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Commit 6aba4a3f authored by Robbert Krebbers's avatar Robbert Krebbers
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Notations `=@{A}` and `≡@{A}` for being explicit about the type of (setoid) equality. 

This followed from discussions in https://gitlab.mpi-sws.org/FP/iris-coq/merge_requests/134
parent 3bcaaf7e
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theories/base.v
View file @ 6aba4a3f
......@@ -159,10 +159,16 @@ Notation "(≠)" := (λ x y, x ≠ y) (only parsing) : stdpp_scope.
Notation "( x ≠)" := (λ y, x ≠ y) (only parsing) : stdpp_scope.
Notation "(≠ x )" := (λ y, y ≠ x) (only parsing) : stdpp_scope.
Infix "=@{ A }" := (@eq A)
(at level 70, only parsing, no associativity) : stdpp_scope.
Notation "(=@{ A } )" := (@eq A) (only parsing) : stdpp_scope.
Notation "(≠@{ A } )" := (λ X Y, ¬X =@{A} Y) (only parsing) : stdpp_scope.
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y) (at level 70, no associativity) : stdpp_scope.
Hint Extern 0 (_ = _) => reflexivity.
Hint Extern 100 (_ ≠ _) => discriminate.
Instance: @PreOrder A (=).
Instance: ∀ A, PreOrder (=@{A}).
Proof. split; repeat intro; congruence. Qed.
(** ** Setoid equality *)
......@@ -174,6 +180,9 @@ Class Equiv A := equiv: relation A.
Hint Mode Equiv ! : typeclass_instances. *)
Infix "≡" := equiv (at level 70, no associativity) : stdpp_scope.
Infix "≡@{ A }" := (@equiv A _)
(at level 70, only parsing, no associativity) : stdpp_scope.
Notation "(≡)" := equiv (only parsing) : stdpp_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(≡ X )" := (λ Y, Y ≡ X) (only parsing) : stdpp_scope.
......@@ -182,6 +191,10 @@ Notation "X ≢ Y":= (¬X ≡ Y) (at level 70, no associativity) : stdpp_scope.
Notation "( X ≢)" := (λ Y, X ≢ Y) (only parsing) : stdpp_scope.
Notation "(≢ X )" := (λ Y, Y ≢ X) (only parsing) : stdpp_scope.
Notation "(≡@{ A } )" := (@equiv A _) (only parsing) : stdpp_scope.
Notation "(≢@{ A } )" := (λ X Y, ¬X ≡@{A} Y) (only parsing) : stdpp_scope.
Notation "X ≢@{ A } Y":= (¬X ≡@{ A } Y) (at level 70, no associativity) : stdpp_scope.
(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
......@@ -189,22 +202,22 @@ reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x ≡ y → x = y.
Hint Mode LeibnizEquiv ! - : typeclass_instances.
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (≡@{A})} (x y : A) :
x ≡ y ↔ x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
Ltac fold_leibniz := repeat
match goal with
| H : context [ @equiv ?A _ _ _ ] |- _ =>
| H : context [ _ ≡@{?A} _ ] |- _ =>
setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
| |- context [ @equiv ?A _ _ _ ] =>
| |- context [ _ ≡@{?A} _ ] =>
setoid_rewrite (leibniz_equiv_iff (A:=A))
end.
Ltac unfold_leibniz := repeat
match goal with
| H : context [ @eq ?A _ _ ] |- _ =>
| H : context [ _ =@{?A} _ ] |- _ =>
setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
| |- context [ @eq ?A _ _ ] =>
| |- context [ _ =@{?A} _ ] =>
setoid_rewrite <-(leibniz_equiv_iff (A:=A))
end.
......@@ -249,7 +262,7 @@ Class RelDecision {A B} (R : A → B → Prop) :=
decide_rel x y :> Decision (R x y).
Hint Mode RelDecision ! ! ! : typeclass_instances.
Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert.
Notation EqDecision A := (RelDecision (@eq A)).
Notation EqDecision A := (RelDecision (=@{A})).
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
......@@ -411,9 +424,9 @@ Lemma exist_proper {A} (P Q : A → Prop) :
(∀ x, P x ↔ Q x) → (∃ x, P x) ↔ (∃ x, Q x).
