Skip to content

GitLab

I
Iris
Commit 73e38f15 authored by Michael Sammler's avatar Michael Sammler
Browse files
Options

seal big_opM, big_opS, big_opMS and big_sepM2

parent 62798412
Hide whitespace changes
Inline Side-by-side
Showing with 103 additions and 92 deletions
theories/algebra/big_op.v
View file @ 73e38f15
......@@ -35,11 +35,13 @@ Notation "'[^' o 'list]' x ∈ l , P" := (big_opL o (λ _ x, P) l)
(at level 200, o at level 1, l at level 10, x at level 1, right associativity,
format "[^ o list] x ∈ l , P") : stdpp_scope.
Definition big_opM `{Monoid M o} `{Countable K} {A} (f : K → A → M)
(m : gmap K A) : M := big_opL o (λ _, curry f) (map_to_list m).
Definition big_opM_def `{Monoid M o} `{Countable K} {A} (f : K → A → M)
(m : gmap K A) : M := big_opL o (λ _, curry f) (map_to_list m).
Definition big_opM_aux : seal (@big_opM_def). by eexists. Qed.
Definition big_opM := big_opM_aux.(unseal).
Arguments big_opM {M} o {_ K _ _ A} _ _.
Definition big_opM_eq : @big_opM = @big_opM_def := big_opM_aux.(seal_eq).
Instance: Params (@big_opM) 7 := {}.
Arguments big_opM {M} o {_ K _ _ A} _ _ : simpl never.
Typeclasses Opaque big_opM.
Notation "'[^' o 'map]' k ↦ x ∈ m , P" := (big_opM o (λ k x, P) m)
(at level 200, o at level 1, m at level 10, k, x at level 1, right associativity,
format "[^ o map] k ↦ x ∈ m , P") : stdpp_scope.
......@@ -47,20 +49,24 @@ Notation "'[^' o 'map]' x ∈ m , P" := (big_opM o (λ _ x, P) m)
(at level 200, o at level 1, m at level 10, x at level 1, right associativity,
format "[^ o map] x ∈ m , P") : stdpp_scope.
Definition big_opS `{Monoid M o} `{Countable A} (f : A → M)
Definition big_opS_def `{Monoid M o} `{Countable A} (f : A → M)
(X : gset A) : M := big_opL o (λ _, f) (elements X).
Definition big_opS_aux : seal (@big_opS_def). by eexists. Qed.
Definition big_opS := big_opS_aux.(unseal).
Arguments big_opS {M} o {_ A _ _} _ _.
Definition big_opS_eq : @big_opS = @big_opS_def := big_opS_aux.(seal_eq).
Instance: Params (@big_opS) 6 := {}.
Arguments big_opS {M} o {_ A _ _} _ _ : simpl never.
Typeclasses Opaque big_opS.
Notation "'[^' o 'set]' x ∈ X , P" := (big_opS o (λ x, P) X)
(at level 200, o at level 1, X at level 10, x at level 1, right associativity,
format "[^ o set] x ∈ X , P") : stdpp_scope.
Definition big_opMS `{Monoid M o} `{Countable A} (f : A → M)
Definition big_opMS_def `{Monoid M o} `{Countable A} (f : A → M)
(X : gmultiset A) : M := big_opL o (λ _, f) (elements X).
Instance: Params (@big_opMS) 7 := {}.
Arguments big_opMS {M} o {_ A _ _} _ _ : simpl never.
Typeclasses Opaque big_opMS.
Definition big_opMS_aux : seal (@big_opMS_def). by eexists. Qed.
Definition big_opMS := big_opMS_aux.(unseal).
Arguments big_opMS {M} o {_ A _ _} _ _.
Definition big_opMS_eq : @big_opMS = @big_opMS_def := big_opMS_aux.(seal_eq).
Instance: Params (@big_opMS) 6 := {}.
Notation "'[^' o 'mset]' x ∈ X , P" := (big_opMS o (λ x, P) X)
(at level 200, o at level 1, X at level 10, x at level 1, right associativity,
format "[^ o mset] x ∈ X , P") : stdpp_scope.
......@@ -171,7 +177,7 @@ Section gmap.
(∀ k x, m !! k = Some x → R (f k x) (g k x)) →
R ([^o map] k ↦ x ∈ m, f k x) ([^o map] k ↦ x ∈ m, g k x).
