Commit 07f5e2fe authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

Put logic stuff in module.

parent b3c56843
......@@ -203,281 +203,275 @@ Infix "↔" := uPred_iff : uPred_scope.
Class TimelessP {M} (P : uPred M) := timelessP x n : {1} x P 1 x P n x.
Section logic.
Module uPred. Section uPred_logic.
Context {M : cmraT}.
Implicit Types P Q : uPred M.
Implicit Types A : Type.
Global Instance uPred_preorder : PreOrder (() : relation (uPred M)).
Global Instance: PreOrder (() : relation (uPred M)).
Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed.
Global Instance uPred_antisym : AntiSymmetric () (() : relation (uPred M)).
Global Instance: AntiSymmetric () (() : relation (uPred M)).
Proof. intros P Q HPQ HQP; split; auto. Qed.
Lemma uPred_equiv_spec P Q : P Q P Q Q P.
Lemma equiv_spec P Q : P Q P Q Q P.
Proof.
split; [|by intros [??]; apply (anti_symmetric ())].
intros HPQ; split; intros x i; apply HPQ.
Qed.
Global Instance uPred_entails_proper :
Global Instance entails_proper :
Proper (() ==> () ==> iff) (() : relation (uPred M)).
Proof.
intros P1 P2 HP Q1 Q2 HQ; rewrite uPred_equiv_spec in HP, HQ; split; intros.
intros P1 P2 HP Q1 Q2 HQ; rewrite equiv_spec in HP, HQ; split; intros.
* by rewrite (proj2 HP), <-(proj1 HQ).
* by rewrite (proj1 HP), <-(proj2 HQ).
Qed.
(** Non-expansiveness and setoid morphisms *)
Global Instance uPred_const_proper : Proper (iff ==> ()) (@uPred_const M).
Global Instance const_proper : Proper (iff ==> ()) (@uPred_const M).
Proof. by intros P Q HPQ ? [|n] ?; try apply HPQ. Qed.
Global Instance uPred_and_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Proof.
intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ.
Qed.
Global Instance uPred_and_proper :
Global Instance and_proper :
Proper (() ==> () ==> ()) (@uPred_and M) := ne_proper_2 _.
Global Instance uPred_or_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
Proof.
intros P P' HP Q Q' HQ; split; intros [?|?];
first [by left; apply HP | by right; apply HQ].
Qed.
Global Instance uPred_or_proper :
Global Instance or_proper :
Proper (() ==> () ==> ()) (@uPred_or M) := ne_proper_2 _.
Global Instance uPred_impl_ne n :
Global Instance impl_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
Proof.
intros P P' HP Q Q' HQ; split; intros HPQ x' n'' ????; apply HQ,HPQ,HP; auto.
Qed.
Global Instance uPred_impl_proper :
Global Instance impl_proper :
Proper (() ==> () ==> ()) (@uPred_impl M) := ne_proper_2 _.
Global Instance uPred_sep_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
Proof.
intros P P' HP Q Q' HQ x n' ? Hx'; split; intros (x1&x2&Hx&?&?);
exists x1, x2; rewrite Hx in Hx'; split_ands; try apply HP; try apply HQ;
eauto using cmra_valid_op_l, cmra_valid_op_r.
Qed.
Global Instance uPred_sep_proper :
Global Instance sep_proper :
Proper (() ==> () ==> ()) (@uPred_sep M) := ne_proper_2 _.
Global Instance uPred_wand_ne n :
Global Instance wand_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
Proof.
intros P P' HP Q Q' HQ x n' ??; split; intros HPQ x' n'' ???;
apply HQ, HPQ, HP; eauto using cmra_valid_op_r.
Qed.
Global Instance uPred_wand_proper :
Global Instance wand_proper :
Proper (() ==> () ==> ()) (@uPred_wand M) := ne_proper_2 _.
Global Instance uPred_eq_ne (A : cofeT) n :
Global Instance eq_ne (A : cofeT) n :
Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
Proof.
intros x x' Hx y y' Hy z n'; split; intros; simpl in *.
* by rewrite <-(dist_le _ _ _ _ Hx), <-(dist_le _ _ _ _ Hy) by auto.