Proof. firstorder. Qed.
Instance: Comm (↔) (@eq A).
Instance: Comm (↔) (=@{A}).
Proof. red; intuition. Qed.
Instance: Comm (↔) (λ x y, @eq A y x).
Instance: Comm (↔) (λ x y, y =@{A} x).
Proof. red; intuition. Qed.
Instance: Comm (↔) (↔).
Proof. red; intuition. Qed.
......@@ -551,7 +564,7 @@ Proof. now intros -> ?. Qed.
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Instance unit_equivalence : Equivalence (≡@{unit}).
Proof. repeat split. Qed.
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
......
theories/coPset.v
View file @ 6aba4a3f
......@@ -192,7 +192,7 @@ Qed.
Instance coPset_elem_of_dec : RelDecision (@elem_of _ coPset _).
Proof. solve_decision. Defined.
Instance coPset_equiv_dec : RelDecision (@equiv coPset _).
Instance coPset_equiv_dec : RelDecision (≡@{coPset}).
Proof. refine (λ X Y, cast_if (decide (X = Y))); abstract (by fold_leibniz). Defined.
Instance mapset_disjoint_dec : RelDecision (@disjoint coPset _).
Proof.
......
theories/collections.v
View file @ 6aba4a3f
......@@ -19,14 +19,14 @@ Typeclasses Opaque collection_equiv collection_subseteq collection_disjoint.
Section setoids_simple.
Context `{SimpleCollection A C}.
Global Instance collection_equivalence: @Equivalence C (≡).
Global Instance collection_equivalence : Equivalence (≡@{C}).
Proof.
split.
- done.
- intros X Y ? x. by symmetry.
- intros X Y Z ?? x; by trans (x ∈ Y).
Qed.
Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)).
Global Instance singleton_proper : Proper ((=) ==> (≡@{C})) singleton.
Proof. apply _. Qed.
Global Instance elem_of_proper :
Proper ((=) ==> (≡) ==> iff) (@elem_of A C _) | 5.
......@@ -35,11 +35,11 @@ Section setoids_simple.
Proof.
intros X1 X2 HX Y1 Y2 HY; apply forall_proper; intros x. by rewrite HX, HY.
Qed.
Global Instance union_proper : Proper ((≡) ==> (≡) ==> (≡)) (@union C _).
Global Instance union_proper : Proper ((≡) ==> (≡) ==> (≡@{C})) union.
Proof. intros X1 X2 HX Y1 Y2 HY x. rewrite !elem_of_union. f_equiv; auto. Qed.
Global Instance union_list_proper: Proper ((≡) ==> (≡)) (union_list (A:=C)).
Global Instance union_list_proper: Proper ((≡) ==> (≡@{C})) union_list.
Proof. by induction 1; simpl; try apply union_proper. Qed.
Global Instance subseteq_proper : Proper ((≡) ==> (≡) ==> iff) ((⊆) : relation C).
Global Instance subseteq_proper : Proper ((≡@{C}) ==> (≡@{C}) ==> iff) (⊆).
Proof.
intros X1 X2 HX Y1 Y2 HY. apply forall_proper; intros x. by rewrite HX, HY.
Qed.
......@@ -50,12 +50,12 @@ Section setoids.
(** * Setoids *)
Global Instance intersection_proper :
Proper ((≡) ==> (≡) ==> (≡)) (@intersection C _).
Proper ((≡) ==> (≡) ==> (≡@{C})) intersection.
Proof.
intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_intersection, HX, HY.
Qed.
Global Instance difference_proper :
Proper ((≡) ==> (≡) ==> (≡)) (@difference C _).
Proper ((≡) ==> (≡) ==> (≡@{C})) difference.
Proof.
intros X1 X2 HX Y1 Y2 HY x. by rewrite !elem_of_difference, HX, HY.
Qed.
......@@ -357,15 +357,15 @@ Section simple_collection.
Lemma union_mono X1 X2 Y1 Y2 : X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof. set_solver. Qed.
Global Instance union_idemp : IdemP ((≡) : relation C) (∪).