Proof.
intros ?? Hf. apply (big_opL_forall R); auto.
intros ?? Hf. rewrite big_opM_eq. apply (big_opL_forall R); auto.
intros k [i x] ?%elem_of_list_lookup_2. by apply Hf, elem_of_map_to_list.
Qed.
......@@ -194,12 +200,12 @@ Section gmap.
Proof. intros f g Hf m ? <-. apply big_opM_forall; apply _ || intros; apply Hf. Qed.
Lemma big_opM_empty f : ([^o map] k↦x ∈ ∅, f k x) = monoid_unit.
Proof. by rewrite /big_opM map_to_list_empty. Qed.
Proof. by rewrite big_opM_eq /big_opM_def map_to_list_empty. Qed.
Lemma big_opM_insert f m i x :
m !! i = None →
([^o map] k↦y ∈ <[i:=x]> m, f k y) ≡ f i x `o` [^o map] k↦y ∈ m, f k y.
Proof. intros ?. by rewrite /big_opM map_to_list_insert. Qed.
Proof. intros ?. by rewrite big_opM_eq /big_opM_def map_to_list_insert. Qed.
Lemma big_opM_delete f m i x :
m !! i = Some x →
......@@ -216,12 +222,12 @@ Section gmap.
Qed.
Lemma big_opM_unit m : ([^o map] k↦y ∈ m, monoid_unit) ≡ (monoid_unit : M).
Proof. induction m using map_ind; rewrite /= ?big_opM_insert ?left_id //. Qed.
Proof. by induction m using map_ind; rewrite /= ?big_opM_insert ?left_id // big_opM_eq. Qed.
Lemma big_opM_fmap {B} (h : A → B) (f : K → B → M) m :
([^o map] k↦y ∈ h <$> m, f k y) ≡ ([^o map] k↦y ∈ m, f k (h y)).
Proof.
rewrite /big_opM map_to_list_fmap big_opL_fmap.
rewrite big_opM_eq /big_opM_def map_to_list_fmap big_opL_fmap.
by apply big_opL_proper=> ? [??].
Qed.
......@@ -262,7 +268,7 @@ Section gmap.
Lemma big_opM_op f g m :
([^o map] k↦x ∈ m, f k x `o` g k x)
≡ ([^o map] k↦x ∈ m, f k x) `o` ([^o map] k↦x ∈ m, g k x).
Proof. rewrite /big_opM -big_opL_op. by apply big_opL_proper=> ? [??]. Qed.
Proof. rewrite big_opM_eq /big_opM_def -big_opL_op. by apply big_opL_proper=> ? [??]. Qed.
End gmap.
......@@ -277,7 +283,7 @@ Section gset.
(∀ x, x ∈ X → R (f x) (g x)) →
R ([^o set] x ∈ X, f x) ([^o set] x ∈ X, g x).
Proof.
intros ?? Hf. apply (big_opL_forall R); auto.
rewrite big_opS_eq. intros ?? Hf. apply (big_opL_forall R); auto.
intros k x ?%elem_of_list_lookup_2. by apply Hf, elem_of_elements.
Qed.
......@@ -298,11 +304,11 @@ Section gset.
Proof. intros f g Hf m ? <-. apply big_opS_forall; apply _ || intros; apply Hf. Qed.
Lemma big_opS_empty f : ([^o set] x ∈ ∅, f x) = monoid_unit.
Proof. by rewrite /big_opS elements_empty. Qed.
Proof. by rewrite big_opS_eq /big_opS_def elements_empty. Qed.
Lemma big_opS_insert f X x :
x ∉ X → ([^o set] y ∈ {[ x ]} ∪ X, f y) ≡ (f x `o` [^o set] y ∈ X, f y).
Proof. intros. by rewrite /big_opS elements_union_singleton. Qed.
Proof. intros. by rewrite big_opS_eq /big_opS_def elements_union_singleton. Qed.
Lemma big_opS_fn_insert {B} (f : A → B → M) h X x b :
x ∉ X →
([^o set] y ∈ {[ x ]} ∪ X, f y (<[x:=b]> h y))
......@@ -334,22 +340,22 @@ Section gset.
Qed.
Lemma big_opS_singleton f x : ([^o set] y ∈ {[ x ]}, f y) ≡ f x.
Proof. intros. by rewrite /big_opS elements_singleton /= right_id. Qed.
Proof. intros. by rewrite big_opS_eq /big_opS_def elements_singleton /= right_id. Qed.
