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Equivalence.v
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Equivalence.v
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Require Import Coq.Strings.String.
Require Export SystemFR.StarLemmas.
Require Export SystemFR.ErasedTermLemmas.
(* Two terms `t1` and `t2` are observationally equivalent if for any context, *)
(* `C[t1]` reduces to true iff `C[t2]` reduces to true *)
Definition equivalent_terms t1 t2 :=
is_erased_term t1 /\
is_erased_term t2 /\
wf t1 0 /\
wf t2 0 /\
pfv t1 term_var = nil /\
pfv t2 term_var = nil /\
forall C,
is_erased_term C ->
wf C 1 ->
pfv C term_var = nil ->
open 0 C t1 ~>* ttrue <->
open 0 C t2 ~>* ttrue.
Notation "[ t1 ≡ t2 ]" := (equivalent_terms t1 t2) (t1 at level 60, t2 at level 60, at level 60).
Ltac equivalence_instantiate C :=
match goal with
| H: [ _ ≡ _ ] |- _ =>
poseNew (Mark (H) "equivalence_instantiate");
unshelve epose proof ((proj2 (proj2 (proj2 (proj2 (proj2 (proj2 H)))))) C _ _ _)
end.
Lemma equivalent_terms_fv1:
forall t1 t2,
[ t1 ≡ t2 ] ->
pfv t1 term_var = nil.
Proof.
unfold equivalent_terms; steps.
Qed.
Lemma equivalent_terms_fv2:
forall t1 t2,
[ t1 ≡ t2 ] ->
pfv t2 term_var = nil.
Proof.
unfold equivalent_terms; steps.
Qed.
#[export]
Hint Immediate equivalent_terms_fv1: fv.
#[export]
Hint Immediate equivalent_terms_fv2: fv.
Lemma equivalent_terms_erased1:
forall t1 t2,
[ t1 ≡ t2 ] ->
is_erased_term t1.
Proof.
unfold equivalent_terms; steps.
Qed.
Lemma equivalent_terms_erased2:
forall t1 t2,
[ t1 ≡ t2 ] ->
is_erased_term t2.
Proof.
unfold equivalent_terms; steps.
Qed.
#[export]
Hint Immediate equivalent_terms_erased1: erased.
#[export]
Hint Immediate equivalent_terms_erased2: erased.
Lemma equivalent_terms_wf1:
forall t1 t2 k,
[ t1 ≡ t2 ] ->
wf t1 k.
Proof.
unfold equivalent_terms; steps; eauto with wf.
Qed.
Lemma equivalent_terms_wf2:
forall t1 t2 k,
[ t1 ≡ t2 ] ->
wf t2 k.
Proof.
unfold equivalent_terms; steps; eauto with wf.
Qed.
#[export]
Hint Immediate equivalent_terms_wf1: wf.
#[export]
Hint Immediate equivalent_terms_wf2: wf.
Lemma equivalent_terms_twf1:
forall t1 t2 k,
[ t1 ≡ t2 ] ->
twf t1 k.
Proof.
unfold equivalent_terms; steps; eauto with twf.
Qed.
Lemma equivalent_terms_twf2:
forall t1 t2 k,
[ t1 ≡ t2 ] ->
twf t2 k.
Proof.
unfold equivalent_terms; steps; eauto with twf.
Qed.
#[export]
Hint Immediate equivalent_terms_twf1: twf.
#[export]
Hint Immediate equivalent_terms_twf2: twf.