Global Instance union_idemp : IdemP (≡@{C}) (∪).
Proof. intros X. set_solver. Qed.
Global Instance union_empty_l : LeftId ((≡) : relation C) ∅ (∪).
Global Instance union_empty_l : LeftId (≡@{C}) ∅ (∪).
Proof. intros X. set_solver. Qed.
Global Instance union_empty_r : RightId ((≡) : relation C) ∅ (∪).
Global Instance union_empty_r : RightId (≡@{C}) ∅ (∪).
Proof. intros X. set_solver. Qed.
Global Instance union_comm : Comm ((≡) : relation C) (∪).
Global Instance union_comm : Comm (≡@{C}) (∪).
Proof. intros X Y. set_solver. Qed.
Global Instance union_assoc : Assoc ((≡) : relation C) (∪).
Global Instance union_assoc : Assoc (≡@{C}) (∪).
Proof. intros X Y Z. set_solver. Qed.
Lemma empty_union X Y : X ∪ Y ≡ ∅ ↔ X ≡ ∅ ∧ Y ≡ ∅.
......@@ -480,15 +480,15 @@ Section simple_collection.
Proof. unfold_leibniz. apply subseteq_union_2. Qed.
(** Union *)
Global Instance union_idemp_L : IdemP (@eq C) (∪).
Global Instance union_idemp_L : IdemP (=@{C}) (∪).
Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
Global Instance union_empty_l_L : LeftId (@eq C) ∅ (∪).
Global Instance union_empty_l_L : LeftId (=@{C}) ∅ (∪).
Proof. intros ?. unfold_leibniz. apply (left_id _ _). Qed.
Global Instance union_empty_r_L : RightId (@eq C) ∅ (∪).
Global Instance union_empty_r_L : RightId (=@{C}) ∅ (∪).
Proof. intros ?. unfold_leibniz. apply (right_id _ _). Qed.
Global Instance union_comm_L : Comm (@eq C) (∪).
Global Instance union_comm_L : Comm (=@{C}) (∪).
Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
Global Instance union_assoc_L : Assoc (@eq C) (∪).
Global Instance union_assoc_L : Assoc (=@{C}) (∪).
Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.
Lemma empty_union_L X Y : X ∪ Y = ∅ ↔ X = ∅ ∧ Y = ∅.
......@@ -527,7 +527,7 @@ Section simple_collection.
End leibniz.
Section dec.
Context `{!RelDecision (@equiv C _)}.
Context `{!RelDecision (≡@{C})}.
Lemma collection_subseteq_inv X Y : X ⊆ Y → X ⊂ Y ∨ X ≡ Y.
Proof. destruct (decide (X ≡ Y)); [by right|left;set_solver]. Qed.
Lemma collection_not_subset_inv X Y : X ⊄ Y → X ⊈ Y ∨ X ≡ Y.
......@@ -580,15 +580,15 @@ Section collection.
X1 ⊆ X2 → Y1 ⊆ Y2 → X1 ∩ Y1 ⊆ X2 ∩ Y2.
Proof. set_solver. Qed.
Global Instance intersection_idemp : IdemP ((≡) : relation C) (∩).
Global Instance intersection_idemp : IdemP (≡@{C}) (∩).
Proof. intros X; set_solver. Qed.
Global Instance intersection_comm : Comm ((≡) : relation C) (∩).
Global Instance intersection_comm : Comm (≡@{C}) (∩).
Proof. intros X Y; set_solver. Qed.
Global Instance intersection_assoc : Assoc ((≡) : relation C) (∩).
Global Instance intersection_assoc : Assoc (≡@{C}) (∩).
Proof. intros X Y Z; set_solver. Qed.
Global Instance intersection_empty_l : LeftAbsorb ((≡) : relation C) ∅ (∩).
Global Instance intersection_empty_l : LeftAbsorb (≡@{C}) ∅ (∩).
Proof. intros X; set_solver. Qed.
Global Instance intersection_empty_r: RightAbsorb ((≡) : relation C) ∅ (∩).
Global Instance intersection_empty_r: RightAbsorb (≡@{C}) ∅ (∩).
Proof. intros X; set_solver. Qed.