Lemma big_opS_unit X : ([^o set] y ∈ X, monoid_unit) ≡ (monoid_unit : M).
Proof.
induction X using set_ind_L; rewrite /= ?big_opS_insert ?left_id //.
by induction X using set_ind_L; rewrite /= ?big_opS_insert ?left_id // big_opS_eq.
Qed.
Lemma big_opS_op f g X :
([^o set] y ∈ X, f y `o` g y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ X, g y).
Proof. by rewrite /big_opS -big_opL_op. Qed.
Proof. by rewrite big_opS_eq /big_opS_def -big_opL_op. Qed.
End gset.
Lemma big_opM_dom `{Countable K} {A} (f : K → M) (m : gmap K A) :
([^o map] k↦_ ∈ m, f k) ≡ ([^o set] k ∈ dom _ m, f k).
Proof.
induction m as [|i x ?? IH] using map_ind; [by rewrite dom_empty_L|].
induction m as [|i x ?? IH] using map_ind; [by rewrite big_opM_eq big_opS_eq dom_empty_L|].
by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom.
Qed.
......@@ -364,7 +370,7 @@ Section gmultiset.
(∀ x, x ∈ X → R (f x) (g x)) →
R ([^o mset] x ∈ X, f x) ([^o mset] x ∈ X, g x).
Proof.
intros ?? Hf. apply (big_opL_forall R); auto.
rewrite big_opMS_eq. intros ?? Hf. apply (big_opL_forall R); auto.
intros k x ?%elem_of_list_lookup_2. by apply Hf, gmultiset_elem_of_elements.
Qed.
......@@ -385,15 +391,15 @@ Section gmultiset.
Proof. intros f g Hf m ? <-. apply big_opMS_forall; apply _ || intros; apply Hf. Qed.
Lemma big_opMS_empty f : ([^o mset] x ∈ ∅, f x) = monoid_unit.
Proof. by rewrite /big_opMS gmultiset_elements_empty. Qed.
Proof. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_empty. Qed.
Lemma big_opMS_disj_union f X Y :
([^o mset] y ∈ X ⊎ Y, f y) ≡ ([^o mset] y ∈ X, f y) `o` [^o mset] y ∈ Y, f y.
Proof. by rewrite /big_opMS gmultiset_elements_disj_union big_opL_app. Qed.
Proof. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_disj_union big_opL_app. Qed.
Lemma big_opMS_singleton f x : ([^o mset] y ∈ {[ x ]}, f y) ≡ f x.
Proof.
intros. by rewrite /big_opMS gmultiset_elements_singleton /= right_id.
intros. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_singleton /= right_id.
Qed.
Lemma big_opMS_delete f X x :
......@@ -405,13 +411,13 @@ Section gmultiset.
Lemma big_opMS_unit X : ([^o mset] y ∈ X, monoid_unit) ≡ (monoid_unit : M).
Proof.
induction X using gmultiset_ind;
rewrite /= ?big_opMS_disj_union ?big_opMS_singleton ?left_id //.
by induction X using gmultiset_ind;
rewrite /= ?big_opMS_disj_union ?big_opMS_singleton ?left_id // big_opMS_eq.
Qed.
Lemma big_opMS_op f g X :
([^o mset] y ∈ X, f y `o` g y) ≡ ([^o mset] y ∈ X, f y) `o` ([^o mset] y ∈ X, g y).
Proof. by rewrite /big_opMS -big_opL_op. Qed.
Proof. by rewrite big_opMS_eq /big_opMS_def -big_opL_op. Qed.
End gmultiset.
End big_op.
......
theories/algebra/cmra_big_op.v
View file @ 73e38f15
......@@ -13,7 +13,7 @@ Qed.
Lemma big_opM_None {M : cmraT} `{Countable K} {A} (f : K → A → option M) m :
([^op map] k↦x ∈ m, f k x) = None ↔ ∀ k x, m !! k = Some x → f k x = None.
Proof.
induction m as [|i x m ? IH] using map_ind=> //=.
induction m as [|i x m ? IH] using map_ind=> /=; first by rewrite big_opM_eq.
rewrite -equiv_None big_opM_insert // equiv_None op_None IH. split.
{ intros [??] k y. rewrite lookup_insert_Some; naive_solver. }
intros Hm; split.
......@@ -23,7 +23,7 @@ Qed.
Lemma big_opS_None {M : cmraT} `{Countable A} (f : A → option M) X :
([^op set] x ∈ X, f x) = None ↔ ∀ x, x ∈ X → f x = None.
