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Commit 2c6727dc authored by Robbert Krebbers's avatar Robbert Krebbers
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Use Φ and Ψ for (value-) indexed uPreds/iProps. 

This avoids ambiguity with P and Q that we were using before for both
uPreds/iProps and indexed uPreds/iProps.
parent 5e9653fb
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algebra/upred.v
View file @ 2c6727dc
......@@ -124,11 +124,11 @@ Next Obligation.
Qed.
Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
Program Definition uPred_forall {M A} (P : A → uPred M) : uPred M :=
{| uPred_holds n x := ∀ a, P a n x |}.
Program Definition uPred_forall {M A} (Ψ : A → uPred M) : uPred M :=
{| uPred_holds n x := ∀ a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
Program Definition uPred_exist {M A} (P : A → uPred M) : uPred M :=
{| uPred_holds n x := ∃ a, P a n x |}.
Program Definition uPred_exist {M A} (Ψ : A → uPred M) : uPred M :=
{| uPred_holds n x := ∃ a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
Program Definition uPred_eq {M} {A : cofeT} (a1 a2 : A) : uPred M :=
......@@ -299,10 +299,10 @@ Global Instance eq_proper (A : cofeT) :
Proper ((≡) ==> (≡) ==> (≡)) (@uPred_eq M A) := ne_proper_2 _.
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.
Proof. by intros n Ψ1 Ψ2 HΨ x n'; split; intros HP a; apply HΨ. Qed.
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.
Proof. by intros Ψ1 Ψ2 HΨ x n'; split; intros HP a; apply HΨ. Qed.
Global Instance exists_ne A :
Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
Proof. by intros n P1 P2 HP x; split; intros [a ?]; exists a; apply HP. Qed.
......@@ -359,20 +359,20 @@ Proof.
Qed.
Lemma impl_elim P Q R : P ⊑ (Q → R) → P ⊑ Q → P ⊑ R.
Proof. by intros HP HP' x n ??; apply HP with x n, HP'. Qed.
Lemma forall_intro {A} P (Q : A → uPred M): (∀ a, P ⊑ Q a) → P ⊑ (∀ a, Q a).
Proof. by intros HPQ x n ?? a; apply HPQ. Qed.
Lemma forall_elim {A} {P : A → uPred M} a : (∀ a, P a) ⊑ P a.
Lemma forall_intro {A} P (Ψ : A → uPred M): (∀ a, P ⊑ Ψ a) → P ⊑ (∀ a, Ψ a).
Proof. by intros HPΨ x n ?? a; apply HPΨ. Qed.
Lemma forall_elim {A} {Ψ : A → uPred M} a : (∀ a, Ψ a) ⊑ Ψ a.
Proof. intros x n ? HP; apply HP. Qed.
Lemma exist_intro {A} {P : A → uPred M} a : P a ⊑ (∃ a, P a).
Lemma exist_intro {A} {Ψ : A → uPred M} a : Ψ a ⊑ (∃ a, Ψ a).
Proof. by intros x n ??; exists a. Qed.
Lemma exist_elim {A} (P : A → uPred M) Q : (∀ a, P a ⊑ Q) → (∃ a, P a) ⊑ Q.
Proof. by intros HPQ x n ? [a ?]; apply HPQ with a. Qed.
Lemma exist_elim {A} (Φ : A → uPred M) Q : (∀ a, Φ a ⊑ Q) → (∃ a, Φ a) ⊑ Q.
Proof. by intros HΦΨ x n ? [a ?]; apply HΦΨ with a. Qed.
Lemma eq_refl {A : cofeT} (a : A) P : P ⊑ (a ≡ a).
Proof. by intros x n ??; simpl. Qed.
Lemma eq_rewrite {A : cofeT} a b (Q : A → uPred M) P
`{HQ:∀ n, Proper (dist n ==> dist n) Q} : P ⊑ (a ≡ b) → P ⊑ Q a → P ⊑ Q b.
Lemma eq_rewrite {A : cofeT} a b (Ψ : A → uPred M) P
`{HΨ : ∀ n, Proper (dist n ==> dist n) Ψ} : P ⊑ (a ≡ b) → P ⊑ Ψ a → P ⊑ Ψ b.
