Commit f4619355 authored by Dan Frumin's avatar Dan Frumin
Browse files

Ordering induced by ⊕ conincides with contextual refinement

parent d698a0c4
......@@ -288,4 +288,75 @@ Section rules.
End rules.
(* TODO: write out the contextual refinements *)
(** Separately, we prove that the ordering induced by ⊕ conincides with
contextual refinement. *)
Lemma Seq_typed Γ e1 e2 τ :
Γ e1 : TUnit
Γ e2 : τ
Γ (e1;;e2) : τ.
Proof. by repeat (econstructor; eauto). Qed.
Lemma or_equiv_refines_1 e t :
( e : TUnit)
( t : TUnit)
( e ctx t : TUnit)
( (e t) =ctx= t : TUnit).
Proof.
intros Te Tt Het. split; last first.
- eapply (ctx_refines_transitive _ _ _ (t e)%V).
+ eapply (refines_sound relocΣ).
iIntros (? Δ).
iApply (or_choice_1_r t t with "[]").
by iApply refines_typed.
+ eapply (refines_sound relocΣ).
iIntros (? Δ). rel_pures_r.
iApply or_comm; by iApply (refines_typed TUnit []).
- eapply (ctx_refines_transitive _ _ _ (t t)%E).
+ ctx_bind_l e.
ctx_bind_r t.
eapply (ctx_refines_congruence _ _ TUnit);
last eassumption.
{ cbn.
econstructor; cbn; eauto.
- econstructor; cbn; eauto.
econstructor; cbn; eauto.
- econstructor; cbn; eauto.
+ econstructor; cbn; eauto.
econstructor; cbn; eauto.
econstructor; cbn; eauto.
change Z0 with (Z.of_nat 0%nat).
econstructor; cbn; eauto.
econstructor; cbn; eauto.
* econstructor; cbn; eauto.
eapply Seq_typed.
** change 1%Z with (Z.of_nat 1%nat).
repeat (econstructor; cbn; eauto).
** repeat (econstructor; cbn; eauto).
* repeat (econstructor; cbn; eauto).
+ econstructor; cbn; eauto.
2:{ econstructor; cbn; eauto. }
enough (typed_ctx_item (CTX_Rec <> <>)
(binder_insert BAnon (TUnit TUnit)%ty
(binder_insert BAnon TUnit ( : stringmap type)))
TUnit (TUnit TUnit)).
{ by simpl in *. }
eapply (TP_CTX_Rec TUnit TUnit BAnon BAnon). }
+ eapply (refines_sound relocΣ).
iIntros (? Δ). rel_pures_l.
iApply or_idemp_l. by iApply (refines_typed TUnit []).
Qed.
Lemma or_equiv_refines_2 e t :
( e : TUnit)
( t : TUnit)
( (e t) =ctx= t : TUnit)
( e ctx t : TUnit).
Proof.
intros Te Tt [Het1 Het2].
eapply (ctx_refines_transitive _ _ _ (e t)%E); last assumption.
eapply (refines_sound relocΣ).
iIntros (? Δ).
rel_pures_r. iApply or_choice_1_r.
by iApply (refines_typed TUnit []).
Qed.
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