Proof.
induction X as [|x X ? IH] using set_ind_L; [done|].
induction X as [|x X ? IH] using set_ind_L; [by rewrite big_opS_eq |].
rewrite -equiv_None big_opS_insert // equiv_None op_None IH. set_solver.
Qed.
Lemma big_opMS_None {M : cmraT} `{Countable A} (f : A → option M) X :
......
theories/base_logic/lib/boxes.v
View file @ 73e38f15
......@@ -100,7 +100,7 @@ Qed.
Lemma box_alloc : box N ∅ True%I.
Proof.
iIntros. iExists (λ _, True)%I. iSplit; by auto.
iIntros. iExists (λ _, True)%I. by rewrite !big_opM_empty.
Qed.
Lemma slice_insert_empty E q f Q P :
......
theories/bi/big_op.v
View file @ 73e38f15
......@@ -50,12 +50,15 @@ Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P" :=
Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P" :=
(big_sepL2 (λ _ x1 x2, P) l1 l2) : bi_scope.
Definition big_sepM2 {PROP : bi} `{Countable K} {A B}
Definition big_sepM2_def {PROP : bi} `{Countable K} {A B}
(Φ : K → A → B → PROP) (m1 : gmap K A) (m2 : gmap K B) : PROP :=
(⌜ ∀ k, is_Some (m1 !! k) ↔ is_Some (m2 !! k) ⌝ ∧
[∗ map] k ↦ xy ∈ map_zip m1 m2, Φ k xy.1 xy.2)%I.
Definition big_sepM2_aux : seal (@big_sepM2_def). by eexists. Qed.
Definition big_sepM2 := big_sepM2_aux.(unseal).
Arguments big_sepM2 {PROP K _ _ A B} _ _ _.
Definition big_sepM2_eq : @big_sepM2 = @big_sepM2_def := big_sepM2_aux.(seal_eq).
Instance: Params (@big_sepM2) 6 := {}.
Typeclasses Opaque big_sepM2.
Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P" :=
(big_sepM2 (λ k x1 x2, P) m1 m2) : bi_scope.
Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P" :=
......@@ -687,7 +690,7 @@ Section map.
Proof. apply big_opM_proper. Qed.
Lemma big_sepM_subseteq `{BiAffine PROP} Φ m1 m2 :
m2 ⊆ m1 → ([∗ map] k ↦ x ∈ m1, Φ k x) ⊢ [∗ map] k ↦ x ∈ m2, Φ k x.
Proof. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed.
Proof. rewrite big_opM_eq. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed.
Global Instance big_sepM_mono' :
Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢))
......@@ -834,7 +837,7 @@ Section map.
[∗ map] k↦x ∈ m, Ψ k x.
Proof.
apply wand_intro_l. induction m as [|i x m ? IH] using map_ind.
{ by rewrite sep_elim_r. }
{ by rewrite big_opM_eq sep_elim_r. }
rewrite !big_sepM_insert // intuitionistically_sep_dup.
rewrite -assoc [(□ _ ∗ _)%I]comm -!assoc assoc. apply sep_mono.
- rewrite (forall_elim i) (forall_elim x) pure_True ?lookup_insert //.
......@@ -848,17 +851,17 @@ Section map.
Global Instance big_sepM_empty_persistent Φ :
Persistent ([∗ map] k↦x ∈ ∅, Φ k x).
Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
Proof. rewrite big_opM_eq /big_opM_def map_to_list_empty. apply _. Qed.
Global Instance big_sepM_persistent Φ m :
(∀ k x, Persistent (Φ k x)) → Persistent ([∗ map] k↦x ∈ m, Φ k x).
Proof. intros. apply big_sepL_persistent=> _ [??]; apply _. Qed.
Proof. rewrite big_opM_eq. intros. apply big_sepL_persistent=> _ [??]; apply _. Qed.
Global Instance big_sepM_empty_affine Φ :
Affine ([∗ map] k↦x ∈ ∅, Φ k x).
Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
Proof. rewrite big_opM_eq /big_opM_def map_to_list_empty. apply _. Qed.
Global Instance big_sepM_affine Φ m :
(∀ k x, Affine (Φ k x)) → Affine ([∗ map] k↦x ∈ m, Φ k x).
Proof. intros. apply big_sepL_affine=> _ [??]; apply _. Qed.
Proof. rewrite big_opM_eq. intros. apply big_sepL_affine=> _ [??]; apply _. Qed.