* by rewrite (dist_le _ _ _ _ Hx), (dist_le _ _ _ _ Hy) by auto.
Qed.
Global Instance uPred_eq_proper (A : cofeT) :
Global Instance eq_proper (A : cofeT) :
Proper (() ==> () ==> ()) (@uPred_eq M A) := ne_proper_2 _.
Global Instance uPred_forall_ne A :
Global Instance forall_ne A :
Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
Global Instance uPred_forall_proper A :
Global Instance forall_proper A :
Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed.
Global Instance uPred_exists_ne A :
Global Instance exists_ne A :
Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
Proof.
by intros n P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
Qed.
Global Instance uPred_exist_proper A :
Global Instance exist_proper A :
Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
Proof.
by intros P1 P2 HP x [|n']; [|split; intros [a ?]; exists a; apply HP].
Qed.
Global Instance uPred_later_contractive : Contractive (@uPred_later M).
Global Instance later_contractive : Contractive (@uPred_later M).
Proof.
intros n P Q HPQ x [|n'] ??; simpl; [done|].
apply HPQ; eauto using cmra_valid_S.
Qed.
Global Instance uPred_later_proper :
Global Instance later_proper :
Proper (() ==> ()) (@uPred_later M) := ne_proper _.
Global Instance uPred_always_ne n: Proper (dist n ==> dist n) (@uPred_always M).
Global Instance always_ne n: Proper (dist n ==> dist n) (@uPred_always M).
Proof. intros P1 P2 HP x n'; split; apply HP; eauto using cmra_unit_valid. Qed.
Global Instance uPred_always_proper :
Global Instance always_proper :
Proper (() ==> ()) (@uPred_always M) := ne_proper _.
Global Instance uPred_own_ne n : Proper (dist n ==> dist n) (@uPred_own M).
Global Instance own_ne n : Proper (dist n ==> dist n) (@uPred_own M).
Proof.
by intros a1 a2 Ha x n'; split; intros [a' ?]; exists a'; simpl; first
[rewrite <-(dist_le _ _ _ _ Ha) by lia|rewrite (dist_le _ _ _ _ Ha) by lia].
Qed.
Global Instance uPred_own_proper :
Proper (() ==> ()) (@uPred_own M) := ne_proper _.
Global Instance uPred_iff_ne n :
Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
Global Instance own_proper : Proper (() ==> ()) (@uPred_own M) := ne_proper _.
Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
Proof. unfold uPred_iff; solve_proper. Qed.
Global Instance uPred_iff_proper :
Global Instance iff_proper :
Proper (() ==> () ==> ()) (@uPred_iff M) := ne_proper_2 _.
(** Introduction and elimination rules *)
Lemma uPred_const_intro P (Q : Prop) : Q P uPred_const Q.
Lemma const_intro P (Q : Prop) : Q P uPred_const Q.
Proof. by intros ?? [|?]. Qed.
Lemma uPred_const_elim (P : Prop) Q R : (P Q R) (Q uPred_const P)%I R.
Lemma const_elim (P : Prop) Q R : (P Q R) (Q uPred_const P)%I R.
Proof. intros HR x [|n] ? [??]; [|apply HR]; auto. Qed.
Lemma uPred_True_intro P : P True%I.
Proof. by apply uPred_const_intro. Qed.
Lemma uPred_False_elim P : False%I P.
Lemma True_intro P : P True%I.
Proof. by apply const_intro. Qed.
Lemma False_elim P : False%I P.
Proof. by intros x [|n] ?. Qed.
Lemma uPred_and_elim_l P Q : (P Q)%I P.
Lemma and_elim_l P Q : (P Q)%I P.
Proof. by intros x n ? [??]. Qed.
Lemma uPred_and_elim_r P Q : (P Q)%I Q.
Lemma and_elim_r P Q : (P Q)%I Q.
Proof. by intros x n ? [??]. Qed.
Lemma uPred_and_intro P Q R : P Q P R P (Q R)%I.
Lemma and_intro P Q R : P Q P R P (Q R)%I.
Proof. intros HQ HR x n ??; split; auto. Qed.
Lemma uPred_or_intro_l P Q R : P Q P (Q R)%I.