Lemma intersection_singletons x : ({[x]} : C) ∩ {[x]} ≡ {[x]}.
......@@ -647,15 +647,15 @@ Section collection.
Lemma subseteq_intersection_2_L X Y : X ∩ Y = X → X ⊆ Y.
Proof. unfold_leibniz. apply subseteq_intersection_2. Qed.
Global Instance intersection_idemp_L : IdemP ((=) : relation C) (∩).
Global Instance intersection_idemp_L : IdemP (=@{C}) (∩).
Proof. intros ?. unfold_leibniz. apply (idemp _). Qed.
Global Instance intersection_comm_L : Comm ((=) : relation C) (∩).
Global Instance intersection_comm_L : Comm (=@{C}) (∩).
Proof. intros ??. unfold_leibniz. apply (comm _). Qed.
Global Instance intersection_assoc_L : Assoc ((=) : relation C) (∩).
Global Instance intersection_assoc_L : Assoc (=@{C}) (∩).
Proof. intros ???. unfold_leibniz. apply (assoc _). Qed.
Global Instance intersection_empty_l_L: LeftAbsorb ((=) : relation C) ∅ (∩).
Global Instance intersection_empty_l_L: LeftAbsorb (=@{C}) ∅ (∩).
Proof. intros ?. unfold_leibniz. apply (left_absorb _ _). Qed.
Global Instance intersection_empty_r_L: RightAbsorb ((=) : relation C) ∅ (∩).
Global Instance intersection_empty_r_L: RightAbsorb (=@{C}) ∅ (∩).
Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed.
Lemma intersection_singletons_L x : {[x]} ∩ {[x]} = ({[x]} : C).
......@@ -776,17 +776,17 @@ Section of_option_list.
SetUnfold (x ∈ l) P → SetUnfold (x ∈ of_list (C:=C) l) P.
Proof. constructor. by rewrite elem_of_of_list, (set_unfold (x ∈ l) P). Qed.
Lemma of_list_nil : of_list (C:=C) [] = ∅.
Lemma of_list_nil : of_list [] =@{C} ∅.
Proof. done. Qed.
Lemma of_list_cons x l : of_list (C:=C) (x :: l) = {[ x ]} ∪ of_list l.
Lemma of_list_cons x l : of_list (x :: l) =@{C} {[ x ]} ∪ of_list l.
Proof. done. Qed.
Lemma of_list_app l1 l2 : of_list (C:=C) (l1 ++ l2) ≡ of_list l1 ∪ of_list l2.
Lemma of_list_app l1 l2 : of_list (l1 ++ l2) ≡@{C} of_list l1 ∪ of_list l2.
Proof. set_solver. Qed.
Global Instance of_list_perm : Proper ((≡ₚ) ==> (≡)) (of_list (C:=C)).
Proof. induction 1; set_solver. Qed.
Context `{!LeibnizEquiv C}.
Lemma of_list_app_L l1 l2 : of_list (C:=C) (l1 ++ l2) = of_list l1 ∪ of_list l2.
Lemma of_list_app_L l1 l2 : of_list (l1 ++ l2) =@{C} of_list l1 ∪ of_list l2.
Proof. set_solver. Qed.
Global Instance of_list_perm_L : Proper ((≡ₚ) ==> (=)) (of_list (C:=C)).
Proof. induction 1; set_solver. Qed.
......@@ -887,10 +887,9 @@ Section fresh.
Context `{FreshSpec A C}.
Implicit Types X Y : C.
Global Instance fresh_proper: Proper ((≡) ==> (=)) (fresh (C:=C)).
Global Instance fresh_proper: Proper ((≡@{C}) ==> (=)) fresh.
Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed.
Global Instance fresh_list_proper n:
Proper ((≡) ==> (=)) (fresh_list (C:=C) n).
Global Instance fresh_list_proper n : Proper ((≡@{C}) ==> (=)) (fresh_list n).
Proof. induction n as [|n IH]; intros ?? E; by setoid_subst. Qed.
Lemma exist_fresh X : ∃ x, x ∉ X.
......@@ -1058,13 +1057,13 @@ Section seq_set.