End map.
(** ** Big ops over two maps *)
......@@ -869,7 +872,7 @@ Section map2.
Lemma big_sepM2_dom Φ m1 m2 :
([∗ map] k↦y1;y2 ∈ m1; m2, Φ k y1 y2) -∗ ⌜ dom (gset K) m1 = dom (gset K) m2 ⌝.
Proof.
rewrite /big_sepM2 and_elim_l. apply pure_mono=>Hm.
rewrite big_sepM2_eq /big_sepM2_def and_elim_l. apply pure_mono=>Hm.
set_unfold=>k. by rewrite !elem_of_dom.
Qed.
......@@ -877,7 +880,8 @@ Section map2.
(∀ k y1 y2, m1 !! k = Some y1 → m2 !! k = Some y2 → Φ k y1 y2 ⊢ Ψ k y1 y2) →
([∗ map] k ↦ y1;y2 ∈ m1;m2, Φ k y1 y2) ⊢ [∗ map] k ↦ y1;y2 ∈ m1;m2, Ψ k y1 y2.
Proof.
intros Hm1m2. rewrite /big_sepM2. apply and_mono_r, big_sepM_mono.
intros Hm1m2. rewrite big_sepM2_eq /big_sepM2_def.
apply and_mono_r, big_sepM_mono.
intros k [x1 x2]. rewrite map_lookup_zip_with.
specialize (Hm1m2 k x1 x2).
destruct (m1 !! k) as [y1|]; last done.
......@@ -897,8 +901,8 @@ Section map2.
==> (=) ==> (=) ==> (dist n))
(big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)).
Proof.
intros Φ1 Φ2 HΦ x1 ? <- x2 ? <-. rewrite /big_sepM2. f_equiv.
f_equiv=> k [y1 y2]. apply HΦ.
intros Φ1 Φ2 HΦ x1 ? <- x2 ? <-. rewrite big_sepM2_eq /big_sepM2_def.
f_equiv. f_equiv=> k [y1 y2]. apply HΦ.
Qed.
Global Instance big_sepM2_mono' :
Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊢)))
......@@ -913,7 +917,7 @@ Section map2.
Lemma big_sepM2_empty Φ : ([∗ map] k↦y1;y2 ∈ ∅; ∅, Φ k y1 y2) ⊣⊢ emp.
Proof.
rewrite /big_sepM2 pure_True ?left_id //.
rewrite big_sepM2_eq /big_sepM2_def big_opM_eq pure_True ?left_id //.
intros k. rewrite !lookup_empty; split; by inversion 1.
Qed.
Lemma big_sepM2_empty' `{BiAffine PROP} P Φ : P ⊢ [∗ map] k↦y1;y2 ∈ ∅;∅, Φ k y1 y2.
......@@ -938,7 +942,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2)
⊣⊢ Φ i x1 x2 ∗ [∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2.
Proof.
intros Hm1 Hm2. rewrite /big_sepM2 -map_insert_zip_with.
intros Hm1 Hm2. rewrite big_sepM2_eq /big_sepM2_def -map_insert_zip_with.
rewrite big_sepM_insert;
last by rewrite map_lookup_zip_with Hm1.
rewrite !persistent_and_affinely_sep_l /=.
......@@ -959,7 +963,7 @@ Section map2.
([∗ map] k↦x;y ∈ m1;m2, Φ k x y) ⊣⊢
Φ i x1 x2 ∗ [∗ map] k↦x;y ∈ delete i m1;delete i m2, Φ k x y.
Proof.
rewrite /big_sepM2 => Hx1 Hx2.
rewrite big_sepM2_eq /big_sepM2_def => Hx1 Hx2.
rewrite !persistent_and_affinely_sep_l /=.
rewrite sep_assoc (sep_comm (Φ _ _ _)) -sep_assoc.
apply sep_proper.
......@@ -993,7 +997,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2) -∗
([∗ map] k↦y1;y2 ∈ <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2).
Proof.
intros Ha. rewrite /big_sepM2.
intros Ha. rewrite big_sepM2_eq /big_sepM2_def.
assert (TCOr (∀ x, Affine (Φ i x.1 x.2)) (Absorbing (Φ i x1 x2))).
{ destruct Ha; try apply _. }
apply wand_intro_r.
......@@ -1027,7 +1031,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2)
⊢ ∃ x2, ⌜m2 !! i = Some x2⌝ ∧ Φ i x1 x2.