Proof.
intros Hab Ha x n ??; apply HQ with n a; auto. by symmetry; apply Hab with x.
intros Hab Ha x n ??; apply HΨ with n a; auto. by symmetry; apply Hab with x.
Qed.
Lemma eq_equiv `{Empty M, !CMRAIdentity M} {A : cofeT} (a b : A) :
True ⊑ (a ≡ b) → a ≡ b.
......@@ -392,7 +392,7 @@ Lemma or_intro_l' P Q R : P ⊑ Q → P ⊑ (Q ∨ R).
Proof. intros ->; apply or_intro_l. Qed.
Lemma or_intro_r' P Q R : P ⊑ R → P ⊑ (Q ∨ R).
Proof. intros ->; apply or_intro_r. Qed.
Lemma exist_intro' {A} P (Q : A → uPred M) a : P ⊑ Q a → P ⊑ (∃ a, Q a).
Lemma exist_intro' {A} P (Ψ : A → uPred M) a : P ⊑ Ψ a → P ⊑ (∃ a, Ψ a).
Proof. intros ->; apply exist_intro. Qed.
Hint Resolve or_elim or_intro_l' or_intro_r'.
......@@ -451,14 +451,14 @@ Proof.
intros HP HQ'; apply impl_intro_l; rewrite -HQ'.
apply impl_elim with P; eauto.
Qed.
Lemma forall_mono {A} (P Q : A → uPred M) :
(∀ a, P a ⊑ Q a) → (∀ a, P a) ⊑ (∀ a, Q a).
Lemma forall_mono {A} (Φ Ψ : A → uPred M) :
(∀ a, Φ a ⊑ Ψ a) → (∀ a, Φ a) ⊑ (∀ a, Ψ a).
Proof.
intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim.
Qed.
Lemma exist_mono {A} (P Q : A → uPred M) :
(∀ a, P a ⊑ Q a) → (∃ a, P a) ⊑ (∃ a, Q a).
Proof. intros HP. apply exist_elim=> a; rewrite (HP a); apply exist_intro. Qed.
Lemma exist_mono {A} (Φ Ψ : A → uPred M) :
(∀ a, Φ a ⊑ Ψ a) → (∃ a, Φ a) ⊑ (∃ a, Ψ a).
Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed.
Global Instance const_mono' : Proper (impl ==> (⊑)) (@uPred_const M).
Proof. intros φ1 φ2; apply const_mono. Qed.
Global Instance and_mono' : Proper ((⊑) ==> (⊑) ==> (⊑)) (@uPred_and M).
......@@ -532,7 +532,7 @@ Proof.
Qed.
Lemma and_or_r P Q R : ((P ∨ Q) ∧ R)%I ≡ (P ∧ R ∨ Q ∧ R)%I.
Proof. by rewrite -!(comm _ R) and_or_l. Qed.
Lemma and_exist_l {A} P (Q : A → uPred M) : (P ∧ ∃ a, Q a)%I ≡ (∃ a, P ∧ Q a)%I.
Lemma and_exist_l {A} P (Ψ : A → uPred M) : (P ∧ ∃ a, Ψ a)%I ≡ (∃ a, P ∧ Ψ a)%I.
Proof.
apply (anti_symm (⊑)).
- apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l.
......@@ -540,10 +540,9 @@ Proof.
- apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l.
by rewrite -(exist_intro a) and_elim_r.
Qed.
Lemma and_exist_r {A} P (Q : A → uPred M) : ((∃ a, Q a) ∧ P)%I ≡ (∃ a, Q a ∧ P)%I.
Lemma and_exist_r {A} P (Φ: A → uPred M) : ((∃ a, Φ a) ∧ P)%I ≡ (∃ a, Φ a ∧ P)%I.
Proof.
rewrite -(comm _ P) and_exist_l.
apply exist_proper=>a. by rewrite comm.
rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm.
Qed.
(* BI connectives *)
......@@ -652,18 +651,18 @@ Proof.
Qed.
Lemma sep_or_r P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I.
Proof. by rewrite -!(comm _ 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.
Lemma sep_exist_l {A} P (Ψ : A → uPred M) : (P ★ ∃ a, Ψ a)%I ≡ (∃ a, P ★ Ψ a)%I.
