Separation type class for singleton maps.

modures/fin_maps.v

@@ -65,7 +65,7 @@ Proof.
   by apply (timeless _); rewrite <-Hm, lookup_insert_ne by done.
 Qed.
 Global Instance map_ra_singleton_timeless `{Cofe A} (i : K) (x : A) :
-  Timeless x → Timeless ({[ i, x ]} : gmap K A) := _.
+  Timeless x → Timeless ({[ i ↦ x ]} : gmap K A) := _.
 Instance map_fmap_ne `{Dist A, Dist B} (f : A → B) n :
   Proper (dist n ==> dist n) f →
   Proper (dist n ==> dist n) (fmap (M:=gmap K) f).
@@ -172,7 +172,7 @@ Proof.
   by specialize (Hm j); simplify_map_equality.
 Qed.
 Global Instance map_ra_singleton_valid_timeless `{Valid A, ValidN A} (i : K) x :
-  ValidTimeless x → ValidTimeless ({[ i, x ]} : gmap K A).
+  ValidTimeless x → ValidTimeless ({[ i ↦ x ]} : gmap K A).
 Proof.
   intros ?; apply (map_ra_insert_valid_timeless _ _ _ _ _); simpl.
   by rewrite lookup_empty.

modures/ra.v

@@ -146,10 +146,9 @@ Proof. unfold big_opM. by rewrite map_to_list_empty. Qed.
 Lemma big_opM_insert (m : M A) i x :
   m !! i = None → big_opM (<[i:=x]> m) ≡ x ⋅ big_opM m.
 Proof. intros ?; unfold big_opM. by rewrite map_to_list_insert by done. Qed.
-Lemma big_opM_singleton i x : big_opM ({[i,x]} : M A) ≡ x.
+Lemma big_opM_singleton i x : big_opM ({[i ↦ x]} : M A) ≡ x.
 Proof.
-  unfold singleton, map_singleton.
-  rewrite big_opM_insert by auto using lookup_empty; simpl.
+  rewrite <-insert_empty, big_opM_insert by auto using lookup_empty; simpl.
   by rewrite big_opM_empty, (right_id _ _).
 Qed.
 Global Instance big_opM_proper : Proper ((≡) ==> (≡)) (big_opM : M A → _).

prelude/base.v

@@ -468,6 +468,11 @@ Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
 Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
 Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
 
+(** The singleton map *)
+Class SingletonM K A M := singletonM: K → A → M.
+Instance: Params (@singletonM) 5.
+Notation "{[ x ↦ y ]}" := (singletonM x y) (at level 1) : C_scope.
+
 (** The function insert [<[k:=a]>m] should update the element at key [k] with
 value [a] in [m]. *)
 Class Insert (K A M : Type) := insert: K → A → M → M.

prelude/fin_map_dom.v

@@ -61,11 +61,8 @@ Proof. rewrite (dom_insert _). solve_elem_of. Qed.
 Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X :
   X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m).
 Proof. intros. transitivity (dom D m); eauto using dom_insert_subseteq. Qed.
-Lemma dom_singleton {A} (i : K) (x : A) : dom D {[(i, x)]} ≡ {[ i ]}.
-Proof.
-  unfold singleton at 1, map_singleton.
-  rewrite dom_insert, dom_empty. solve_elem_of.
-Qed.
+Lemma dom_singleton {A} (i : K) (x : A) : dom D {[i ↦ x]} ≡ {[ i ]}.
+Proof. rewrite <-insert_empty, dom_insert, dom_empty; solve_elem_of. Qed.
 Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}.
 Proof.
   apply elem_of_equiv. intros j. rewrite elem_of_difference, !elem_of_dom.
@@ -121,7 +118,7 @@ Lemma dom_alter_L {A} f (m : M A) i : dom D (alter f i m) = dom D m.
 Proof. unfold_leibniz; apply dom_alter. Qed.
 Lemma dom_insert_L {A} (m : M A) i x : dom D (<[i:=x]>m) = {[ i ]} ∪ dom D m.
 Proof. unfold_leibniz; apply dom_insert. Qed.
-Lemma dom_singleton_L {A} (i : K) (x : A) : dom D {[(i, x)]} = {[ i ]}.
+Lemma dom_singleton_L {A} (i : K) (x : A) : dom D {[i ↦ x]} = {[ i ]}.
 Proof. unfold_leibniz; apply dom_singleton. Qed.
 Lemma dom_delete_L {A} (m : M A) i : dom D (delete i m) = dom D m ∖ {[ i ]}.
 Proof. unfold_leibniz; apply dom_delete. Qed.

prelude/hashset.v

@@ -23,7 +23,7 @@ Instance hashset_elem_of: ElemOf A (hashset hash) := λ x m, ∃ l,
 Program Instance hashset_empty: Empty (hashset hash) := Hashset ∅ _.
 Next Obligation. by intros n X; simpl_map. Qed.
 Program Instance hashset_singleton: Singleton A (hashset hash) := λ x,
-  Hashset {[ hash x, [x] ]} _.
+  Hashset {[ hash x ↦ [x] ]} _.
 Next Obligation.
   intros n l. rewrite lookup_singleton_Some. intros [<- <-].
   rewrite Forall_singleton; auto using NoDup_singleton.

prelude/mapset.v

@@ -17,7 +17,7 @@ Instance mapset_elem_of: ElemOf K (mapset M) := λ x X,
   mapset_car X !! x = Some ().
 Instance mapset_empty: Empty (mapset M) := Mapset ∅.
 Instance mapset_singleton: Singleton K (mapset M) := λ x,
-  Mapset {[ (x,()) ]}.
+  Mapset {[ x ↦ () ]}.
 Instance mapset_union: Union (mapset M) := λ X1 X2,
   let (m1) := X1 in let (m2) := X2 in Mapset (m1 ∪ m2).
 Instance mapset_intersection: Intersection (mapset M) := λ X1 X2,