Proof. intros x. rewrite elem_of_singleton, elem_of_seq_set. omega. Qed.
Lemma seq_set_S_union start len :
seq_set start (C:=C) (S len) ≡ {[ start + len ]} ∪ seq_set start len.
seq_set start (S len) ≡@{C} {[ start + len ]} ∪ seq_set start len.
Proof.
intros x. rewrite elem_of_union, elem_of_singleton, !elem_of_seq_set. omega.
Qed.
Lemma seq_set_S_union_L `{!LeibnizEquiv C} start len :
seq_set start (C:=C) (S len) = {[ start + len ]} ∪ seq_set start len.
seq_set start (S len) =@{C} {[ start + len ]} ∪ seq_set start len.
Proof. unfold_leibniz. apply seq_set_S_union. Qed.
End seq_set.
......@@ -1078,7 +1077,7 @@ Section minimal.
Context `{SimpleCollection A C} {R : relation A}.
Implicit Types X Y : C.
Global Instance minimal_proper x : Proper (@equiv C _ ==> iff) (minimal R x).
Global Instance minimal_proper x : Proper ((≡@{C}) ==> iff) (minimal R x).
Proof. intros X X' y; unfold minimal; set_solver. Qed.
Lemma minimal_anti_symm_1 `{!AntiSymm (=) R} X x y :
......
theories/fin_maps.v
View file @ 6aba4a3f
......@@ -147,38 +147,34 @@ Section setoid.
m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y.
Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed.
Global Instance map_equivalence :
Equivalence ((≡) : relation A) → Equivalence ((≡) : relation (M A)).
Global Instance map_equivalence : Equivalence (≡@{A}) → Equivalence (≡@{M A}).
Proof.
split.
- by intros m i.
- by intros m1 m2 ? i.
- by intros m1 m2 m3 ?? i; trans (m2 !! i).
Qed.
Global Instance lookup_proper (i : K) :
Proper ((≡) ==> (≡)) (lookup (M:=M A) i).
Global Instance lookup_proper (i : K) : Proper ((≡@{M A}) ==> (≡)) (lookup i).
Proof. by intros m1 m2 Hm. Qed.
Global Instance partial_alter_proper :
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (partial_alter (M:=M A)).
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡@{M A})) partial_alter.
Proof.
by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
try apply Hf; apply lookup_proper.
Qed.
Global Instance insert_proper (i : K) :
Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=M A) i).
Proper ((≡) ==> (≡) ==> (≡@{M A})) (insert i).
Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
Global Instance singleton_proper k :
Proper ((≡) ==> (≡)) (singletonM k : A → M A).
Global Instance singleton_proper k : Proper ((≡) ==> (≡@{M A})) (singletonM k).
Proof.
intros ???; apply insert_proper; [done|].
intros ?. rewrite lookup_empty; constructor.
Qed.
Global Instance delete_proper (i : K) :
Proper ((≡) ==> (≡)) (delete (M:=M A) i).
Global Instance delete_proper (i : K) : Proper ((≡) ==> (≡@{M A})) (delete i).
Proof. by apply partial_alter_proper; [constructor|]. Qed.
Global Instance alter_proper :
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (alter (A:=A) (M:=M A)).
Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡@{M A})) alter.
Proof.
intros ?? Hf; apply partial_alter_proper.
by destruct 1; constructor; apply Hf.
......@@ -186,12 +182,12 @@ Section setoid.
Lemma merge_ext `{Equiv B, Equiv C} (f g : option A → option B → option C)
`{!DiagNone f, !DiagNone g} :
((≡) ==> (≡) ==> (≡))%signature f g →
((≡) ==> (≡) ==> (≡))%signature (merge (M:=M) f) (merge g).
((≡) ==> (≡) ==> (≡@{M _}))%signature (merge f) (merge g).
Proof.
by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
Qed.
Global Instance union_with_proper :
Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==>(≡)) (union_with (M:=M A)).
Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==>(≡@{M A})) union_with.
Proof.
intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
by do 2 destruct 1; first [apply Hf | constructor].
......@@ -205,7 +201,7 @@ Section setoid.
- intros ?. rewrite lookup_empty; constructor.
Qed.