Proof.
intros; apply (anti_symm (⊑)).
- apply wand_elim_r', exist_elim=>a. apply wand_intro_l.
by rewrite -(exist_intro a).
- 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.
Lemma sep_exist_r {A} (Φ: A → uPred M) Q: ((∃ a, Φ a) ★ Q)%I ≡ (∃ a, Φ a ★ Q)%I.
Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed.
Lemma sep_forall_l {A} P (Q : A → uPred M) : (P ★ ∀ a, Q a) ⊑ (∀ a, P ★ Q a).
Lemma sep_forall_l {A} P (Ψ : A → uPred M) : (P ★ ∀ a, Ψ a) ⊑ (∀ a, P ★ Ψ a).
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.
Lemma sep_forall_r {A} (P : A → uPred M) Q : ((∀ a, P a) ★ Q) ⊑ (∀ a, P a ★ Q).
Lemma sep_forall_r {A} (Φ : A → uPred M) Q : ((∀ a, Φ a) ★ Q) ⊑ (∀ a, Φ a ★ Q).
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.
(* Later *)
......@@ -685,12 +684,12 @@ Lemma later_and P Q : (▷ (P ∧ Q))%I ≡ (▷ P ∧ ▷ Q)%I.
Proof. by intros x [|n]; split. Qed.
Lemma later_or P Q : (▷ (P ∨ Q))%I ≡ (▷ P ∨ ▷ Q)%I.
Proof. intros x [|n]; simpl; tauto. Qed.
Lemma later_forall {A} (P : A → uPred M) : (▷ ∀ a, P a)%I ≡ (∀ a, ▷ P a)%I.
Lemma later_forall {A} (Φ : A → uPred M) : (▷ ∀ a, Φ a)%I ≡ (∀ a, ▷ Φ a)%I.
Proof. by intros x [|n]. Qed.
Lemma later_exist_1 {A} (P : A → uPred M) : (∃ a, ▷ P a) ⊑ (▷ ∃ a, P a).
Lemma later_exist_1 {A} (Φ : A → uPred M) : (∃ a, ▷ Φ a) ⊑ (▷ ∃ a, Φ a).
Proof. by intros x [|[|n]]. Qed.
Lemma later_exist `{Inhabited A} (P : A → uPred M) :
(▷ ∃ a, P a)%I ≡ (∃ a, ▷ P a)%I.
Lemma later_exist `{Inhabited A} (Φ : A → uPred M) :
(▷ ∃ a, Φ a)%I ≡ (∃ a, ▷ Φ a)%I.
Proof. intros x [|[|n]]; split; done || by exists inhabitant; simpl. Qed.
Lemma later_sep P Q : (▷ (P ★ Q))%I ≡ (▷ P ★ ▷ Q)%I.
Proof.
......@@ -718,10 +717,10 @@ Lemma later_wand P Q : ▷ (P -★ Q) ⊑ (▷ P -★ ▷ Q).
Proof. apply wand_intro_r;rewrite -later_sep; apply later_mono,wand_elim_l. Qed.
Lemma later_iff P Q : (▷ (P ↔ Q)) ⊑ (▷P ↔ ▷Q).
Proof. by rewrite /uPred_iff later_and !later_impl. Qed.
Lemma löb_all_1 {A} (P Q : A → uPred M) :
(∀ a, (▷(∀ b, P b → Q b) ∧ P a) ⊑ Q a) → ∀ a, P a ⊑ Q a.
Lemma löb_all_1 {A} (Φ Ψ : A → uPred M) :
(∀ a, (▷ (∀ b, Φ b → Ψ b) ∧ Φ a) ⊑ Ψ a) → ∀ a, Φ a ⊑ Ψ a.
Proof.
intros Hlöb a. apply impl_entails. transitivity (∀ a, P a → Q a)%I; last first.
intros Hlöb a. apply impl_entails. transitivity (∀ a, Φ a → Ψ a)%I; last first.
{ by rewrite (forall_elim a). } clear a.
etransitivity; last by eapply löb.
apply impl_intro_l, forall_intro=>a. rewrite right_id. by apply impl_intro_r.
