Commit 4f9deccf authored by Robbert Krebbers's avatar Robbert Krebbers

Prove (P -∗ Q) ⊣⊢ ∃ R, R ∗ □ (P ∗ R → Q) and dually for →.

parent 066ce7e2
......@@ -536,6 +536,24 @@ Proof. intros; rewrite -always_and_sep_r'; auto. Qed.
Lemma always_laterN n P : ^n P ⊣⊢ ^n P.
Proof. induction n as [|n IH]; simpl; auto. by rewrite always_later IH. Qed.
Lemma wand_alt P Q : (P - Q) ⊣⊢ R, R (P R Q).
Proof.
apply (anti_symm ()).
- rewrite -(right_id True%I uPred_sep (P - Q)%I) -(exist_intro (P - Q)%I).
apply sep_mono_r. rewrite -always_pure. apply always_mono, impl_intro_l.
by rewrite wand_elim_r right_id.
- apply exist_elim=> R. apply wand_intro_l. rewrite assoc -always_and_sep_r'.
by rewrite always_elim impl_elim_r.
Qed.
Lemma impl_alt P Q : (P Q) ⊣⊢ R, R (P R - Q).
Proof.
apply (anti_symm ()).
- rewrite -(right_id True%I uPred_and (P Q)%I) -(exist_intro (P Q)%I).
apply and_mono_r. rewrite -always_pure. apply always_mono, wand_intro_l.
by rewrite impl_elim_r right_id.
- apply exist_elim=> R. apply impl_intro_l. rewrite assoc always_and_sep_r'.
by rewrite always_elim wand_elim_r.
Qed.
(* Later derived *)
Lemma later_proper P Q : (P ⊣⊢ Q) P ⊣⊢ Q.
......
......@@ -79,4 +79,7 @@ Qed.
Lemma vs_alloc N P : P ={N}=> inv N P.
Proof. iIntros "!# HP". by iApply inv_alloc. Qed.
Lemma wand_fupd_alt E1 E2 P Q : (P ={E1,E2}= Q) ⊣⊢ R, R (P R ={E1,E2}=> Q).
Proof. rewrite uPred.wand_alt. by setoid_rewrite <-uPred.always_wand_impl. Qed.
End vs.
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