Lemma or_intro_l P Q R : P Q P (Q R)%I.
Proof. intros HQ x n ??; left; auto. Qed.
Lemma uPred_or_intro_r P Q R : P R P (Q R)%I.
Lemma or_intro_r P Q R : P R P (Q R)%I.
Proof. intros HR x n ??; right; auto. Qed.
Lemma uPred_or_elim R P Q : P R Q R (P Q)%I R.
Lemma or_elim R P Q : P R Q R (P Q)%I R.
Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed.
Lemma uPred_impl_intro P Q R : (R P)%I Q R (P Q)%I.
Lemma impl_intro P Q R : (R P)%I Q R (P Q)%I.
Proof.
intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using uPred_weaken.
Qed.
Lemma uPred_impl_elim P Q R : P (Q R)%I P Q P R.
Lemma impl_elim P Q R : P (Q R)%I P Q P R.
Proof. by intros HP HP' x n ??; apply HP with x n, HP'. Qed.
Lemma uPred_forall_intro P `(Q: A uPred M): ( a, P Q a) P ( a, Q a)%I.
Lemma forall_intro P `(Q: A uPred M): ( a, P Q a) P ( a, Q a)%I.
Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
Lemma uPred_forall_elim `(P : A uPred M) a : ( a, P a)%I P a.
Lemma forall_elim `(P : A uPred M) a : ( a, P a)%I P a.
Proof. intros x n ? HP; apply HP. Qed.
Lemma uPred_exist_intro `(P : A uPred M) a : P a ( a, P a)%I.
Lemma exist_intro `(P : A uPred M) a : P a ( a, P a)%I.
Proof. by intros x [|n] ??; [done|exists a]. Qed.
Lemma uPred_exist_elim `(P : A uPred M) Q : ( a, P a Q) ( a, P a)%I Q.
Lemma exist_elim `(P : A uPred M) Q : ( a, P a Q) ( a, P a)%I Q.
Proof. by intros HPQ x [|n] ?; [|intros [a ?]; apply HPQ with a]. Qed.
Lemma uPred_eq_refl {A : cofeT} (a : A) P : P (a a)%I.
Lemma eq_refl {A : cofeT} (a : A) P : P (a a)%I.
Proof. by intros x n ??; simpl. Qed.
Lemma uPred_eq_rewrite {A : cofeT} P (Q : A uPred M)
Lemma eq_rewrite {A : cofeT} P (Q : A uPred M)
`{HQ : ! n, Proper (dist n ==> dist n) Q} a b :
P (a b)%I P Q a P Q b.
Proof.
intros Hab Ha x n ??; apply HQ with n a; auto. by symmetry; apply Hab with x.
Qed.
Lemma uPred_eq_equiv `{Empty M, !RAEmpty M} {A : cofeT} (a b : A) :
Lemma eq_equiv `{Empty M, !RAEmpty M} {A : cofeT} (a b : A) :
True%I (a b : uPred M)%I a b.
Proof.
intros Hab; apply equiv_dist; intros n; apply Hab with .
* apply cmra_valid_validN, ra_empty_valid.
* by destruct n.
Qed.
Lemma uPred_iff_equiv P Q : True%I (P Q)%I P Q.
Lemma iff_equiv P Q : True%I (P Q)%I P Q.
Proof. by intros HPQ x [|n] ?; [|split; intros; apply HPQ with x (S n)]. Qed.
(* Derived logical stuff *)
Lemma uPred_and_elim_l' P Q R : P R (P Q)%I R.
Proof. by rewrite uPred_and_elim_l. Qed.
Lemma uPred_and_elim_r' P Q R : Q R (P Q)%I R.
Proof. by rewrite uPred_and_elim_r. Qed.
Lemma and_elim_l' P Q R : P R (P Q)%I R.
Proof. by rewrite and_elim_l. Qed.
Lemma and_elim_r' P Q R : Q R (P Q)%I R.
Proof. by rewrite and_elim_r. Qed.
Hint Resolve uPred_or_elim uPred_or_intro_l uPred_or_intro_r.
Hint Resolve uPred_and_intro uPred_and_elim_l' uPred_and_elim_r'.
