
(closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step
R_xml_0_deep_rew.R_xml_0_rules)
) l1 l2) ->
marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term
f
l2).
Proof.
unfold marked_measure in *.
apply InterpZ.marked_measure_star_monotonic.
intros ;apply P_id_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_ACTIVATE_monotonic;assumption.
intros ;apply P_id_SIEVE_monotonic;assumption.
intros ;apply P_id_FILTER_monotonic;assumption.
intros ;apply P_id_NATS_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_3_large_scc_1_strict_in_lt :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_3_large_scc_1_strict lt.
Proof.
unfold Relation_Definitions.inclusion, lt in *.
intros a b H;destruct H;
match goal with
| |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_lt_compat_right (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma DP_R_xml_0_scc_3_large_scc_1_large_in_le :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_3_large_scc_1_large le.
Proof.
unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *.
intros a b H;destruct H;
match goal with
| |- _ <= marked_measure (algebra.Alg.Term ?f ?l) =>
let l'' := algebra.Alg_ext.find_replacement l in
((apply (interp.le_trans (interp.o_Z 0)) with
(marked_measure (algebra.Alg.Term f l''));[idtac|
apply marked_measure_star_monotonic;
repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos);
(assumption)||(constructor 1)]))
end
;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp );
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition wf_DP_R_xml_0_scc_3_large_scc_1_large :=
WF_DP_R_xml_0_scc_3_large_scc_1_large.wf.
Lemma wf :
well_founded WF_DP_R_xml_0_scc_3_large.DP_R_xml_0_scc_3_large_scc_1.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_3_large_scc_1_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';apply DP_R_xml_0_scc_3_large_scc_1_strict_in_lt;
econstructor eassumption)||
((apply IHx;[econstructor eassumption|
intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ;
apply DP_R_xml_0_scc_3_large_scc_1_large_in_le;
econstructor eassumption])).
apply wf_DP_R_xml_0_scc_3_large_scc_1_large.
Qed.
End WF_DP_R_xml_0_scc_3_large_scc_1.
Definition wf_DP_R_xml_0_scc_3_large_scc_1 :=
WF_DP_R_xml_0_scc_3_large_scc_1.wf.
Lemma acc_DP_R_xml_0_scc_3_large_scc_1 :
forall x y,
(DP_R_xml_0_scc_3_large_scc_1 x y) ->
Acc WF_DP_R_xml_0_scc_3.DP_R_xml_0_scc_3_large x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_3_large_scc_1).
intros x' _ Hrec y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply Hrec;econstructor eassumption)||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
apply wf_DP_R_xml_0_scc_3_large_scc_1.
Qed.
Inductive DP_R_xml_0_scc_3_large_non_scc_2 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <sieve(cons(s(N_),Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_non_scc_2_0 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large_non_scc_2 (algebra.Alg.Term
algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
.
Lemma acc_DP_R_xml_0_scc_3_large_non_scc_2 :
forall x y,
(DP_R_xml_0_scc_3_large_non_scc_2 x y) ->
Acc WF_DP_R_xml_0_scc_3.DP_R_xml_0_scc_3_large x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
Qed.
Inductive DP_R_xml_0_scc_3_large_non_scc_3 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <sieve(cons(0,Y_)),activate(Y_)> *)
| DP_R_xml_0_scc_3_large_non_scc_3_0 :
forall x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_0 nil)::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large_non_scc_3 (algebra.Alg.Term
algebra.F.id_activate (x2::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
.
Lemma acc_DP_R_xml_0_scc_3_large_non_scc_3 :
forall x y,
(DP_R_xml_0_scc_3_large_non_scc_3 x y) ->
Acc WF_DP_R_xml_0_scc_3.DP_R_xml_0_scc_3_large x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
Qed.