......@@ -744,9 +743,9 @@ Lemma always_and P Q : (□ (P ∧ Q))%I ≡ (□ P ∧ □ Q)%I.
Proof. done. Qed.
Lemma always_or P Q : (□ (P ∨ Q))%I ≡ (□ P ∨ □ Q)%I.
Proof. done. Qed.
Lemma always_forall {A} (P : A → uPred M) : (□ ∀ a, P a)%I ≡ (∀ a, □ P a)%I.
Lemma always_forall {A} (Ψ : A → uPred M) : (□ ∀ a, Ψ a)%I ≡ (∀ a, □ Ψ a)%I.
Proof. done. Qed.
Lemma always_exist {A} (P : A → uPred M) : (□ ∃ a, P a)%I ≡ (∃ a, □ P a)%I.
Lemma always_exist {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a)%I ≡ (∃ a, □ Ψ a)%I.
Proof. done. Qed.
Lemma always_and_sep_1 P Q : □ (P ∧ Q) ⊑ □ (P ★ Q).
Proof.
......@@ -898,14 +897,14 @@ Proof.
apply HP, HPQ, uPred_weaken with x' (S n');
eauto 3 using cmra_validN_le, cmra_validN_op_r.
Qed.
Global Instance forall_timeless {A} (P : A → uPred M) :
(∀ x, TimelessP (P x)) → TimelessP (∀ x, P x).
Proof. by setoid_rewrite timelessP_spec=>HP x n ?? a; apply HP. Qed.
Global Instance exist_timeless {A} (P : A → uPred M) :
(∀ x, TimelessP (P x)) → TimelessP (∃ x, P x).
Global Instance forall_timeless {A} (Ψ : A → uPred M) :
(∀ x, TimelessP (Ψ x)) → TimelessP (∀ x, Ψ x).
Proof. by setoid_rewrite timelessP_spec=> HΨ x n ?? a; apply HΨ. Qed.
Global Instance exist_timeless {A} (Ψ : A → uPred M) :
(∀ x, TimelessP (Ψ x)) → TimelessP (∃ x, Ψ x).
Proof.
by setoid_rewrite timelessP_spec=>HP x [|n] ?;
[|intros [a ?]; exists a; apply HP].
by setoid_rewrite timelessP_spec=> HΨ x [|n] ?;
[|intros [a ?]; exists a; apply HΨ].
Qed.
Global Instance always_timeless P : TimelessP P → TimelessP (□ P).
Proof.
......@@ -940,11 +939,11 @@ Global Instance or_always_stable P Q : AS P → AS Q → AS (P ∨ Q).
Proof. by intros; rewrite /AlwaysStable always_or; apply or_mono. Qed.
Global Instance sep_always_stable P Q: AS P → AS Q → AS (P ★ Q).
Proof. by intros; rewrite /AlwaysStable always_sep; apply sep_mono. Qed.
Global Instance forall_always_stable {A} (P : A → uPred M) :
(∀ x, AS (P x)) → AS (∀ x, P x).
Global Instance forall_always_stable {A} (Ψ : A → uPred M) :
(∀ x, AS (Ψ x)) → AS (∀ x, Ψ x).
Proof. by intros; rewrite /AlwaysStable always_forall; apply forall_mono. Qed.
Global Instance exist_always_stable {A} (P : A → uPred M) :
(∀ x, AS (P x)) → AS (∃ x, P x).
Global Instance exist_always_stable {A} (Ψ : A → uPred M) :
(∀ x, AS (Ψ x)) → AS (∃ x, Ψ x).
Proof. by intros; rewrite /AlwaysStable always_exist; apply exist_mono. Qed.
Global Instance eq_always_stable {A : cofeT} (a b : A) : AS (a ≡ b : uPred M)%I.
Proof. by intros; rewrite /AlwaysStable always_eq. Qed.
......@@ -954,8 +953,8 @@ Global Instance later_always_stable P : AS P → AS (▷ P).
Proof. by intros; rewrite /AlwaysStable always_later; apply later_mono. Qed.
Global Instance ownM_unit_always_stable (a : M) : AS (uPred_ownM (unit a)).
Proof. by rewrite /AlwaysStable always_ownM_unit. Qed.
