(* Copyright (c) 2012, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file collects general purpose definitions and theorems on the option data type that are not in the Coq standard library. *) Require Export base tactics decidable. (** * General definitions and theorems *) (** Basic properties about equality. *) Lemma None_ne_Some `(a : A) : None ≠ Some a. Proof. congruence. Qed. Lemma Some_ne_None `(a : A) : Some a ≠ None. Proof. congruence. Qed. Lemma eq_None_ne_Some `(x : option A) a : x = None → x ≠ Some a. Proof. congruence. Qed. Instance Some_inj {A} : Injective (=) (=) (@Some A). Proof. congruence. Qed. (** The non dependent elimination principle on the option type. *) Definition option_case {A B} (f : A → B) (b : B) (x : option A) : B := match x with | None => b | Some a => f a end. (** The [maybe] function allows us to get the value out of the option type by specifying a default value. *) Definition maybe {A} (a : A) (x : option A) : A := match x with | None => a | Some b => b end. (** An alternative, but equivalent, definition of equality on the option data type. This theorem is useful to prove that two options are the same. *) Lemma option_eq {A} (x y : option A) : x = y ↔ ∀ a, x = Some a ↔ y = Some a. Proof. split. { intros. by subst. } intros E. destruct x, y. + by apply E. + symmetry. by apply E. + by apply E. + done. Qed. Inductive is_Some {A} : option A → Prop := make_is_Some x : is_Some (Some x). Lemma make_is_Some_alt `(x : option A) a : x = Some a → is_Some x. Proof. intros. by subst. Qed. Hint Resolve make_is_Some_alt. Lemma is_Some_None {A} : ¬is_Some (@None A). Proof. by inversion 1. Qed. Hint Resolve is_Some_None. Lemma is_Some_alt `(x : option A) : is_Some x ↔ ∃ y, x = Some y. Proof. split. inversion 1; eauto. intros [??]. by subst. Qed. Ltac inv_is_Some := repeat match goal with | H : is_Some _ |- _ => inversion H; clear H; subst end. Definition is_Some_sigT `(x : option A) : is_Some x → { a | x = Some a } := match x with | None => False_rect _ ∘ is_Some_None | Some a => λ _, a ↾ eq_refl end. Instance is_Some_dec `(x : option A) : Decision (is_Some x) := match x with | Some x => left (make_is_Some x) | None => right is_Some_None end. Lemma eq_None_not_Some `(x : option A) : x = None ↔ ¬is_Some x. Proof. split. by destruct 2. destruct x. by intros []. done. Qed. Lemma not_eq_None_Some `(x : option A) : x ≠ None ↔ is_Some x. Proof. rewrite eq_None_not_Some. intuition auto using dec_stable. Qed. Lemma make_eq_Some {A} (x : option A) a : is_Some x → (∀ b, x = Some b → b = a) → x = Some a. Proof. destruct 1. intros. f_equal. auto. Qed. (** Equality on [option] is decidable. *) Instance option_eq_dec `{dec : ∀ x y : A, Decision (x = y)} (x y : option A) : Decision (x = y) := match x with | Some a => match y with | Some b => match dec a b with | left H => left (f_equal _ H) | right H => right (H ∘ injective Some _ _) end | None => right (Some_ne_None _) end | None => match y with | Some _ => right (None_ne_Some _) | None => left eq_refl end end. (** * Monadic operations *) Instance option_ret: MRet option := @Some. Instance option_bind: MBind option := λ A B f x, match x with | Some a => f a | None => None end. Instance option_join: MJoin option := λ A x, match x with | Some x => x | None => None end. Instance option_fmap: FMap option := @option_map. Instance option_guard: MGuard option := λ P dec A x, if dec then x else None. Lemma option_fmap_is_Some {A B} (f : A → B) (x : option A) : is_Some x ↔ is_Some (f <\$> x). Proof. split; inversion 1. done. by destruct x. Qed. Lemma option_fmap_is_None {A B} (f : A → B) (x : option A) : x = None ↔ f <\$> x = None. Proof. unfold fmap, option_fmap. by destruct x. Qed. Tactic Notation "simplify_option_equality" "by" tactic3(tac) := repeat match goal with | _ => first [progress simpl in * | progress simplify_equality] | H : mbind (M:=option) (A:=?A) ?f ?o = ?x |- _ => let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; let Hx := fresh in assert (o = Some x') as Hx by tac; rewrite Hx in H; clear Hx | 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; destruct o eqn:? | |- mbind (M:=option) (A:=?A) ?f ?o = ?x => let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; let Hx := fresh in assert (o = Some x') as Hx by tac; rewrite Hx; clear Hx | H : context C [@mguard option _ ?P ?dec _ ?x] |- _ => let X := context C [ if dec then x else None ] in change X in H; destruct dec | |- context C [@mguard option _ ?P ?dec _ ?x] => let X := context C [ if dec then x else None ] in change X; destruct dec end. Tactic Notation "simplify_option_equality" := simplify_option_equality by eauto. (** * Union, intersection and difference *) Instance option_union: UnionWith option := λ A f x y, match x, y with | Some a, Some b => Some (f a b) | Some a, None => Some a | None, Some b => Some b | None, None => None end. Instance option_intersection: IntersectionWith option := λ A f x y, match x, y with | Some a, Some b => Some (f a b) | _, _ => None end. Instance option_difference: DifferenceWith option := λ A 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. Context {A} (f : A → A → 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. intros ? [?|] [?|]; compute; try reflexivity. by rewrite (commutative f). Qed. Global Instance: Associative (=) f → Associative (=) (union_with f). Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. by rewrite (associative f). Qed. Global Instance: Idempotent (=) f → Idempotent (=) (union_with f). Proof. intros ? [?|]; compute; try reflexivity. by rewrite (idempotent f). Qed. Global Instance: Commutative (=) f → Commutative (=) (intersection_with f). Proof. intros ? [?|] [?|]; compute; try reflexivity. by rewrite (commutative f). Qed. Global Instance: Associative (=) f → Associative (=) (intersection_with f). Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. by rewrite (associative f). Qed. Global Instance: Idempotent (=) f → Idempotent (=) (intersection_with f). Proof. intros ? [?|]; compute; try reflexivity. by rewrite (idempotent f). Qed. End option_union_intersection. Section option_difference. Context {A} (f : A → A → option A). Global Instance: RightId (=) None (difference_with f). Proof. by intros [?|]. Qed. End option_difference.