(* 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 orders. (** * 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. now subst. } intros E. destruct x, y. + now apply E. + symmetry. now apply E. + now apply E. + easy. Qed. (** We define [is_Some] as a [sig] instead of a [sigT] as extraction of witnesses can be derived (see [is_Some_sigT] below). *) Definition is_Some `(x : option A) : Prop := ∃ a, x = Some a. Hint Extern 10 (is_Some _) => solve [eexists; eauto]. Ltac simplify_is_Some := repeat intro; repeat match goal with | _ => progress simplify_equality | H : is_Some _ |- _ => destruct H as [??] | |- is_Some _ => eauto end. Lemma Some_is_Some `(a : A) : is_Some (Some a). Proof. simplify_is_Some. Qed. Lemma None_not_is_Some {A} : ¬is_Some (@None A). Proof. simplify_is_Some. Qed. Definition is_Some_sigT `(x : option A) : is_Some x → { a | x = Some a } := match x with | None => False_rect _ ∘ ex_ind None_ne_Some | Some a => λ _, a↾eq_refl end. Lemma eq_Some_is_Some `(x : option A) a : x = Some a → is_Some x. Proof. simplify_is_Some. Qed. Lemma eq_None_not_Some `(x : option A) : x = None ↔ ¬is_Some x. Proof. destruct x; simpl; firstorder congruence. Qed. Lemma make_eq_Some {A} (x : option A) a : is_Some x → (∀ b, x = Some b → b = a) → x = Some a. Proof. intros [??] H. subst. 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. Lemma option_fmap_is_Some {A B} (f : A → B) (x : option A) : is_Some x ↔ is_Some (f <\$> x). Proof. destruct x; split; intros [??]; subst; compute; eauto; discriminate. Qed. Lemma option_fmap_is_None {A B} (f : A → B) (x : option A) : x = None ↔ f <\$> x = None. Proof. unfold fmap, option_fmap. destruct x; simpl; split; congruence. Qed. Ltac simplify_option_bind := repeat match goal with | |- context C [mbind (M:=option) ?f None] => let X := (context C [ None ]) in change X | H : context C [mbind (M:=option) ?f None] |- _ => let X := (context C [ None ]) in change X in H | |- context C [mbind (M:=option) ?f (Some ?a)] => let X := (context C [ f a ]) in let X' := eval simpl in X in change X' | H : context C [mbind (M:=option) ?f (Some ?a)] |- _ => let X := context C [ f a ] in let X' := eval simpl in X in change X' in H | _ => progress simplify_equality | H : mbind (M:=option) ?f ?o = ?x |- _ => destruct o eqn:? | H : context [ ?o = _ ] |- mbind (M:=option) ?f ?o = ?x => erewrite H by eauto end. (** * 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. now intros [?|]. Qed. Global Instance: RightId (=) None (union_with f). Proof. now intros [?|]. Qed. Global Instance: Commutative (=) f → Commutative (=) (union_with f). Proof. intros ? [?|] [?|]; compute; try reflexivity. now rewrite (commutative f). Qed. Global Instance: Associative (=) f → Associative (=) (union_with f). Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. now rewrite (associative f). Qed. Global Instance: Idempotent (=) f → Idempotent (=) (union_with f). Proof. intros ? [?|]; compute; try reflexivity. now rewrite (idempotent f). Qed. Global Instance: Commutative (=) f → Commutative (=) (intersection_with f). Proof. intros ? [?|] [?|]; compute; try reflexivity. now rewrite (commutative f). Qed. Global Instance: Associative (=) f → Associative (=) (intersection_with f). Proof. intros ? [?|] [?|] [?|]; compute; try reflexivity. now rewrite (associative f). Qed. Global Instance: Idempotent (=) f → Idempotent (=) (intersection_with f). Proof. intros ? [?|]; compute; try reflexivity. now 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. now intros [?|]. Qed. End option_difference.