Global Instance default_always_stable {A} P (Q : A → uPred M) (mx : option A) :
AS P → (∀ x, AS (Q x)) → AS (default P mx Q).
Global Instance default_always_stable {A} P (Ψ : A → uPred M) (mx : option A) :
AS P → (∀ x, AS (Ψ x)) → AS (default P mx Ψ).
Proof. destruct mx; apply _. Qed.
(* Derived lemmas for always stable *)
......
algebra/upred_big_op.v
View file @ 2c6727dc
......@@ -16,17 +16,17 @@ Notation "'Π★' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope.
(** * Other big ops *)
(** We use a type class to obtain overloaded notations *)
Definition uPred_big_sepM {M} `{Countable K} {A}
(m : gmap K A) (P : K → A → uPred M) : uPred M :=
uPred_big_sep (curry P <$> map_to_list m).
(m : gmap K A) (Φ : K → A → uPred M) : uPred M :=
uPred_big_sep (curry Φ <$> map_to_list m).
Instance: Params (@uPred_big_sepM) 6.
Notation "'Π★{map' m } P" := (uPred_big_sepM m P)
(at level 20, m at level 10, format "Π★{map m } P") : uPred_scope.
Notation "'Π★{map' m } Φ" := (uPred_big_sepM m Φ)
(at level 20, m at level 10, format "Π★{map m } Φ") : uPred_scope.
Definition uPred_big_sepS {M} `{Countable A}
(X : gset A) (P : A → uPred M) : uPred M := uPred_big_sep (P <$> elements X).
(X : gset A) (Φ : A → uPred M) : uPred M := uPred_big_sep (Φ <$> elements X).
Instance: Params (@uPred_big_sepS) 5.
Notation "'Π★{set' X } P" := (uPred_big_sepS X P)
(at level 20, X at level 10, format "Π★{set X } P") : uPred_scope.
Notation "'Π★{set' X } Φ" := (uPred_big_sepS X Φ)
(at level 20, X at level 10, format "Π★{set X } Φ") : uPred_scope.
(** * Always stability for lists *)
Class AlwaysStableL {M} (Ps : list (uPred M)) :=
......@@ -97,56 +97,56 @@ Proof. induction 1; simpl; auto with I. Qed.
Section gmap.
Context `{Countable K} {A : Type}.
Implicit Types m : gmap K A.
Implicit Types P : K → A → uPred M.
Implicit Types Φ Ψ : K → A → uPred M.
Lemma big_sepM_mono P Q m1 m2 :
m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → P k x ⊑ Q k x) →
(Π★{map m1} P) ⊑ (Π★{map m2} Q).
Lemma big_sepM_mono Φ Ψ m1 m2 :
m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → Φ k x ⊑ Ψ k x) →
(Π★{map m1} Φ) ⊑ (Π★{map m2} Ψ).
Proof.
intros HX HP. transitivity (Π★{map m2} P)%I.
intros HX HΦ. transitivity (Π★{map m2} Φ)%I.
- by apply big_sep_contains, fmap_contains, map_to_list_contains.
- apply big_sep_mono', Forall2_fmap, Forall2_Forall.
apply Forall_forall=> -[i x] ? /=. by apply HP, elem_of_map_to_list.
apply Forall_forall=> -[i x] ? /=. by apply HΦ, elem_of_map_to_list.
Qed.
Global Instance big_sepM_ne m n :
Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n))
(uPred_big_sepM (M:=M) m).
Proof.
intros P1 P2 HP. apply big_sep_ne, Forall2_fmap.
apply Forall2_Forall, Forall_true=> -[i x]; apply HP.
intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap.
apply Forall2_Forall, Forall_true=> -[i x]; apply HΦ.
Qed.
Global Instance big_sepM_proper m :
Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (≡))
(uPred_big_sepM (M:=M) m).
Proof.
intros P1 P2 HP; apply equiv_dist=> n.
apply big_sepM_ne=> k x; apply equiv_dist, HP.
intros Φ1 Φ2 HΦ; apply equiv_dist=> n.
apply big_sepM_ne=> k x; apply equiv_dist, HΦ.
Qed.