Inductive DP_R_xml_0_scc_3_large_non_scc_4 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <sieve(cons(s(N_),Y_)),filter(activate(Y_),N_,N_)> *)
| DP_R_xml_0_scc_3_large_non_scc_4_0 :
forall x4 x2 x9,
(closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules)
(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term
algebra.F.id_s (x4::nil))::x2::nil)) x9) ->
DP_R_xml_0_scc_3_large_non_scc_4 (algebra.Alg.Term
algebra.F.id_filter
((algebra.Alg.Term
algebra.F.id_activate (x2::nil))::
x4::x4::nil))
(algebra.Alg.Term algebra.F.id_sieve (x9::nil))
.
Lemma acc_DP_R_xml_0_scc_3_large_non_scc_4 :
forall x y,
(DP_R_xml_0_scc_3_large_non_scc_4 x y) ->
Acc WF_DP_R_xml_0_scc_3.DP_R_xml_0_scc_3_large x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
Qed.
Lemma wf : well_founded WF_DP_R_xml_0_scc_3.DP_R_xml_0_scc_3_large.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3_large_non_scc_4;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_non_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_non_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_large_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3_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_3_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_filter (x9:Z) (x10:Z) (x11:Z) := 1* x9.
Definition P_id_nats (x9:Z) := 0.
Definition P_id_activate (x9:Z) := 1* x9.
Definition P_id_0 := 0.
Definition P_id_zprimes := 2.
Definition P_id_sieve (x9:Z) := 1 + 2* x9.
Definition P_id_cons (x9:Z) (x10:Z) := 1* x10.
Definition P_id_n__nats (x9:Z) := 0.
Definition P_id_s (x9:Z) := 0.
Definition P_id_n__filter (x9:Z) (x10:Z) (x11:Z) := 1* x9.
Definition P_id_n__sieve (x9:Z) := 1 + 2* x9.
Lemma P_id_filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_filter x10 x12 x14 <= P_id_filter x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_nats_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_nats x10 <= P_id_nats x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_activate_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_activate x10 <= P_id_activate x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_sieve_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_sieve x10 <= P_id_sieve x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_cons_monotonic :
forall x12 x10 x9 x11,
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->P_id_cons x10 x12 <= P_id_cons x9 x11.
Proof.
intros x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__nats_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_n__nats x10 <= P_id_n__nats x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_s_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_s x10 <= P_id_s x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__filter_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_n__filter x10 x12 x14 <= P_id_n__filter x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__sieve_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_n__sieve x10 <= P_id_n__sieve x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_filter_bounded :
forall x10 x9 x11,
(0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_filter x11 x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_nats_bounded : forall x9, (0 <= x9) ->0 <= P_id_nats x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_activate_bounded :
forall x9, (0 <= x9) ->0 <= P_id_activate x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_0_bounded : 0 <= P_id_0 .
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_zprimes_bounded : 0 <= P_id_zprimes .
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_sieve_bounded : forall x9, (0 <= x9) ->0 <= P_id_sieve x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_cons_bounded :
forall x10 x9, (0 <= x9) ->(0 <= x10) ->0 <= P_id_cons x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__nats_bounded : forall x9, (0 <= x9) ->0 <= P_id_n__nats x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_s_bounded : forall x9, (0 <= x9) ->0 <= P_id_s x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__filter_bounded :
forall x10 x9 x11,
(0 <= x9) ->(0 <= x10) ->(0 <= x11) ->0 <= P_id_n__filter x11 x10 x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_n__sieve_bounded :
forall x9, (0 <= x9) ->0 <= P_id_n__sieve x9.
Proof.
intros .
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition measure :=
InterpZ.measure 0 P_id_filter P_id_nats P_id_activate P_id_0
P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s P_id_n__filter
P_id_n__sieve.
Lemma measure_equation :
forall t,
measure t = match t with
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_nats (measure x9)
| (algebra.Alg.Term algebra.F.id_activate (x9::nil)) =>
P_id_activate (measure x9)
| (algebra.Alg.Term algebra.F.id_0 nil) => P_id_0
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_zprimes
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_sieve (measure x9)
| (algebra.Alg.Term algebra.F.id_cons (x10::x9::nil)) =>
P_id_cons (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__nats (x9::nil)) =>
P_id_n__nats (measure x9)
| (algebra.Alg.Term algebra.F.id_s (x9::nil)) =>
P_id_s (measure x9)
| (algebra.Alg.Term algebra.F.id_n__filter (x11::x10::
x9::nil)) =>
P_id_n__filter (measure x11) (measure x10) (measure x9)
| (algebra.Alg.Term algebra.F.id_n__sieve (x9::nil)) =>
P_id_n__sieve (measure x9)
| _ => 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_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
Qed.
