(* Copyright (c) 2012-2015, 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 prelude.base prelude.tactics prelude.decidable.
Inductive option_reflect {A} (P : A → Prop) (Q : Prop) : option A → Type :=
| ReflectSome x : P x → option_reflect P Q (Some x)
| ReflectNone : Q → option_reflect P Q None.
(** * General definitions and theorems *)
(** Basic properties about equality. *)
Lemma None_ne_Some {A} (a : A) : None ≠ Some a.
Proof. congruence. Qed.
Lemma Some_ne_None {A} (a : A) : Some a ≠ None.
Proof. congruence. Qed.
Lemma eq_None_ne_Some {A} (x : option A) a : x = None → x ≠ Some a.
Proof. congruence. Qed.
Instance Some_inj {A} : Inj (=) (=) (@Some A).
Proof. congruence. Qed.
(** The non dependent elimination principle on the option type. *)
Definition default {A B} (b : B) (x : option A) (f : A → B) : 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; [by intros; by subst |]. destruct x, y; naive_solver. Qed.
Lemma option_eq_1 {A} (x y : option A) a : x = y → x = Some a → y = Some a.
Proof. congruence. Qed.
Lemma option_eq_1_alt {A} (x y : option A) a : x = y → y = Some a → x = Some a.
Proof. congruence. Qed.
Definition is_Some {A} (x : option A) := ∃ y, x = Some y.
Lemma mk_is_Some {A} (x : option A) y : x = Some y → is_Some x.
Proof. intros; red; subst; eauto. Qed.
Hint Resolve mk_is_Some.
Lemma is_Some_None {A} : ¬is_Some (@None A).
Proof. by destruct 1. Qed.
Hint Resolve is_Some_None.
Instance is_Some_pi {A} (x : option A) : ProofIrrel (is_Some x).
Proof.
set (P (y : option A) := match y with Some _ => True | _ => False end).
set (f x := match x return P x → is_Some x with
Some _ => λ _, ex_intro _ _ eq_refl | None => False_rect _ end).
set (g x (H : is_Some x) :=
match H return P x with ex_intro _ p => eq_rect _ _ I _ (eq_sym p) end).
assert (∀ x H, f x (g x H) = H) as f_g by (by intros ? [??]; subst).
intros p1 p2. rewrite <-(f_g _ p1), <-(f_g _ p2). by destruct x, p1.
Qed.
Instance is_Some_dec {A} (x : option A) : Decision (is_Some x) :=
match x with
| Some x => left (ex_intro _ x eq_refl)
| None => right is_Some_None
end.
Definition is_Some_proj {A} {x : option A} : is_Some x → A :=
match x with Some a => λ _, a | None => False_rect _ ∘ is_Some_None end.
Definition Some_dec {A} (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.
Lemma eq_None_not_Some {A} (x : option A) : x = None ↔ ¬is_Some x.
Proof. destruct x; unfold is_Some; naive_solver. 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.
(** Lifting a relation point-wise to option *)
Inductive option_Forall2 {A B} (P: A → B → Prop) : option A → option B → Prop :=
| Some_Forall2 x y : P x y → option_Forall2 P (Some x) (Some y)
| None_Forall2 : option_Forall2 P None None.
Definition option_relation {A B} (R: A → B → Prop) (P: A → Prop) (Q: B → Prop)
(mx : option A) (my : option B) : Prop :=
match mx, my with
| Some x, Some y => R x y
| Some x, None => P x
| None, Some y => Q y
| None, None => True
end.
(** Setoids *)
Section setoids.
Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
Global Instance option_equiv : Equiv (option A) := option_Forall2 (≡).
Global Instance option_equivalence : Equivalence ((≡) : relation (option A)).
Proof.
split.
* by intros []; constructor.
* by destruct 1; constructor.
* destruct 1; inversion 1; constructor; etransitivity; eauto.
Qed.
Global Instance Some_proper : Proper ((≡) ==> (≡)) (@Some A).
Proof. by constructor. Qed.
Global Instance option_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (option A).
Proof. intros x y; destruct 1; fold_leibniz; congruence. Qed.
Lemma equiv_None (mx : option A) : mx ≡ None ↔ mx = None.
Proof. split; [by inversion_clear 1|by intros ->]. Qed.
Lemma equiv_Some (mx my : option A) x :
mx ≡ my → mx = Some x → ∃ y, my = Some y ∧ x ≡ y.
Proof. destruct 1; naive_solver. Qed.
