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Commit b441f29b authored by Robbert Krebbers's avatar Robbert Krebbers
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Symmetric variant of wand_intro.

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...@@ -538,7 +538,7 @@ Proof. ...@@ -538,7 +538,7 @@ Proof.
+ by rewrite (associative op) -Hy -Hx. + by rewrite (associative op) -Hy -Hx.
+ by exists y2, x2. + by exists y2, x2.
Qed. Qed.
Lemma wand_intro P Q R : (P ★ Q) ⊑ R → P ⊑ (Q -★ R). Lemma wand_intro_r P Q R : (P ★ Q) ⊑ R → P ⊑ (Q -★ R).
Proof. Proof.
intros HPQR x n ?? x' n' ???; apply HPQR; auto. intros HPQR x n ?? x' n' ???; apply HPQR; auto.
exists x, x'; split_ands; auto. exists x, x'; split_ands; auto.
...@@ -556,7 +556,7 @@ Hint Resolve sep_mono. ...@@ -556,7 +556,7 @@ Hint Resolve sep_mono.
Global Instance sep_mono' : Proper ((⊑) ==> (⊑) ==> (⊑)) (@uPred_sep M). Global Instance sep_mono' : Proper ((⊑) ==> (⊑) ==> (⊑)) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Lemma wand_mono P P' Q Q' : Q ⊑ P → P' ⊑ Q' → (P -★ P') ⊑ (Q -★ Q'). Lemma wand_mono P P' Q Q' : Q ⊑ P → P' ⊑ Q' → (P -★ P') ⊑ (Q -★ Q').
Proof. intros HP HQ; apply wand_intro; rewrite HP -HQ; apply wand_elim_l. Qed. Proof. intros HP HQ; apply wand_intro_r; rewrite HP -HQ; apply wand_elim_l. Qed.
Global Instance wand_mono' : Proper (flip (⊑) ==> (⊑) ==> (⊑)) (@uPred_wand M). Global Instance wand_mono' : Proper (flip (⊑) ==> (⊑) ==> (⊑)) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
...@@ -571,6 +571,8 @@ Proof. intros ->; apply sep_elim_l. Qed. ...@@ -571,6 +571,8 @@ Proof. intros ->; apply sep_elim_l. Qed.
Lemma sep_elim_r' P Q R : Q ⊑ R → (P ★ Q) ⊑ R. Lemma sep_elim_r' P Q R : Q ⊑ R → (P ★ Q) ⊑ R.
Proof. intros ->; apply sep_elim_r. Qed. Proof. intros ->; apply sep_elim_r. Qed.
Hint Resolve sep_elim_l' sep_elim_r'. Hint Resolve sep_elim_l' sep_elim_r'.
Lemma wand_intro_l P Q R : (Q ★ P) ⊑ R → P ⊑ (Q -★ R).
Proof. rewrite (commutative _); apply wand_intro_r. Qed.
Lemma wand_elim_r P Q : (P ★ (P -★ Q)) ⊑ Q. Lemma wand_elim_r P Q : (P ★ (P -★ Q)) ⊑ Q.
Proof. rewrite (commutative _ P); apply wand_elim_l. Qed. Proof. rewrite (commutative _ P); apply wand_elim_l. Qed.
Lemma wand_elim_l' P Q R : P ⊑ (Q -★ R) → (P ★ Q) ⊑ R. Lemma wand_elim_l' P Q R : P ⊑ (Q -★ R) → (P ★ Q) ⊑ R.
...@@ -580,7 +582,7 @@ Proof. intros ->; apply wand_elim_r. Qed. ...@@ -580,7 +582,7 @@ Proof. intros ->; apply wand_elim_r. Qed.
Lemma sep_and P Q : (P ★ Q) ⊑ (P ∧ Q). Lemma sep_and P Q : (P ★ Q) ⊑ (P ∧ Q).
Proof. auto. Qed. Proof. auto. Qed.
Lemma impl_wand P Q : (P → Q) ⊑ (P -★ Q). Lemma impl_wand P Q : (P → Q) ⊑ (P -★ Q).