Global Instance big_sepM_mono' m :
Proper (pointwise_relation _ (pointwise_relation _ (⊑)) ==> (⊑))
(uPred_big_sepM (M:=M) m).
Proof. intros P1 P2 HP. apply big_sepM_mono; intros; [done|apply HP]. Qed.
Proof. intros Φ1 Φ2 HΦ. apply big_sepM_mono; intros; [done|apply HΦ]. Qed.
Lemma big_sepM_empty P : (Π★{map ∅} P)%I ≡ True%I.
Lemma big_sepM_empty Φ : (Π★{map ∅} Φ)%I ≡ True%I.
Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed.
Lemma big_sepM_insert P (m : gmap K A) i x :
m !! i = None → (Π★{map <[i:=x]> m} P)%I ≡ (P i x ★ Π★{map m} P)%I.
Lemma big_sepM_insert Φ (m : gmap K A) i x :
m !! i = None → (Π★{map <[i:=x]> m} Φ)%I ≡ (Φ i x ★ Π★{map m} Φ)%I.
Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed.
Lemma big_sepM_singleton P i x : (Π★{map {[i := x]}} P)%I ≡ (P i x)%I.
Lemma big_sepM_singleton Φ i x : (Π★{map {[i := x]}} Φ)%I ≡ (Φ i x)%I.
Proof.
rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty.
by rewrite big_sepM_empty right_id.
Qed.
Lemma big_sepM_sepM P Q m :
(Π★{map m} (λ i x, P i x ★ Q i x))%I ≡ (Π★{map m} P ★ Π★{map m} Q)%I.
Lemma big_sepM_sepM Φ Ψ m :
(Π★{map m} (λ i x, Φ i x ★ Ψ i x))%I ≡ (Π★{map m} Φ ★ Π★{map m} Ψ)%I.
Proof.
rewrite /uPred_big_sepM.
induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //.
by rewrite IH -!assoc (assoc _ (Q _ _)) [(Q _ _ ★ _)%I]comm -!assoc.
by rewrite IH -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%I]comm -!assoc.
Qed.
Lemma big_sepM_later P m : (▷ Π★{map m} P)%I ≡ (Π★{map m} (λ i x, ▷ P i x))%I.
Lemma big_sepM_later Φ m : (▷ Π★{map m} Φ)%I ≡ (Π★{map m} (λ i x, ▷ Φ i x))%I.
Proof.
rewrite /uPred_big_sepM.
induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //.
......@@ -158,56 +158,56 @@ End gmap.
Section gset.
Context `{Countable A}.
Implicit Types X : gset A.
Implicit Types P : A → uPred M.
Implicit Types Φ : A → uPred M.
Lemma big_sepS_mono P Q X Y :
Y ⊆ X → (∀ x, x ∈ Y → P x ⊑ Q x) → (Π★{set X} P) ⊑ (Π★{set Y} Q).
Lemma big_sepS_mono Φ Ψ X Y :
Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊑ Ψ x) → (Π★{set X} Φ) ⊑ (Π★{set Y} Ψ).
Proof.
intros HX HP. transitivity (Π★{set Y} P)%I.
intros HX HΦ. transitivity (Π★{set Y} Φ)%I.
- by apply big_sep_contains, fmap_contains, elements_contains.
- apply big_sep_mono', Forall2_fmap, Forall2_Forall.
apply Forall_forall=> x ? /=. by apply HP, elem_of_elements.
apply Forall_forall=> x ? /=. by apply HΦ, elem_of_elements.
Qed.
Lemma big_sepS_ne X n :
Proper (pointwise_relation _ (dist n) ==> dist n) (uPred_big_sepS (M:=M) X).
Proof.
intros P1 P2 HP. apply big_sep_ne, Forall2_fmap.
apply Forall2_Forall, Forall_true=> x; apply HP.
intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap.
apply Forall2_Forall, Forall_true=> x; apply HΦ.
Qed.
Lemma big_sepS_proper X :
Proper (pointwise_relation _ (≡) ==> (≡)) (uPred_big_sepS (M:=M) X).
Proof.
intros P1 P2 HP; apply equiv_dist=> n.
apply big_sepS_ne=> x; apply equiv_dist, HP.
intros Φ1 Φ2 HΦ; apply equiv_dist=> n.
apply big_sepS_ne=> x; apply equiv_dist, HΦ.