Definition P_id_ACTIVATE (x9:Z) := 2* x9.
Definition P_id_ZPRIMES := 0.
Definition P_id_SIEVE (x9:Z) := 2* x9.
Definition P_id_FILTER (x9:Z) (x10:Z) (x11:Z) := 2* x9.
Definition P_id_NATS (x9:Z) := 0.
Lemma P_id_ACTIVATE_monotonic :
forall x10 x9,
(0 <= x10)/\ (x10 <= x9) ->P_id_ACTIVATE x10 <= P_id_ACTIVATE x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_SIEVE_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_SIEVE x10 <= P_id_SIEVE x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_FILTER_monotonic :
forall x12 x10 x14 x9 x13 x11,
(0 <= x14)/\ (x14 <= x13) ->
(0 <= x12)/\ (x12 <= x11) ->
(0 <= x10)/\ (x10 <= x9) ->
P_id_FILTER x10 x12 x14 <= P_id_FILTER x9 x11 x13.
Proof.
intros x14 x13 x12 x11 x10 x9.
intros [H_1 H_0].
intros [H_3 H_2].
intros [H_5 H_4].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Lemma P_id_NATS_monotonic :
forall x10 x9, (0 <= x10)/\ (x10 <= x9) ->P_id_NATS x10 <= P_id_NATS x9.
Proof.
intros x10 x9.
intros [H_1 H_0].
cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition;
(auto with zarith)||(repeat (translate_vars );prove_ineq ).
Qed.
Definition marked_measure :=
InterpZ.marked_measure 0 P_id_filter P_id_nats P_id_activate P_id_0
P_id_zprimes P_id_sieve P_id_cons P_id_n__nats P_id_s P_id_n__filter
P_id_n__sieve P_id_ACTIVATE P_id_ZPRIMES P_id_SIEVE P_id_FILTER
P_id_NATS.
Lemma marked_measure_equation :
forall t,
marked_measure t = match t with
| (algebra.Alg.Term algebra.F.id_activate
(x9::nil)) =>
P_id_ACTIVATE (measure x9)
| (algebra.Alg.Term algebra.F.id_zprimes nil) =>
P_id_ZPRIMES
| (algebra.Alg.Term algebra.F.id_sieve (x9::nil)) =>
P_id_SIEVE (measure x9)
| (algebra.Alg.Term algebra.F.id_filter (x11::x10::
x9::nil)) =>
P_id_FILTER (measure x11) (measure x10)
(measure x9)
| (algebra.Alg.Term algebra.F.id_nats (x9::nil)) =>
P_id_NATS (measure x9)
| _ => measure t
end.
Proof.
reflexivity .
Qed.
Lemma marked_measure_star_monotonic :
forall f l1 l2,
(closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step
R_xml_0_deep_rew.R_xml_0_rules)
) l1 l2) ->
marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term
f l2).
Proof.
unfold marked_measure in *.
apply InterpZ.marked_measure_star_monotonic.
intros ;apply P_id_filter_monotonic;assumption.
intros ;apply P_id_nats_monotonic;assumption.
intros ;apply P_id_activate_monotonic;assumption.
intros ;apply P_id_sieve_monotonic;assumption.
intros ;apply P_id_cons_monotonic;assumption.
intros ;apply P_id_n__nats_monotonic;assumption.
intros ;apply P_id_s_monotonic;assumption.
intros ;apply P_id_n__filter_monotonic;assumption.
intros ;apply P_id_n__sieve_monotonic;assumption.
intros ;apply P_id_filter_bounded;assumption.
intros ;apply P_id_nats_bounded;assumption.
intros ;apply P_id_activate_bounded;assumption.