Proof.
intros Hm1. rewrite /big_sepM2.
intros Hm1. rewrite big_sepM2_eq /big_sepM2_def.
rewrite persistent_and_sep_1.
apply wand_elim_l'. apply pure_elim'=>Hm.
assert (is_Some (m2 !! i)) as [x2 Hm2].
......@@ -1044,7 +1048,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2)
⊢ ∃ x1, ⌜m1 !! i = Some x1⌝ ∧ Φ i x1 x2.
Proof.
intros Hm2. rewrite /big_sepM2.
intros Hm2. rewrite big_sepM2_eq /big_sepM2_def.
rewrite persistent_and_sep_1.
apply wand_elim_l'. apply pure_elim'=>Hm.
assert (is_Some (m1 !! i)) as [x1 Hm1].
......@@ -1059,7 +1063,8 @@ Section map2.
Lemma big_sepM2_singleton Φ i x1 x2 :
([∗ map] k↦y1;y2 ∈ {[ i := x1 ]}; {[ i := x2 ]}, Φ k y1 y2) ⊣⊢ Φ i x1 x2.
Proof.
rewrite /big_sepM2 map_zip_with_singleton big_sepM_singleton.
rewrite big_sepM2_eq /big_sepM2_def.
rewrite map_zip_with_singleton big_sepM_singleton.
apply (anti_symm _).
- apply and_elim_r.
- rewrite <- (left_id True%I (∧)%I (Φ i x1 x2)).
......@@ -1072,7 +1077,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ f <$> m1; g <$> m2, Φ k y1 y2)
⊣⊢ ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k (f y1) (g y2)).
Proof.
rewrite /big_sepM2. rewrite map_fmap_zip.
rewrite big_sepM2_eq /big_sepM2_def. rewrite map_fmap_zip.
apply and_proper.
- apply pure_proper. split.
+ intros Hm k. specialize (Hm k). revert Hm.
......@@ -1096,7 +1101,7 @@ Section map2.
([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2 ∗ Ψ k y1 y2)
⊣⊢ ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2) ∗ ([∗ map] k↦y1;y2 ∈ m1;m2, Ψ k y1 y2).
Proof.
rewrite /big_sepM2.
rewrite big_sepM2_eq /big_sepM2_def.
rewrite -{1}(and_idem ⌜∀ k : K, is_Some (m1 !! k) ↔ is_Some (m2 !! k)⌝%I).
rewrite -and_assoc.
rewrite !persistent_and_affinely_sep_l /=.
......@@ -1114,7 +1119,7 @@ Section map2.
<pers> ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2)
⊣⊢ [∗ map] k↦y1;y2 ∈ m1;m2, <pers> (Φ k y1 y2).
Proof.
by rewrite /big_sepM2 persistently_and
by rewrite big_sepM2_eq /big_sepM2_def persistently_and
persistently_pure big_sepM_persistently.
Qed.
......@@ -1124,7 +1129,7 @@ Section map2.
⌜∀ k : K, is_Some (m1 !! k) ↔ is_Some (m2 !! k)⌝
∧ (∀ k x1 x2, ⌜m1 !! k = Some x1⌝ → ⌜m2 !! k = Some x2⌝ → Φ k x1 x2).
Proof.
rewrite /big_sepM2=> ?. apply and_proper=>//.
rewrite big_sepM2_eq /big_sepM2_def=> ?. apply and_proper=>//.
rewrite big_sepM_forall. apply forall_proper=>k.
apply (anti_symm _).
- apply forall_intro=> x1. apply forall_intro=> x2.
......@@ -1145,7 +1150,7 @@ Section map2.
[∗ map] k↦y1;y2 ∈ m1;m2, Ψ k y1 y2.
Proof.
apply wand_intro_l.
rewrite /big_sepM2.
rewrite big_sepM2_eq /big_sepM2_def.
rewrite !persistent_and_affinely_sep_l /=.
rewrite sep_assoc. rewrite (sep_comm (□ _)%I) -sep_assoc.
apply sep_mono_r. apply wand_elim_r'.
......@@ -1164,7 +1169,7 @@ Section map2.
Global Instance big_sepM2_persistent Φ m1 m2 :
(∀ k x1 x2, Persistent (Φ k x1 x2)) →
Persistent ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2).
Proof. rewrite /big_sepM2. apply _. Qed.
Proof. rewrite big_sepM2_eq /big_sepM2_def. apply _. Qed.