Qed.
Lemma big_sepS_mono' X :
Proper (pointwise_relation _ (⊑) ==> (⊑)) (uPred_big_sepS (M:=M) X).
Proof. intros P1 P2 HP. apply big_sepS_mono; naive_solver. Qed.
Proof. intros Φ1 Φ2 HΦ. apply big_sepS_mono; naive_solver. Qed.
Lemma big_sepS_empty P : (Π★{set ∅} P)%I ≡ True%I.
Lemma big_sepS_empty Φ : (Π★{set ∅} Φ)%I ≡ True%I.
Proof. by rewrite /uPred_big_sepS elements_empty. Qed.
Lemma big_sepS_insert P X x :
x ∉ X → (Π★{set {[ x ]} ∪ X} P)%I ≡ (P x ★ Π★{set X} P)%I.
Lemma big_sepS_insert Φ X x :
x ∉ X → (Π★{set {[ x ]} ∪ X} Φ)%I ≡ (Φ x ★ Π★{set X} Φ)%I.
Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed.
Lemma big_sepS_delete P X x :
x ∈ X → (Π★{set X} P)%I ≡ (P x ★ Π★{set X ∖ {[ x ]}} P)%I.
Lemma big_sepS_delete Φ X x :
x ∈ X → (Π★{set X} Φ)%I ≡ (Φ x ★ Π★{set X ∖ {[ x ]}} Φ)%I.
Proof.
intros. rewrite -big_sepS_insert; last set_solver.
by rewrite -union_difference_L; last set_solver.
Qed.
Lemma big_sepS_singleton P x : (Π★{set {[ x ]}} P)%I ≡ (P x)%I.
Lemma big_sepS_singleton Φ x : (Π★{set {[ x ]}} Φ)%I ≡ (Φ x)%I.
Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed.
Lemma big_sepS_sepS P Q X :
(Π★{set X} (λ x, P x ★ Q x))%I ≡ (Π★{set X} P ★ Π★{set X} Q)%I.
Lemma big_sepS_sepS Φ Ψ X :
(Π★{set X} (λ x, Φ x ★ Ψ x))%I ≡ (Π★{set X} Φ ★ Π★{set X} Ψ)%I.
Proof.
rewrite /uPred_big_sepS.
induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id.
by rewrite IH -!assoc (assoc _ (Q _)) [(Q _ ★ _)%I]comm -!assoc.
by rewrite IH -!assoc (assoc _ (Ψ _)) [(Ψ _ ★ _)%I]comm -!assoc.
Qed.
Lemma big_sepS_later P X : (▷ Π★{set X} P)%I ≡ (Π★{set X} (λ x, ▷ P x))%I.
Lemma big_sepS_later Φ X : (▷ Π★{set X} Φ)%I ≡ (Π★{set X} (λ x, ▷ Φ x))%I.
Proof.
rewrite /uPred_big_sepS.
induction (elements X) as [|x l IH]; csimpl; first by rewrite ?later_True.
......
barrier/barrier.v
View file @ 2c6727dc
......@@ -123,12 +123,12 @@ Section proof.
(∃ γ, barrier_ctx γ l P ★ sts_ownS γ low_states {[ Send ]})%I.
Definition recv (l : loc) (R : iProp) : iProp :=
(∃ γ (P Q : iProp) i, barrier_ctx γ l P ★ sts_ownS γ (i_states i) {[ Change i ]} ★
(∃ γ P Q i, barrier_ctx γ l P ★ sts_ownS γ (i_states i) {[ Change i ]} ★
saved_prop_own i Q ★ ▷(Q -★ R))%I.
Lemma newchan_spec (P : iProp) (Q : val → iProp) :
(heap_ctx heapN ★ ∀ l, recv l P ★ send l P -★ Q (LocV l))
⊑ wp ⊤ (newchan '()) Q.
Lemma newchan_spec (P : iProp) (Φ : val → iProp) :
(heap_ctx heapN ★ ∀ l, recv l P ★ send l P -★ Φ (LocV l))
⊑ wp ⊤ (newchan '()) Φ.
