Commit 8e0effe3 authored by Robbert Krebbers's avatar Robbert Krebbers

Prove that -∗ and → with a pure LHS are persistent.

parent 84481877
......@@ -808,9 +808,43 @@ Global Instance from_option_timeless {A} P (Ψ : A → uPred M) (mx : option A)
( x, TimelessP (Ψ x)) TimelessP P TimelessP (from_option Ψ P mx).
Proof. destruct mx; apply _. Qed.
(* Derived lemmas for persistence *)
Lemma always_always P `{!PersistentP P} : P P.
Proof. apply (anti_symm ()); auto using always_elim. Qed.
Lemma always_if_always p P `{!PersistentP P} : ?p P P.
Proof. destruct p; simpl; auto using always_always. Qed.
Lemma always_intro P Q `{!PersistentP P} : (P Q) P Q.
Proof. rewrite -(always_always P); apply always_intro'. Qed.
Lemma always_and_sep_l P Q `{!PersistentP P} : P Q P Q.
Proof. by rewrite -(always_always P) always_and_sep_l'. Qed.
Lemma always_and_sep_r P Q `{!PersistentP Q} : P Q P Q.
Proof. by rewrite -(always_always Q) always_and_sep_r'. Qed.
Lemma always_sep_dup P `{!PersistentP P} : P P P.
Proof. by rewrite -(always_always P) -always_sep_dup'. Qed.
Lemma always_entails_l P Q `{!PersistentP Q} : (P Q) P Q P.
Proof. by rewrite -(always_always Q); apply always_entails_l'. Qed.
Lemma always_entails_r P Q `{!PersistentP Q} : (P Q) P P Q.
Proof. by rewrite -(always_always Q); apply always_entails_r'. Qed.
Lemma always_impl_wand P `{!PersistentP P} Q : (P Q) (P - Q).
Proof.
apply (anti_symm _); auto using impl_wand.
apply impl_intro_l. by rewrite always_and_sep_l wand_elim_r.
Qed.
(* Persistence *)
Global Instance pure_persistent φ : PersistentP (⌜φ⌝ : uPred M)%I.
Proof. by rewrite /PersistentP always_pure. Qed.
Global Instance pure_impl_persistent φ Q :
PersistentP Q PersistentP (⌜φ⌝ Q)%I.
Proof.
rewrite /PersistentP pure_impl_forall always_forall. auto using forall_mono.
Qed.
Global Instance pure_wand_persistent φ Q :
PersistentP Q PersistentP (⌜φ⌝ - Q)%I.
Proof.
rewrite /PersistentP -always_impl_wand pure_impl_forall always_forall.
auto using forall_mono.
Qed.
Global Instance always_persistent P : PersistentP ( P).
Proof. by intros; apply always_intro'. Qed.
Global Instance and_persistent P Q :
......@@ -843,23 +877,5 @@ Proof. intros. by rewrite /PersistentP always_ownM. Qed.
Global Instance from_option_persistent {A} P (Ψ : A uPred M) (mx : option A) :
( x, PersistentP (Ψ x)) PersistentP P PersistentP (from_option Ψ P mx).
Proof. destruct mx; apply _. Qed.
(* Derived lemmas for persistence *)
Lemma always_always P `{!PersistentP P} : P P.
Proof. apply (anti_symm ()); auto using always_elim. Qed.
Lemma always_if_always p P `{!PersistentP P} : ?p P P.
Proof. destruct p; simpl; auto using always_always. Qed.
Lemma always_intro P Q `{!PersistentP P} : (P Q) P Q.
Proof. rewrite -(always_always P); apply always_intro'. Qed.
Lemma always_and_sep_l P Q `{!PersistentP P} : P Q P Q.
Proof. by rewrite -(always_always P) always_and_sep_l'. Qed.
Lemma always_and_sep_r P Q `{!PersistentP Q} : P Q P Q.
Proof. by rewrite -(always_always Q) always_and_sep_r'. Qed.
Lemma always_sep_dup P `{!PersistentP P} : P P P.
Proof. by rewrite -(always_always P) -always_sep_dup'. Qed.
Lemma always_entails_l P Q `{!PersistentP Q} : (P Q) P Q P.
Proof. by rewrite -(always_always Q); apply always_entails_l'. Qed.
Lemma always_entails_r P Q `{!PersistentP Q} : (P Q) P P Q.
Proof. by rewrite -(always_always Q); apply always_entails_r'. Qed.
End derived.
End uPred.
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