### Add monoid operation on option.

parent 71214d32
 ... ... @@ -138,6 +138,46 @@ Qed. Lemma bind_with_Some {A} (x : option A) : x ≫= Some = x. Proof. by destruct x. Qed. (** * Union, intersection and difference *) Instance option_union_with {A} : UnionWith A (option A) := λ f x y, match x, y with | Some a, Some b => f a b | Some a, None => Some a | None, Some b => Some b | None, None => None end. Instance option_intersection_with {A} : IntersectionWith A (option A) := λ f x y, match x, y with Some a, Some b => f a b | _, _ => None end. Instance option_difference_with {A} : DifferenceWith A (option A) := λ f x y, match x, y with | Some a, Some b => f a b | Some a, None => Some a | None, _ => None end. Instance option_union {A} : Union (option A) := union_with (λ x _, Some x). Lemma option_union_Some {A} (x y : option A) z : x ∪ y = Some z → x = Some z ∨ y = Some z. Proof. destruct x, y; intros; simplify_equality; auto. Qed. Section option_union_intersection_difference. Context {A} (f : A → A → option A). Global Instance: LeftId (=) None (union_with f). Proof. by intros [?|]. Qed. Global Instance: RightId (=) None (union_with f). Proof. by intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (union_with f). Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed. Global Instance: LeftAbsorb (=) None (intersection_with f). Proof. by intros [?|]. Qed. Global Instance: RightAbsorb (=) None (intersection_with f). Proof. by intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (intersection_with f). Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed. Global Instance: RightId (=) None (difference_with f). Proof. by intros [?|]. Qed. End option_union_intersection_difference. (** * Tactics *) Tactic Notation "case_option_guard" "as" ident(Hx) := match goal with | H : context C [@mguard option _ ?P ?dec _ ?x] |- _ => ... ... @@ -199,6 +239,7 @@ Tactic Notation "simplify_option_equality" "by" tactic3(tac) := repeat match goal with | _ => progress simplify_equality' | _ => progress simpl_option_monad by tac | H : _ ∪ _ = Some _ |- _ => apply option_union_Some in H; destruct H | H : mbind (M:=option) ?f ?o = ?x |- _ => match o with Some _ => fail 1 | None => fail 1 | _ => idtac end; match x with Some _ => idtac | None => idtac | _ => fail 1 end; ... ... @@ -223,38 +264,3 @@ Tactic Notation "simplify_option_equality" "by" tactic3(tac) := | _ => progress case_option_guard end. Tactic Notation "simplify_option_equality" := simplify_option_equality by eauto. (** * Union, intersection and difference *) Instance option_union_with {A} : UnionWith A (option A) := λ f x y, match x, y with | Some a, Some b => f a b | Some a, None => Some a | None, Some b => Some b | None, None => None end. Instance option_intersection_with {A} : IntersectionWith A (option A) := λ f x y, match x, y with Some a, Some b => f a b | _, _ => None end. Instance option_difference_with {A} : DifferenceWith A (option A) := λ f x y, match x, y with | Some a, Some b => f a b | Some a, None => Some a | None, _ => None end. Section option_union_intersection_difference. Context {A} (f : A → A → option A). Global Instance: LeftId (=) None (union_with f). Proof. by intros [?|]. Qed. Global Instance: RightId (=) None (union_with f). Proof. by intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (union_with f). Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed. Global Instance: LeftAbsorb (=) None (intersection_with f). Proof. by intros [?|]. Qed. Global Instance: RightAbsorb (=) None (intersection_with f). Proof. by intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (intersection_with f). Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed. Global Instance: RightId (=) None (difference_with f). Proof. by intros [?|]. Qed. End option_union_intersection_difference.
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