Proof.
rewrite /newchan. wp_rec. (* TODO: wp_seq. *)
rewrite -wp_pvs. wp> eapply wp_alloc; eauto with I ndisj.
......@@ -177,8 +177,8 @@ Section proof.
rewrite !mkSet_elem_of /=. set_solver.
Qed.
Lemma signal_spec l P (Q : val → iProp) :
heapN ⊥ N → (send l P ★ P ★ Q '()) ⊑ wp ⊤ (signal (LocV l)) Q.
Lemma signal_spec l P (Φ : val → iProp) :
heapN ⊥ N → (send l P ★ P ★ Φ '()) ⊑ wp ⊤ (signal (LocV l)) Φ.
Proof.
intros Hdisj. rewrite /signal /send /barrier_ctx. rewrite sep_exist_r.
apply exist_elim=>γ. wp_rec. (* FIXME wp_let *)
......@@ -201,19 +201,19 @@ Section proof.
apply sep_mono; last first.
{ apply wand_intro_l. eauto with I. }
(* Now we come to the core of the proof: Updating from waiting to ress. *)
rewrite /waiting /ress sep_exist_l. apply exist_elim=>{Q} Q.
rewrite /waiting /ress sep_exist_l. apply exist_elim=>{Φ} Φ.
rewrite later_wand {1}(later_intro P) !assoc wand_elim_r.
rewrite big_sepS_later -big_sepS_sepS. apply big_sepS_mono'=>i.
rewrite -(exist_intro (Q i)) comm. done.
rewrite -(exist_intro (Φ i)) comm. done.
Qed.
Lemma wait_spec l P (Q : val → iProp) :
heapN ⊥ N → (recv l P ★ (P -★ Q '())) ⊑ wp ⊤ (wait (LocV l)) Q.
Lemma wait_spec l P (Φ : val → iProp) :
heapN ⊥ N → (recv l P ★ (P -★ Φ '())) ⊑ wp ⊤ (wait (LocV l)) Φ.
Proof.
Abort.
Lemma split_spec l P1 P2 Q :
(recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Q '())) ⊑ wp ⊤ Skip Q.
Lemma split_spec l P1 P2 Φ :
(recv l (P1 ★ P2) ★ (recv l P1 ★ recv l P2 -★ Φ '())) ⊑ wp ⊤ Skip Φ.
Proof.
Abort.
......
heap_lang/derived.v
View file @ 2c6727dc
......@@ -12,49 +12,49 @@ Notation Skip := (Seq (Lit LitUnit) (Lit LitUnit)).
Section derived.
Context {Σ : iFunctor}.
Implicit Types P : iProp heap_lang Σ.
Implicit Types Q : val → iProp heap_lang Σ.
Implicit Types P Q : iProp heap_lang Σ.
Implicit Types Φ : val → iProp heap_lang Σ.
(** Proof rules for the sugar *)
Lemma wp_lam' E x ef e v Q :
to_val e = Some v → ▷ wp E (subst ef x v) Q ⊑ wp E (App (Lam x ef) e) Q.
Lemma wp_lam' E x ef e v Φ :
to_val e = Some v → ▷ wp E (subst ef x v) Φ ⊑ wp E (App (Lam x ef) e) Φ.
Proof. intros. by rewrite -wp_rec' ?subst_empty. Qed.
Lemma wp_let' E x e1 e2 v Q :
to_val e1 = Some v → ▷ wp E (subst e2 x v) Q ⊑ wp E (Let x e1 e2) Q.
Lemma wp_let' E x e1 e2 v Φ :
to_val e1 = Some v → ▷ wp E (subst e2 x v) Φ ⊑ wp E (Let x e1 e2) Φ.
Proof. apply wp_lam'. Qed.
Lemma wp_seq E e1 e2 Q : wp E e1 (λ _, ▷ wp E e2 Q) ⊑ wp E (Seq e1 e2) Q.
Lemma wp_seq E e1 e2 Φ : wp E e1 (λ _, ▷ wp E e2 Φ) ⊑ wp E (Seq e1 e2) Φ.
Proof.
rewrite -(wp_bind [LetCtx "" e2]). apply wp_mono=>v.
by rewrite -wp_let' //= ?to_of_val ?subst_empty.