intros ;apply P_id_0_bounded;assumption.
intros ;apply P_id_zprimes_bounded;assumption.
intros ;apply P_id_sieve_bounded;assumption.
intros ;apply P_id_cons_bounded;assumption.
intros ;apply P_id_n__nats_bounded;assumption.
intros ;apply P_id_s_bounded;assumption.
intros ;apply P_id_n__filter_bounded;assumption.
intros ;apply P_id_n__sieve_bounded;assumption.
apply rules_monotonic.
intros ;apply P_id_ACTIVATE_monotonic;assumption.
intros ;apply P_id_SIEVE_monotonic;assumption.
intros ;apply P_id_FILTER_monotonic;assumption.
intros ;apply P_id_NATS_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_3_strict_in_lt :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_3_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_3_large_in_le :
Relation_Definitions.inclusion _ DP_R_xml_0_scc_3_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_3_large := WF_DP_R_xml_0_scc_3_large.wf.
Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_3.
Proof.
intros x.
apply (well_founded_ind wf_lt).
clear x.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_3_large).
clear x.
intros x _ IHx IHx'.
constructor.
intros y H.
destruct H;
(apply IHx';apply DP_R_xml_0_scc_3_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_3_large_in_le;econstructor eassumption])).
apply wf_DP_R_xml_0_scc_3_large.
Qed.
End WF_DP_R_xml_0_scc_3.
Definition wf_DP_R_xml_0_scc_3 := WF_DP_R_xml_0_scc_3.wf.
Lemma acc_DP_R_xml_0_scc_3 :
forall x y, (DP_R_xml_0_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x.
pattern x.
apply (@Acc_ind _ DP_R_xml_0_scc_3).
intros x' _ Hrec y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply Hrec;econstructor eassumption)||
((eapply acc_DP_R_xml_0_non_scc_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_3.
Qed.
Inductive DP_R_xml_0_non_scc_4 :
algebra.Alg.term ->algebra.Alg.term ->Prop :=
(* <zprimes,sieve(nats(s(s(0))))> *)
| DP_R_xml_0_non_scc_4_0 :
DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_sieve
((algebra.Alg.Term algebra.F.id_nats
((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_s
((algebra.Alg.Term algebra.F.id_0
nil)::nil))::nil))::nil))::nil))
(algebra.Alg.Term algebra.F.id_zprimes nil)
.
Lemma acc_DP_R_xml_0_non_scc_4 :
forall x y,
(DP_R_xml_0_non_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x.
Proof.
intros x y h.
inversion h;clear h;subst;
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||
(eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))).
Qed.
Lemma wf : well_founded WF_R_xml_0_deep_rew.DP_R_xml_0.
Proof.
constructor;intros _y _h;inversion _h;clear _h;subst;
(eapply acc_DP_R_xml_0_non_scc_4;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_non_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_3;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_2;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_1;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((eapply acc_DP_R_xml_0_scc_0;
econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))||
((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail)))))))))).
Qed.
End WF_DP_R_xml_0.
Definition wf_H := WF_DP_R_xml_0.wf.
Lemma wf :
well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules).
Proof.
apply ddp.dp_criterion.
apply R_xml_0_deep_rew.R_xml_0_non_var.
apply R_xml_0_deep_rew.R_xml_0_reg.
intros ;
apply (ddp.constructor_defined_dec _ _
R_xml_0_deep_rew.R_xml_0_rules_included).
refine (Inclusion.wf_incl _ _ _ _ wf_H).
intros x y H.
destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]].
destruct (ddp.dp_list_complete _ _
R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3)
as [x' [y' [sigma [h1 [h2 h3]]]]].
clear H3.
subst.
vm_compute in h3|-.
let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e).
Qed.
End WF_R_xml_0_deep_rew.
(*
*** Local Variables: ***
*** coq-prog-name: "coqtop" ***
*** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") ***
*** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/ c_output/strat/tpdb-5.0___TRS___TRCSR___Ex9_BLR02_Z.trs/a3pat.v" ***
*** End: ***
*)