Proof. apply wand_intro, impl_elim with P; auto. Qed. Proof. apply wand_intro_r, impl_elim with P; auto. Qed.
Global Instance sep_False : LeftAbsorb (≡) False%I (@uPred_sep M). Global Instance sep_False : LeftAbsorb (≡) False%I (@uPred_sep M).
Proof. intros P; apply (anti_symmetric _); auto. Qed. Proof. intros P; apply (anti_symmetric _); auto. Qed.
...@@ -651,7 +653,7 @@ Proof. ...@@ -651,7 +653,7 @@ Proof.
apply later_mono, impl_elim with P; auto. apply later_mono, impl_elim with P; auto.
Qed. Qed.
Lemma later_wand P Q : ▷ (P -★ Q) ⊑ (▷ P -★ ▷ Q). Lemma later_wand P Q : ▷ (P -★ Q) ⊑ (▷ P -★ ▷ Q).
Proof. apply wand_intro; rewrite -later_sep; apply later_mono,wand_elim_l. Qed. Proof. apply wand_intro_r;rewrite -later_sep; apply later_mono,wand_elim_l. Qed.
(* Always *) (* Always *)
Lemma always_const φ : (□ ■ φ : uPred M)%I ≡ (■ φ)%I. Lemma always_const φ : (□ ■ φ : uPred M)%I ≡ (■ φ)%I.
...@@ -713,7 +715,7 @@ Proof. by rewrite !(commutative _ P) always_and_sep_l. Qed. ...@@ -713,7 +715,7 @@ Proof. by rewrite !(commutative _ P) always_and_sep_l. Qed.
Lemma always_sep P Q : (□ (P ★ Q))%I ≡ (□ P ★ □ Q)%I. Lemma always_sep P Q : (□ (P ★ Q))%I ≡ (□ P ★ □ Q)%I.
Proof. by rewrite -always_and_sep -always_and_sep_l always_and. Qed. Proof. by rewrite -always_and_sep -always_and_sep_l always_and. Qed.
Lemma always_wand P Q : □ (P -★ Q) ⊑ (□ P -★ □ Q). Lemma always_wand P Q : □ (P -★ Q) ⊑ (□ P -★ □ Q).
Proof. by apply wand_intro; rewrite -always_sep wand_elim_l. Qed. Proof. by apply wand_intro_r; rewrite -always_sep wand_elim_l. Qed.
Lemma always_sep_dup P : (□ P)%I ≡ (□ P ★ □ P)%I. Lemma always_sep_dup P : (□ P)%I ≡ (□ P ★ □ P)%I.
Proof. by rewrite -always_sep -always_and_sep (idempotent _). Qed. Proof. by rewrite -always_sep -always_and_sep (idempotent _). Qed.
Lemma always_wand_impl P Q : (□ (P -★ Q))%I ≡ (□ (P → Q))%I. Lemma always_wand_impl P Q : (□ (P -★ Q))%I ≡ (□ (P → Q))%I.
...@@ -821,8 +823,9 @@ Qed. ...@@ -821,8 +823,9 @@ Qed.
Global Instance sep_timeless P Q: TimelessP P → TimelessP Q → TimelessP (P ★ Q). Global Instance sep_timeless P Q: TimelessP P → TimelessP Q → TimelessP (P ★ Q).
Proof. Proof.
intros; rewrite /TimelessP later_sep (timelessP P) (timelessP Q). intros; rewrite /TimelessP later_sep (timelessP P) (timelessP Q).
apply wand_elim_l',or_elim;apply wand_intro; auto. apply wand_elim_l', or_elim; apply wand_intro_r; auto.
apply wand_elim_r',or_elim;apply wand_intro; rewrite ?(commutative _ Q); auto. apply wand_elim_r', or_elim; apply wand_intro_r; auto.
rewrite ?(commutative _ Q); auto.
Qed. Qed.
Global Instance wand_timeless P Q : TimelessP Q → TimelessP (P -★ Q). Global Instance wand_timeless P Q : TimelessP Q → TimelessP (P -★ Q).
Proof. Proof.
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