Global Instance map_fmap_proper `{Equiv B} (f : A → B) :
Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f).
Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡@{M _})) (fmap f).
Proof.
intros ? m m' ? k; rewrite !lookup_fmap. by apply option_fmap_proper.
Qed.
......@@ -921,13 +917,13 @@ Section map_of_to_collection.
Proof. unfold map_to_collection; simpl. by rewrite map_to_list_empty. Qed.
Lemma map_to_collection_insert (f : K → A → B)(m : M A) i x :
m !! i = None →
map_to_collection (C:=C) f (<[i:=x]>m) ≡ {[f i x]} ∪ map_to_collection f m.
map_to_collection f (<[i:=x]>m) ≡@{C} {[f i x]} ∪ map_to_collection f m.
Proof.
intros. unfold map_to_collection; simpl. by rewrite map_to_list_insert.
Qed.
Lemma map_to_collection_insert_L `{!LeibnizEquiv C} (f : K → A → B) (m : M A) i x :
m !! i = None →
map_to_collection (C:=C) f (<[i:=x]>m) = {[f i x]} ∪ map_to_collection f m.
map_to_collection f (<[i:=x]>m) =@{C} {[f i x]} ∪ map_to_collection f m.
Proof. unfold_leibniz. apply map_to_collection_insert. Qed.
End map_of_to_collection.
......@@ -1166,19 +1162,19 @@ Lemma merge_comm m1 m2 :
(∀ i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) →
merge f m1 m2 = merge f m2 m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance merge_comm' : Comm (=) f → Comm (=) (merge (M:=M) f).
Global Instance merge_comm' : Comm (=) f → Comm (=@{M _}) (merge f).
Proof. intros ???. apply merge_comm. intros. by apply (comm f). Qed.
Lemma merge_assoc m1 m2 m3 :
(∀ i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
f (f (m1 !! i) (m2 !! i)) (m3 !! i)) →
merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance merge_assoc' : Assoc (=) f → Assoc (=) (merge (M:=M) f).
Global Instance merge_assoc' : Assoc (=) f → Assoc (=@{M _}) (merge f).
Proof. intros ????. apply merge_assoc. intros. by apply (assoc_L f). Qed.
Lemma merge_idemp m1 :
(∀ i, f (m1 !! i) (m1 !! i) = m1 !! i) → merge f m1 m1 = m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance merge_idemp' : IdemP (=) f → IdemP (=) (merge (M:=M) f).
Global Instance merge_idemp' : IdemP (=) f → IdemP (=@{M _}) (merge f).
Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed.
End merge.
......@@ -1433,9 +1429,9 @@ Proof.
rewrite lookup_union_with.
destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Global Instance: LeftId (@eq (M A)) ∅ (union_with f).
Global Instance: LeftId (=@{M A}) ∅ (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Global Instance: RightId (@eq (M A)) ∅ (union_with f).
Global Instance: RightId (=@{M A}) ∅ (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Lemma union_with_comm m1 m2 :
(∀ i x y, m1 !! i = Some x → m2 !! i = Some y → f x y = f y x) →
......@@ -1444,7 +1440,7 @@ Proof.
intros. apply (merge_comm _). intros i.
destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Global Instance: Comm (=) f → Comm (@eq (M A)) (union_with f).
Global Instance: Comm (=) f → Comm (=@{M A}) (union_with f).
Proof. intros ???. apply union_with_comm. eauto. Qed.
Lemma union_with_idemp m :
(∀ i x, m !! i = Some x → f x x = Some x) → union_with f m m = m.
......@@ -1502,15 +1498,15 @@ Qed.
End union_with.
(** ** Properties of the [union] operation *)
Global Instance: LeftId (@eq (M A)) ∅ (∪) := _.
Global Instance: RightId (@eq (M A)) ∅ (∪) := _.
Global Instance: Assoc (@eq (M A)) (∪).
Global Instance: LeftId (=@{M A}) ∅ (∪) := _.
Global Instance: RightId (=@{M A}) ∅ (∪) := _.
Global Instance: Assoc (=@{M A}) (∪).