Hint Immediate uPred_True_intro uPred_False_elim.
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 uPred_and_idem : Idempotent () (@uPred_and M).
Global Instance and_idem : Idempotent () (@uPred_and M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_or_idem : Idempotent () (@uPred_or M).
Global Instance or_idem : Idempotent () (@uPred_or M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_and_comm : Commutative () (@uPred_and M).
Global Instance and_comm : Commutative () (@uPred_and M).
Proof. intros P Q; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_True_and : LeftId () True%I (@uPred_and M).
Global Instance True_and : LeftId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_and_True : RightId () True%I (@uPred_and M).
Global Instance and_True : RightId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_False_and : LeftAbsorb () False%I (@uPred_and M).
Global Instance False_and : LeftAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_and_False : RightAbsorb () False%I (@uPred_and M).
Global Instance and_False : RightAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_True_or : LeftAbsorb () True%I (@uPred_or M).
Global Instance True_or : LeftAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_or_True : RightAbsorb () True%I (@uPred_or M).
Global Instance or_True : RightAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_False_or : LeftId () False%I (@uPred_or M).
Global Instance False_or : LeftId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_or_False : RightId () False%I (@uPred_or M).
Global Instance or_False : RightId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_and_assoc : Associative () (@uPred_and M).
Global Instance and_assoc : Associative () (@uPred_and M).
Proof. intros P Q R; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_or_comm : Commutative () (@uPred_or M).
Global Instance or_comm : Commutative () (@uPred_or M).
Proof. intros P Q; apply (anti_symmetric ()); auto. Qed.
Global Instance uPred_or_assoc : Associative () (@uPred_or M).
Global Instance or_assoc : Associative () (@uPred_or M).
Proof. intros P Q R; apply (anti_symmetric ()); auto. Qed.
Lemma uPred_const_mono (P Q: Prop) : (P Q) uPred_const P @uPred_const M Q.
Lemma const_mono (P Q: Prop) : (P Q) uPred_const P @uPred_const M Q.
Proof.
intros; rewrite <-(left_id True%I _ (uPred_const P)).
eauto using uPred_const_elim, uPred_const_intro.
eauto using const_elim, const_intro.
Qed.
Lemma uPred_and_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Lemma and_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Proof. auto. Qed.
Lemma uPred_or_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Lemma or_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Proof. auto. Qed.
Lemma uPred_impl_mono P P' Q Q' : Q P P' Q' (P P')%I (Q Q')%I.
Lemma impl_mono P P' Q Q' : Q P P' Q' (P P')%I (Q Q')%I.
Proof.
intros HP HQ'; apply uPred_impl_intro; rewrite <-HQ'.
transitivity ((P P') P)%I; eauto using uPred_impl_elim.
intros HP HQ'; apply impl_intro; rewrite <-HQ'.
transitivity ((P P') P)%I; eauto using impl_elim.
Qed.
Lemma uPred_forall_mono {A} (P Q : A uPred M) :
Lemma forall_mono {A} (P Q : A uPred M) :
( a, P a Q a) ( a, P a)%I ( a, Q a)%I.
Proof.
intros HPQ. apply uPred_forall_intro; intros a; rewrite <-(HPQ a).
apply uPred_forall_elim.
intros HPQ. apply forall_intro; intros a; rewrite <-(HPQ a).
apply forall_elim.
Qed.
Lemma uPred_exist_mono {A} (P Q : A uPred M) :
Lemma exist_mono {A} (P Q : A uPred M) :
( a, P a Q a) ( a, P a)%I ( a, Q a)%I.
Proof.
intros HPQ. apply uPred_exist_elim; intros a; rewrite (HPQ a).
apply uPred_exist_intro.
intros HPQ. apply exist_elim; intros a; rewrite (HPQ a); apply exist_intro.
Qed.
Global Instance uPred_const_mono' : Proper (impl ==> ()) (@uPred_const M).
Proof. intros P Q; apply uPred_const_mono. Qed.
Global Instance uPred_and_mono' : Proper (() ==> () ==> ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply uPred_and_mono. Qed.
Global Instance uPred_or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply uPred_or_mono. Qed.
