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Commit 542fe0c1 authored by Robbert Krebbers's avatar Robbert Krebbers
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More logic lemmas: distributivity, symmetric variants of existing lemmas.

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......@@ -356,9 +356,9 @@ Lemma or_intro_l P Q : P ⊑ (P ∨ Q).
Proof. intros x n ??; left; auto. Qed.
Lemma or_intro_r P Q : Q ⊑ (P ∨ Q).
Proof. intros x n ??; right; auto. Qed.
Lemma or_elim R P Q : P ⊑ R → Q ⊑ R → (P ∨ Q) ⊑ R.
Lemma or_elim P Q R : P ⊑ R → Q ⊑ R → (P ∨ Q) ⊑ R.
Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed.
Lemma impl_intro P Q R : (R ∧ P) ⊑ Q → R ⊑ (P → Q).
Lemma impl_intro P Q R : (P ∧ Q) ⊑ R → P ⊑ (Q → R).
Proof.
intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using uPred_weaken.
Qed.
......@@ -405,44 +405,29 @@ Hint Resolve or_elim or_intro_l' or_intro_r'.
Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.
Global Instance and_idem : Idempotent (≡) (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_idem : Idempotent (≡) (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_comm : Commutative (≡) (@uPred_and M).
Proof. intros P Q; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance True_and : LeftId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_True : RightId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance False_and : LeftAbsorb (≡) False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_False : RightAbsorb (≡) False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance True_or : LeftAbsorb (≡) True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_True : RightAbsorb (≡) True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance False_or : LeftId (≡) False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_False : RightId (≡) False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_assoc : Associative (≡) (@uPred_and M).
Proof. intros P Q R; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_comm : Commutative (≡) (@uPred_or M).
Proof. intros P Q; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_assoc : Associative (≡) (@uPred_or M).
Proof. intros P Q R; apply (anti_symmetric (⊑)); auto. Qed.
Lemma impl_elim_l P Q : ((P → Q) ∧ P) ⊑ Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_r P Q : (P ∧ (P → Q)) ⊑ Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_l' P Q R : P ⊑ (Q → R) → (P ∧ Q) ⊑ R.
Proof. intros; apply impl_elim with Q; auto. Qed.
Lemma impl_elim_r' P Q R : Q ⊑ (P → R) → (P ∧ Q) ⊑ R.
Proof. intros; apply impl_elim with P; auto. Qed.
Lemma const_elim_l (P : Prop) Q R : (P → Q ⊑ R) → (■ P ∧ Q) ⊑ R.
Proof. intros; apply const_elim with P; eauto. Qed.
Lemma const_elim_r (P : Prop) Q R : (P → Q ⊑ R) → (Q ∧ ■ P) ⊑ R.
Proof. intros; apply const_elim with P; eauto. Qed.
Lemma const_mono (P Q: Prop) : (P → Q) → ■ P ⊑ ■ Q.
Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊑ (a ≡ b).
Proof. intros ->; apply eq_refl. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b) ⊑ (b ≡ a).
Proof.
intros; rewrite <-(left_id True%I _ (■ P)%I).
eauto using const_elim_r, const_intro.
refine (eq_rewrite _ (λ b, b ≡ a)%I a b _ _); auto using eq_refl.
intros n; solve_proper.
Qed.
Lemma const_mono (P Q: Prop) : (P → Q) → ■ P ⊑ ■ Q.
Proof. intros; apply const_elim with P; eauto using const_intro. Qed.
Lemma and_mono P P' Q Q' : P ⊑ Q → P' ⊑ Q' → (P ∧ P') ⊑ (Q ∧ Q').
Proof. auto. Qed.
Lemma or_mono P P' Q Q' : P ⊑ Q → P' ⊑ Q' → (P ∨ P') ⊑ (Q ∨ Q').
......@@ -462,7 +447,6 @@ Lemma exist_mono {A} (P Q : A → uPred M) :
Proof.
intros HPQ. apply exist_elim=> a; rewrite ->(HPQ a); apply exist_intro.
Qed.
Global Instance const_mono' : Proper (impl ==> (⊑)) (@uPred_const M).
Proof. intros P Q; apply const_mono. Qed.
Global Instance and_mono' : Proper ((⊑) ==> (⊑) ==> (⊑)) (@uPred_and M).
......@@ -479,13 +463,49 @@ Global Instance exist_mono' A :
Proper (pointwise_relation _ (⊑) ==> (⊑)) (@uPred_exist M A).
Proof. intros P1 P2; apply exist_mono. Qed.
Lemma equiv_eq {A : cofeT} P (a b : A) : a ≡ b → P ⊑ (a ≡ b).
Proof. intros ->; apply eq_refl. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a ≡ b) ⊑ (b ≡ a).
Global Instance and_idem : Idempotent (≡) (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_idem : Idempotent (≡) (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_comm : Commutative (≡) (@uPred_and M).
