(* Copyright (c) 2012-2013, 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 [from_option] function allows us to get the value out of the option type by specifying a default value. *) Definition from_option {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_proj `{x : option A} : is_Some x → A := match x with | Some a => λ _, a | None => False_rect _ ∘ is_Some_None end. Definition Some_dec `(x : option A) : { a | x = Some a } + { x = None } := match x return { a | x = Some a } + { x = None } with | Some a => inleft (a ↾ eq_refl _) | None => inright 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. Instance None_dec `(x : option A) : Decision (x = None) := match x with | Some x => right (Some_ne_None x) | None => left eq_refl 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. split. apply dec_stable. tauto. 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, y with | Some a, Some b => match dec a b with | left H => left (f_equal _ H) | right H => right (H ∘ injective Some _ _) end | Some _, None => right (Some_ne_None _) | None, Some _ => right (None_ne_Some _) | None, None => left eq_refl 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 fmap_is_Some {A B} (f : A → B) (x : option A) : is_Some (f <\$> x) ↔ is_Some x. Proof. split; inversion 1. by destruct x. done. Qed. Lemma fmap_Some {A B} (f : A → B) (x : option A) y : f <\$> x = Some y ↔ ∃ x', x = Some x' ∧ y = f x'. Proof. unfold fmap, option_fmap. destruct x; naive_solver. Qed. Lemma fmap_None {A B} (f : A → B) (x : option A) : f <\$> x = None ↔ x = None. Proof. unfold fmap, option_fmap. by destruct x. Qed. Lemma option_fmap_id {A} (x : option A) : id <\$> x = x. Proof. by destruct x. Qed. Lemma option_bind_assoc {A B C} (f : A → option B) (g : B → option C) (x : option A) : (x ≫= f) ≫= g = x ≫= (mbind g ∘ f). Proof. by destruct x; simpl. Qed. Tactic Notation "simplify_option_equality" "by" tactic3(tac) := repeat match goal with | _ => first [progress simpl in * | progress simplify_equality] | H : context [mbind (M:=option) (A:=?A) ?f ?o] |- _ => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx in H; clear Hx | H : context [fmap (M:=option) (A:=?A) ?f ?o] |- _ => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx in H; clear Hx | H : context [ match ?o with _ => _ end ] |- _ => match type of o with | option ?A => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx in H; clear Hx end | 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:? | H : ?x = mbind (M:=option) ?f ?o |- _ => match o with Some _ => fail 1 | None => fail 1 | _ => idtac end; match x with Some _ => idtac | None => idtac | _ => fail 1 end; destruct o eqn:? | H : fmap (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:? | H : ?x = fmap (M:=option) ?f ?o |- _ => match o with Some _ => fail 1 | None => fail 1 | _ => idtac end; match x with Some _ => idtac | None => idtac | _ => fail 1 end; destruct o eqn:? | |- context [mbind (M:=option) (A:=?A) ?f ?o] => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx; clear Hx | |- context [fmap (M:=option) (A:=?A) ?f ?o] => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx; clear Hx | |- context [ match ?o with _ => _ end ] => match type of o with | option ?A => let Hx := fresh in first [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x; assert (o = Some x') as Hx by tac | assert (o = None) as Hx by tac ]; rewrite Hx; clear Hx end | H : context C [@mguard option _ ?P ?dec _ ?x] |- _ => let X := context C [ if dec then x else None ] in change X in H; destruct_decide dec | |- context C [@mguard option _ ?P ?dec _ ?x] => let X := context C [ if dec then x else None ] in change X; destruct_decide dec | H1 : ?o = Some ?x, H2 : ?o = Some ?y |- _ => assert (y = x) by congruence; clear H2 | H1 : ?o = Some ?x, H2 : ?o = None |- _ => congruence end. Tactic Notation "simplify_option_equality" := simplify_option_equality by eauto. Hint Extern 100 => simplify_option_equality : simplify_option_equality. (** * 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.