Global Instance uPred_impl_mono' :
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).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
Proper (flip () ==> () ==> ()) (@uPred_impl M).
Proof. by intros P P' HP Q Q' HQ; apply uPred_impl_mono. Qed.
Global Instance uPred_forall_mono' A :
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance forall_mono' A :
Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
Proof. intros P1 P2; apply uPred_forall_mono. Qed.
Global Instance uPred_exist_mono' A :
Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance exist_mono' A :
Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
Proof. intros P1 P2; apply uPred_exist_mono. Qed.
Proof. intros P1 P2; apply exist_mono. Qed.
Lemma uPred_equiv_eq {A : cofeT} P (a b : A) : a b P (a b)%I.
Proof. intros ->; apply uPred_eq_refl. Qed.
Lemma uPred_eq_sym {A : cofeT} (a b : A) : (a b)%I (b a : uPred M)%I.
Lemma equiv_eq {A : cofeT} P (a b : A) : a b P (a b)%I.
Proof. intros ->; apply eq_refl. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a b)%I (b a : uPred M)%I.
Proof.
refine (uPred_eq_rewrite _ (λ b, b a)%I a b _ _); auto using uPred_eq_refl.
refine (eq_rewrite _ (λ b, b a)%I a b _ _); auto using eq_refl.
intros n; solve_proper.
Qed.
(* BI connectives *)
Lemma uPred_sep_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Lemma sep_mono P P' Q Q' : P Q P' Q' (P P')%I (Q Q')%I.
Proof.
intros HQ HQ' x n' Hx' (x1&x2&Hx&?&?); exists x1, x2;
rewrite Hx in Hx'; eauto 7 using cmra_valid_op_l, cmra_valid_op_r.
Qed.
Global Instance uPred_True_sep : LeftId () True%I (@uPred_sep M).
Global Instance True_sep : LeftId () True%I (@uPred_sep M).
Proof.
intros P x n Hvalid; split.
* intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *.
eauto using uPred_weaken, ra_included_r.
* by destruct n as [|n]; [|intros ?; exists (unit x), x; rewrite ra_unit_l].
Qed.
Global Instance uPred_sep_commutative : Commutative () (@uPred_sep M).
Global Instance sep_commutative : Commutative () (@uPred_sep M).
Proof.
by intros P Q x n ?; split;
intros (x1&x2&?&?&?); exists x2, x1; rewrite (commutative op).
Qed.
Global Instance uPred_sep_associative : Associative () (@uPred_sep M).
Global Instance sep_associative : Associative () (@uPred_sep M).
Proof.
intros P Q R x n ?; split.
* intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1 y1), y2; split_ands; auto.
......@@ -487,23 +481,23 @@ Proof.
+ by rewrite (associative op), <-Hy, <-Hx.
+ by exists y2, x2.
Qed.
Lemma uPred_wand_intro P Q R : (R P)%I Q R (P - Q)%I.
Lemma wand_intro P Q R : (R P)%I Q R (P - Q)%I.
Proof.
intros HPQ x n ?? x' n' ???; apply HPQ; auto.
exists x, x'; split_ands; auto.
eapply uPred_weaken with x n; eauto using cmra_valid_op_l.
Qed.
Lemma uPred_wand_elim P Q : ((P - Q) P)%I Q.
Lemma wand_elim P Q : ((P - Q) P)%I Q.
Proof.
by intros x n Hvalid (x1&x2&Hx&HPQ&?); rewrite Hx in Hvalid |- *; apply HPQ.
Qed.
Lemma uPred_or_sep_distr P Q R : ((P Q) R)%I ((P R) (Q R))%I.
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 uPred_exist_sep_distr `(P : A uPred M) Q :
Lemma exist_sep_distr `(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|]].
......@@ -511,63 +505,62 @@ Proof.
Qed.
(* Derived BI Stuff *)
Hint Resolve uPred_sep_mono.
Global Instance uPred_sep_mono' : Proper (() ==> () ==> ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply uPred_sep_mono. Qed.
Lemma uPred_wand_mono P P' Q Q' : Q P P' Q' (P - P')%I (Q - Q')%I.
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')%I (Q - Q')%I.