Proof. intros P Q; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance True_and : LeftId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_True : RightId (≡) True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance False_and : LeftAbsorb (≡) False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_False : RightAbsorb (≡) False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance True_or : LeftAbsorb (≡) True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_True : RightAbsorb (≡) True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance False_or : LeftId (≡) False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_False : RightId (≡) False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance and_assoc : Associative (≡) (@uPred_and M).
Proof. intros P Q R; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_comm : Commutative (≡) (@uPred_or M).
Proof. intros P Q; apply (anti_symmetric (⊑)); auto. Qed.
Global Instance or_assoc : Associative (≡) (@uPred_or M).
Proof. intros P Q R; apply (anti_symmetric (⊑)); auto. Qed.
Lemma or_and_l P Q R : (P ∨ Q ∧ R)%I ≡ ((P ∨ Q) ∧ (P ∨ R))%I.
Proof.
refine (eq_rewrite _ (λ b, b ≡ a)%I a b _ _); auto using eq_refl.
intros n; solve_proper.
apply (anti_symmetric (⊑)); first auto.
apply impl_elim_l', or_elim; apply impl_intro; first auto.
apply impl_elim_r', or_elim; apply impl_intro; auto.
Qed.
Lemma or_and_r P Q R : (P ∧ Q ∨ R)%I ≡ ((P ∨ R) ∧ (Q ∨ R))%I.
Proof. by rewrite -!(commutative _ R) or_and_l. Qed.
Lemma and_or_l P Q R : (P ∧ (Q ∨ R))%I ≡ (P ∧ Q ∨ P ∧ R)%I.
Proof.
apply (anti_symmetric (⊑)); last auto.
apply impl_elim_r', or_elim; apply impl_intro; auto.
Qed.
Lemma and_or_r P Q R : ((P ∨ Q) ∧ R)%I ≡ (P ∧ R ∨ Q ∧ R)%I.
Proof. by rewrite -!(commutative _ R) and_or_l. Qed.
(* BI connectives *)
Lemma sep_mono P P' Q Q' : P ⊑ Q → P' ⊑ Q' → (P ★ P') ⊑ (Q ★ Q').
......@@ -514,33 +534,27 @@ Proof.
+ by rewrite (associative op) -Hy -Hx.
+ by exists y2, x2.
Qed.
Lemma wand_intro P Q R : (R ★ P) ⊑ Q → R ⊑ (P -★ Q).
Lemma wand_intro P Q R : (P ★ Q) ⊑ R → P ⊑ (Q -★ R).
Proof.
intros HPQ x n ?? x' n' ???; apply HPQ; auto.
intros HPQR x n ?? x' n' ???; apply HPQR; auto.
exists x, x'; split_ands; auto.
eapply uPred_weaken with x n; eauto using cmra_valid_op_l.
Qed.
Lemma wand_elim P Q : ((P -★ Q) ★ P) ⊑ Q.
Lemma wand_elim_l P Q : ((P -★ Q) ★ P) ⊑ Q.
Proof. by intros x n ? (x1&x2&Hx&HPQ&?); cofe_subst; apply HPQ. Qed.
Lemma or_sep_distr P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
Proof.
split; [by intros (x1&x2&Hx&[?|?]&?); [left|right]; exists x1, x2|].
intros [(x1&x2&Hx&?&?)|(x1&x2&Hx&?&?)]; exists x1, x2; split_ands;
first [by left | by right | done].
Qed.
Lemma exist_sep_distr {A} (P : A → uPred M) Q :
((∃ b, P b) ★ Q)%I ≡ (∃ b, P b ★ Q)%I.
Proof.
intros x [|n]; [done|split; [by intros (x1&x2&Hx&[a ?]&?); exists a,x1,x2|]].
intros [a (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists a.
Qed.
Lemma sep_or_l_1 P Q R : (P ★ (Q ∨ R)) ⊑ (P ★ Q ∨ P ★ R).
Proof. by intros x n ? (x1&x2&Hx&?&[?|?]); [left|right]; exists x1, x2. Qed.
Lemma sep_exist_l_1 {A} P (Q : A → uPred M) : (P ★ ∃ b, Q b) ⊑ (∃ b, P ★ Q b).
Proof. by intros x [|n] ?; [done|intros (x1&x2&?&?&[a ?]); exists a,x1,x2]. Qed.
(* Derived BI Stuff *)
Hint Resolve sep_mono.
Global Instance sep_mono' : Proper ((⊑) ==> (⊑) ==> (⊑)) (@uPred_sep M).
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').
Proof. intros HP HQ; apply wand_intro; rewrite ->HP, <-HQ; apply wand_elim. Qed.
Proof.
intros HP HQ; apply wand_intro; rewrite ->HP, <-HQ; apply wand_elim_l.
Qed.
Global Instance wand_mono' : Proper (flip (⊑) ==> (⊑) ==> (⊑)) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
......@@ -555,18 +569,40 @@ Proof. intros ->; apply sep_elim_l. Qed.
Lemma sep_elim_r' P Q R : Q ⊑ R → (P ★ Q) ⊑ R.
Proof. intros ->; apply sep_elim_r. Qed.
Hint Resolve sep_elim_l' sep_elim_r'.