Global Instance big_sepM2_empty_affine Φ :
Affine ([∗ map] k↦y1;y2 ∈ ∅; ∅, Φ k y1 y2).
......@@ -1172,7 +1177,7 @@ Section map2.
Global Instance big_sepM2_affine Φ m1 m2 :
(∀ k x1 x2, Affine (Φ k x1 x2)) →
Affine ([∗ map] k↦y1;y2 ∈ m1;m2, Φ k y1 y2).
Proof. rewrite /big_sepM2. apply _. Qed.
Proof. rewrite big_sepM2_eq /big_sepM2_def. apply _. Qed.
End map2.
(** ** Big ops over finite sets *)
......@@ -1191,7 +1196,7 @@ Section gset.
Proof. apply big_opS_proper. Qed.
Lemma big_sepS_subseteq `{BiAffine PROP} Φ X Y :
Y ⊆ X → ([∗ set] x ∈ X, Φ x) ⊢ [∗ set] x ∈ Y, Φ x.
Proof. intros. by apply big_sepL_submseteq, elements_submseteq. Qed.
Proof. rewrite big_opS_eq. intros. by apply big_sepL_submseteq, elements_submseteq. Qed.
Global Instance big_sepS_mono' :
Proper (pointwise_relation _ (⊢) ==> (=) ==> (⊢)) (big_opS (@bi_sep PROP) (A:=A)).
......@@ -1309,7 +1314,7 @@ Section gset.
[∗ set] x ∈ X, Ψ x.
Proof.
apply wand_intro_l. induction X as [|x X ? IH] using set_ind_L.
{ by rewrite sep_elim_r. }
{ by rewrite big_opS_eq sep_elim_r. }
rewrite !big_sepS_insert // intuitionistically_sep_dup.
rewrite -assoc [(□ _ ∗ _)%I]comm -!assoc assoc. apply sep_mono.
- rewrite (forall_elim x) pure_True; last set_solver.
......@@ -1321,16 +1326,16 @@ Section gset.
Global Instance big_sepS_empty_persistent Φ :
Persistent ([∗ set] x ∈ ∅, Φ x).
Proof. rewrite /big_opS elements_empty. apply _. Qed.
Proof. rewrite big_opS_eq /big_opS_def elements_empty. apply _. Qed.
Global Instance big_sepS_persistent Φ X :
(∀ x, Persistent (Φ x)) → Persistent ([∗ set] x ∈ X, Φ x).
Proof. rewrite /big_opS. apply _. Qed.
Proof. rewrite big_opS_eq /big_opS_def. apply _. Qed.
Global Instance big_sepS_empty_affine Φ : Affine ([∗ set] x ∈ ∅, Φ x).
Proof. rewrite /big_opS elements_empty. apply _. Qed.
Proof. rewrite big_opS_eq /big_opS_def elements_empty. apply _. Qed.
Global Instance big_sepS_affine Φ X :
(∀ x, Affine (Φ x)) → Affine ([∗ set] x ∈ X, Φ x).
Proof. rewrite /big_opS. apply _. Qed.
Proof. rewrite big_opS_eq /big_opS_def. apply _. Qed.
End gset.
Lemma big_sepM_dom `{Countable K} {A} (Φ : K → PROP) (m : gmap K A) :
......@@ -1353,7 +1358,7 @@ Section gmultiset.
Proof. apply big_opMS_proper. Qed.
Lemma big_sepMS_subseteq `{BiAffine PROP} Φ X Y :
Y ⊆ X → ([∗ mset] x ∈ X, Φ x) ⊢ [∗ mset] x ∈ Y, Φ x.
Proof. intros. by apply big_sepL_submseteq, gmultiset_elements_submseteq. Qed.
Proof. rewrite big_opMS_eq. intros. by apply big_sepL_submseteq, gmultiset_elements_submseteq. Qed.
Global Instance big_sepMS_mono' :
Proper (pointwise_relation _ (⊢) ==> (=) ==> (⊢)) (big_opMS (@bi_sep PROP) (A:=A)).
......@@ -1400,16 +1405,16 @@ Section gmultiset.
Global Instance big_sepMS_empty_persistent Φ :
Persistent ([∗ mset] x ∈ ∅, Φ x).
Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
Proof. rewrite big_opMS_eq /big_opMS_def gmultiset_elements_empty. apply _. Qed.
Global Instance big_sepMS_persistent Φ X :
(∀ x, Persistent (Φ x)) → Persistent ([∗ mset] x ∈ X, Φ x).