Proof.
intros HP HQ; apply uPred_wand_intro; rewrite HP, <-HQ; apply uPred_wand_elim.
intros HP HQ; apply wand_intro; rewrite HP, <-HQ; apply wand_elim.
Qed.
Global Instance uPred_wand_mono' :
Proper (flip () ==> () ==> ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply uPred_wand_mono. Qed.
Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
Global Instance uPred_sep_True : RightId () True%I (@uPred_sep M).
Global Instance sep_True : RightId () True%I (@uPred_sep M).
Proof. by intros P; rewrite (commutative _), (left_id _ _). Qed.
Lemma uPred_sep_elim_l P Q R : P R (P Q)%I R.
Proof. by intros HR; rewrite <-(right_id _ () R)%I, HR, (uPred_True_intro Q). Qed.
Lemma uPred_sep_elim_r P Q : (P Q)%I Q.
Proof. by rewrite (commutative ())%I; apply uPred_sep_elim_l. Qed.
Hint Resolve uPred_sep_elim_l uPred_sep_elim_r.
Lemma uPred_sep_and P Q : (P Q)%I (P Q)%I.
Lemma sep_elim_l P Q R : P R (P Q)%I R.
Proof. by intros HR; rewrite <-(right_id _ () R)%I, HR, (True_intro Q). Qed.
Lemma sep_elim_r P Q : (P Q)%I Q.
Proof. by rewrite (commutative ())%I; apply sep_elim_l. Qed.
Hint Resolve sep_elim_l sep_elim_r.
Lemma sep_and P Q : (P Q)%I (P Q)%I.
Proof. auto. Qed.
Global Instance uPred_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.
Global Instance uPred_False_sep : RightAbsorb () False%I (@uPred_sep M).
Global Instance False_sep : RightAbsorb () False%I (@uPred_sep M).
Proof. intros P; apply (anti_symmetric _); auto. Qed.
Lemma uPred_impl_wand P Q : (P Q)%I (P - Q)%I.
Proof. apply uPred_wand_intro, uPred_impl_elim with P; auto. Qed.
Lemma uPred_and_sep_distr P Q R : ((P Q) R)%I ((P R) (Q R))%I.
Lemma impl_wand P Q : (P Q)%I (P - Q)%I.
Proof. apply wand_intro, impl_elim with P; auto. Qed.
Lemma and_sep_distr P Q R : ((P Q) R)%I ((P R) (Q R))%I.
Proof. auto. Qed.
Lemma uPred_forall_sep_distr `(P : A uPred M) Q :
Lemma forall_sep_distr `(P : A uPred M) Q :
(( a, P a) Q)%I ( a, P a Q)%I.
Proof. by apply uPred_forall_intro; intros a; rewrite uPred_forall_elim. Qed.
Proof. by apply forall_intro; intros a; rewrite forall_elim. Qed.
(* Later *)
Lemma uPred_later_mono P Q : P Q ( P)%I ( Q)%I.
Lemma later_mono P Q : P Q ( P)%I ( Q)%I.
Proof. intros HP x [|n] ??; [done|apply HP; auto using cmra_valid_S]. Qed.
Lemma uPred_later_intro P : P ( P)%I.
Lemma later_intro P : P ( P)%I.
Proof.
intros x [|n] ??; simpl in *; [done|].
apply uPred_weaken with x (S n); auto using cmra_valid_S.
Qed.
Lemma uPred_lub P : ( P P)%I P.
Lemma lub P : ( P P)%I P.
Proof.
intros x n ? HP; induction n as [|n IH]; [by apply HP|].
apply HP, IH, uPred_weaken with x (S n); eauto using cmra_valid_S.
Qed.
Lemma uPred_later_and P Q : ( (P Q))%I ( P Q)%I.
Lemma later_and P Q : ( (P Q))%I ( P Q)%I.
Proof. by intros x [|n]; split. Qed.
Lemma uPred_later_or P Q : ( (P Q))%I ( P Q)%I.
Lemma later_or P Q : ( (P Q))%I ( P Q)%I.
Proof. intros x [|n]; simpl; tauto. Qed.
Lemma uPred_later_forall `(P : A uPred M) : (