Lemma wand_elim_r P Q : (P ★ (P -★ Q)) ⊑ Q.
Proof. rewrite (commutative _ P); apply wand_elim_l. Qed.
Lemma wand_elim_l' P Q R : P ⊑ (Q -★ R) → (P ★ Q) ⊑ R.
Proof. intros ->; apply wand_elim_l. Qed.
Lemma wand_elim_r' P Q R : Q ⊑ (P -★ R) → (P ★ Q) ⊑ R.
Proof. intros ->; apply wand_elim_r. Qed.
Lemma sep_and P Q : (P ★ Q) ⊑ (P ∧ Q).
Proof. auto. Qed.
Lemma impl_wand P Q : (P → Q) ⊑ (P -★ Q).
Proof. apply wand_intro, impl_elim with P; auto. Qed.
Global Instance sep_False : LeftAbsorb (≡) False%I (@uPred_sep M).
Proof. intros P; apply (anti_symmetric _); auto. Qed.
Global Instance False_sep : RightAbsorb (≡) False%I (@uPred_sep M).
Proof. intros P; apply (anti_symmetric _); auto. Qed.
Lemma impl_wand P Q : (P → Q) ⊑ (P -★ Q).
Proof. apply wand_intro, impl_elim with P; auto. Qed.
Lemma and_sep_distr P Q R : ((P ∧ Q) ★ R) ⊑ ((P ★ R) ∧ (Q ★ R)).
Lemma sep_and_l P Q R : (P ★ (Q ∧ R)) ⊑ ((P ★ Q) ∧ (P ★ R)).
Proof. auto. Qed.
Lemma forall_sep_distr {A} (P : A → uPred M) Q :
((∀ a, P a) ★ Q) ⊑ (∀ a, P a ★ Q).
Lemma sep_and_r P Q R : ((P ∧ Q) ★ R) ⊑ ((P ★ R) ∧ (Q ★ R)).
Proof. auto. Qed.
Lemma sep_or_l P Q R : (P ★ (Q ∨ R))%I ≡ ((P ★ Q) ∨ (P ★ R))%I.
Proof. apply (anti_symmetric (⊑)); eauto 10 using sep_or_l_1. Qed.
Lemma sep_or_r P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
Proof. by rewrite -!(commutative _ R) sep_or_l. Qed.
Lemma sep_exist_l {A} P (Q : A → uPred M) : (P ★ ∃ a, Q a)%I ≡ (∃ a, P ★ Q a)%I.
Proof.
intros; apply (anti_symmetric (⊑)); eauto using sep_exist_l_1.
apply exist_elim=> a; apply sep_mono; auto using exist_intro.
Qed.
Lemma sep_exist_r {A} (P: A → uPred M) Q: ((∃ a, P a) ★ Q)%I ≡ (∃ a, P a ★ Q)%I.
Proof. setoid_rewrite (commutative _ _ Q); apply sep_exist_l. Qed.
Lemma sep_forall_l {A} P (Q : A → uPred M) : (P ★ ∀ a, Q a) ⊑ (∀ a, P ★ Q a).
Proof. by apply forall_intro; intros a; rewrite ->forall_elim. Qed.
Lemma sep_forall_r {A} (P : A → uPred M) Q : ((∀ a, P a) ★ Q) ⊑ (∀ a, P a ★ Q).
Proof. by apply forall_intro; intros a; rewrite ->forall_elim. Qed.
(* Later *)
......@@ -613,7 +649,7 @@ Proof.
apply later_mono, impl_elim with P; auto.
Qed.
Lemma later_wand P Q : ▷ (P -★ Q) ⊑ (▷ P -★ ▷ Q).
Proof. apply wand_intro; rewrite <-later_sep; apply later_mono, wand_elim. Qed.
Proof. apply wand_intro; rewrite <-later_sep; apply later_mono,wand_elim_l. Qed.
(* Always *)
Lemma always_const (P : Prop) : (□ ■ P : uPred M)%I ≡ (■ P)%I.
......@@ -677,7 +713,7 @@ Proof.
* rewrite <-always_and_sep', always_and; auto.
Qed.
Lemma always_wand P Q : □ (P -★ Q) ⊑ (□ P -★ □ Q).
Proof. by apply wand_intro; rewrite <-always_sep, wand_elim. Qed.
Proof. by apply wand_intro; rewrite <-always_sep, wand_elim_l. Qed.
Lemma always_sep_and P Q : (□ (P ★ Q))%I ≡ (□ (P ∧ Q))%I.
Proof. apply (anti_symmetric (⊑)); auto using always_and_sep'. Qed.
......@@ -687,7 +723,7 @@ Lemma always_wand_impl P Q : (□ (P -★ Q))%I ≡ (□ (P → Q))%I.
Proof.
apply (anti_symmetric (⊑)); [|by rewrite <-impl_wand].
apply always_intro, impl_intro.
by rewrite ->always_and_sep_l, always_elim, wand_elim.
by rewrite ->always_and_sep_l, always_elim, wand_elim_l.
Qed.
(* Own *)
......
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