Proof. rewrite /big_opMS. apply _. Qed.
Proof. rewrite big_opMS_eq /big_opMS_def. apply _. Qed.
Global Instance big_sepMS_empty_affine Φ : Affine ([∗ mset] x ∈ ∅, Φ x).
Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
Proof. rewrite big_opMS_eq /big_opMS_def gmultiset_elements_empty. apply _. Qed.
Global Instance big_sepMS_affine Φ X :
(∀ x, Affine (Φ x)) → Affine ([∗ mset] x ∈ X, Φ x).
Proof. rewrite /big_opMS. apply _. Qed.
Proof. rewrite big_opMS_eq /big_opMS_def. apply _. Qed.
End gmultiset.
End bi_big_op.
......@@ -1513,10 +1518,10 @@ Section gmap.
Global Instance big_sepM_empty_timeless `{!Timeless (emp%I : PROP)} Φ :
Timeless ([∗ map] k↦x ∈ ∅, Φ k x).
Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
Proof. rewrite big_opM_eq /big_opM_def map_to_list_empty. apply _. Qed.
Global Instance big_sepM_timeless `{!Timeless (emp%I : PROP)} Φ m :
(∀ k x, Timeless (Φ k x)) → Timeless ([∗ map] k↦x ∈ m, Φ k x).
Proof. intros. apply big_sepL_timeless=> _ [??]; apply _. Qed.
Proof. rewrite big_opM_eq. intros. apply big_sepL_timeless=> _ [??]; apply _. Qed.
End gmap.
Section gmap2.
......@@ -1527,7 +1532,7 @@ Section gmap2.
(▷ [∗ map] k↦x1;x2 ∈ m1;m2, Φ k x1 x2)
⊢ ◇ ([∗ map] k↦x1;x2 ∈ m1;m2, ▷ Φ k x1 x2).
Proof.
rewrite /big_sepM2 later_and (timeless ⌜_⌝%I).
rewrite big_sepM2_eq /big_sepM2_def later_and (timeless ⌜_⌝%I).
rewrite big_sepM_later except_0_and.
auto using and_mono_r, except_0_intro.
Qed.
......@@ -1535,7 +1540,7 @@ Section gmap2.
([∗ map] k↦x1;x2 ∈ m1;m2, ▷ Φ k x1 x2)
⊢ ▷ [∗ map] k↦x1;x2 ∈ m1;m2, Φ k x1 x2.
Proof.
rewrite /big_sepM2 later_and -(later_intro ⌜_⌝%I).
rewrite big_sepM2_eq /big_sepM2_def later_and -(later_intro ⌜_⌝%I).
apply and_mono_r. by rewrite big_opM_commute.
Qed.
......@@ -1551,18 +1556,18 @@ Section gmap2.
Lemma big_sepM2_flip Φ m1 m2 :
([∗ map] k↦y1;y2 ∈ m2; m1, Φ k y2 y1) ⊣⊢ ([∗ map] k↦y1;y2 ∈ m1; m2, Φ k y1 y2).
Proof.
rewrite /big_sepM2. apply and_proper; [apply pure_proper; naive_solver |].
rewrite big_sepM2_eq /big_sepM2_def. apply and_proper; [apply pure_proper; naive_solver |].
rewrite -map_zip_with_flip map_zip_with_map_zip big_sepM_fmap.
apply big_sepM_proper. by intros k [b a].
Qed.
Global Instance big_sepM2_empty_timeless `{!Timeless (emp%I : PROP)} Φ :
Timeless ([∗ map] k↦x1;x2 ∈ ∅;∅, Φ k x1 x2).
Proof. rewrite /big_sepM2 map_zip_with_empty. apply _. Qed.
Proof. rewrite big_sepM2_eq /big_sepM2_def map_zip_with_empty. apply _. Qed.
Global Instance big_sepM2_timeless `{!Timeless (emp%I : PROP)} Φ m1 m2 :
(∀ k x1 x2, Timeless (Φ k x1 x2)) →
Timeless ([∗ map] k↦x1;x2 ∈ m1;m2, Φ k x1 x2).
Proof. intros. rewrite /big_sepM2. apply _. Qed.
Proof. intros. rewrite big_sepM2_eq /big_sepM2_def. apply _. Qed.
End gmap2.
(** ** Big ops over finite sets *)
......@@ -1587,10 +1592,10 @@ Section gset.