(* 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.