Proof.
intros A m1 m2 m3. unfold union, map_union, union_with, map_union_with.
apply (merge_assoc _). intros i.
by destruct (m1 !! i), (m2 !! i), (m3 !! i).
Qed.
Global Instance: IdemP (@eq (M A)) (∪).
Global Instance: IdemP (=@{M A}) (∪).
Proof. intros A ?. by apply union_with_idemp. Qed.
Lemma lookup_union_Some_raw {A} (m1 m2 : M A) i x :
(m1 ∪ m2) !! i = Some x ↔
......@@ -1766,9 +1762,9 @@ Proof. intro. by rewrite map_disjoint_of_list_zip_r. Qed.
Section intersection_with.
Context {A} (f : A → A → option A).
Implicit Type (m: M A).
Global Instance : LeftAbsorb (@eq (M A)) ∅ (intersection_with f).
Global Instance : LeftAbsorb (=@{M A}) ∅ (intersection_with f).
Proof. unfold intersection_with, map_intersection_with. apply _. Qed.
Global Instance: RightAbsorb (@eq (M A)) ∅ (intersection_with f).
Global Instance: RightAbsorb (=@{M A}) ∅ (intersection_with f).
Proof. unfold intersection_with, map_intersection_with. apply _. Qed.
Lemma lookup_intersection_with m1 m2 i :
intersection_with f m1 m2 !! i = intersection_with f (m1 !! i) (m2 !! i).
......@@ -1787,7 +1783,7 @@ Proof.
intros. apply (merge_comm _). intros i.
destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Global Instance: Comm (=) f → Comm (@eq (M A)) (intersection_with f).
Global Instance: Comm (=) f → Comm (=@{M A}) (intersection_with f).
Proof. intros ???. apply intersection_with_comm. eauto. Qed.
Lemma intersection_with_idemp m :
(∀ i x, m !! i = Some x → f x x = Some x) → intersection_with f m m = m.
......@@ -1817,16 +1813,16 @@ Proof. by intros; apply (partial_alter_merge _). Qed.
End intersection_with.
(** ** Properties of the [intersection] operation *)
Global Instance: LeftAbsorb (@eq (M A)) ∅ (∩) := _.
Global Instance: RightAbsorb (@eq (M A)) ∅ (∩) := _.
Global Instance: Assoc (@eq (M A)) (∩).
Global Instance: LeftAbsorb (=@{M A}) ∅ (∩) := _.
Global Instance: RightAbsorb (=@{M A}) ∅ (∩) := _.
Global Instance: Assoc (=@{M A}) (∩).
Proof.
intros A m1 m2 m3.
unfold intersection, map_intersection, intersection_with, map_intersection_with.
apply (merge_assoc _). intros i.
by destruct (m1 !! i), (m2 !! i), (m3 !! i).
Qed.
Global Instance: IdemP (@eq (M A)) (∩).
Global Instance: IdemP (=@{M A}) (∩).
Proof. intros A ?. by apply intersection_with_idemp. Qed.
Lemma lookup_intersection_Some {A} (m1 m2 : M A) i x :
......
theories/gmultiset.v
View file @ 6aba4a3f
......@@ -102,20 +102,20 @@ Global Instance gmultiset_elem_of_dec : RelDecision (@elem_of _ (gmultiset A) _)
Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined.
(* Algebraic laws *)
Global Instance gmultiset_comm : Comm (@eq (gmultiset A)) (∪).
Global Instance gmultiset_comm : Comm (=@{gmultiset A}) (∪).
Proof.
intros X Y. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
Qed.
Global Instance gmultiset_assoc : Assoc (@eq (gmultiset A)) (∪).
Global Instance gmultiset_assoc : Assoc (=@{gmultiset A}) (∪).
Proof.
intros X Y Z. apply gmultiset_eq; intros x. rewrite !multiplicity_union; omega.
Qed.
Global Instance gmultiset_left_id : LeftId (@eq (gmultiset A)) ∅ (∪).
Global Instance gmultiset_left_id : LeftId (=@{gmultiset A}) ∅ (∪).
Proof.
intros X. apply gmultiset_eq; intros x.
by rewrite multiplicity_union, multiplicity_empty.