Global Instance is_Some_proper : Proper ((≡) ==> iff) (@is_Some A).
Proof. inversion_clear 1; split; eauto. Qed.
End setoids.
(** Equality on [option] is decidable. *)
Instance option_eq_None_dec {A} (x : option A) : Decision (x = None) :=
match x with Some _ => right (Some_ne_None _) | None => left eq_refl end.
Instance option_None_eq_dec {A} (x : option A) : Decision (None = x) :=
match x with Some _ => right (None_ne_Some _) | None => left eq_refl end.
Instance option_eq_dec `{dec : ∀ x y : A, Decision (x = y)}
(x y : option A) : Decision (x = y).
Proof.
refine
match x, y with
| Some a, Some b => cast_if (decide (a = b))
| None, None => left _ | _, _ => right _
end; clear dec; abstract congruence.
Defined.
(** * 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,
match dec with left H => x H | _ => None end.
Lemma fmap_is_Some {A B} (f : A → B) x : is_Some (f <$> x) ↔ is_Some x.
Proof. unfold is_Some; destruct x; naive_solver. Qed.
Lemma fmap_Some {A B} (f : A → B) x y :
f <$> x = Some y ↔ ∃ x', x = Some x' ∧ y = f x'.
Proof. destruct x; naive_solver. Qed.
Lemma fmap_None {A B} (f : A → B) x : f <$> x = None ↔ x = None.
Proof. by destruct x. Qed.
Lemma option_fmap_id {A} (x : option A) : id <$> x = x.
Proof. by destruct x. Qed.
Lemma option_fmap_compose {A B} (f : A → B) {C} (g : B → C) x :
g ∘ f <$> x = g <$> f <$> x.
Proof. by destruct x. Qed.
Lemma option_fmap_ext {A B} (f g : A → B) x :
(∀ y, f y = g y) → f <$> x = g <$> x.
Proof. destruct x; simpl; auto with f_equal. Qed.
Lemma option_fmap_setoid_ext `{Equiv A, Equiv B} (f g : A → B) x :
(∀ y, f y ≡ g y) → f <$> x ≡ g <$> x.
Proof. destruct x; constructor; auto. Qed.
Lemma option_fmap_bind {A B C} (f : A → B) (g : B → option C) x :
(f <$> x) ≫= g = x ≫= g ∘ f.
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.
Lemma option_bind_ext {A B} (f g : A → option B) x y :
(∀ a, f a = g a) → x = y → x ≫= f = y ≫= g.
Proof. intros. destruct x, y; simplify_equality; csimpl; auto. Qed.
Lemma option_bind_ext_fun {A B} (f g : A → option B) x :
(∀ a, f a = g a) → x ≫= f = x ≫= g.
Proof. intros. by apply option_bind_ext. Qed.
Lemma bind_Some {A B} (f : A → option B) (x : option A) b :
x ≫= f = Some b ↔ ∃ a, x = Some a ∧ f a = Some b.
Proof. split. by destruct x as [a|]; [exists a|]. by intros (?&->&?). Qed.
Lemma bind_None {A B} (f : A → option B) (x : option A) :
x ≫= f = None ↔ x = None ∨ ∃ a, x = Some a ∧ f a = None.
Proof.
split; [|by intros [->|(?&->&?)]].
destruct x; intros; simplify_equality'; eauto.
Qed.
Lemma bind_with_Some {A} (x : option A) : x ≫= Some = x.
Proof. by destruct x. Qed.
(** ** Inverses of constructors *)
(** We can do this in a fancy way using dependent types, but rewrite does
not particularly like type level reductions. *)
Class Maybe {A B : Type} (c : A → B) :=
maybe : B → option A.
Arguments maybe {_ _} _ {_} !_ /.
Class Maybe2 {A1 A2 B : Type} (c : A1 → A2 → B) :=
maybe2 : B → option (A1 * A2).
Arguments maybe2 {_ _ _} _ {_} !_ /.
Class Maybe3 {A1 A2 A3 B : Type} (c : A1 → A2 → A3 → B) :=
maybe3 : B → option (A1 * A2 * A3).
Arguments maybe3 {_ _ _ _} _ {_} !_ /.
Class Maybe4 {A1 A2 A3 A4 B : Type} (c : A1 → A2 → A3 → A4 → B) :=
maybe4 : B → option (A1 * A2 * A3 * A4).
Arguments maybe4 {_ _ _ _ _} _ {_} !_ /.
Instance maybe_comp `{Maybe B C c1, Maybe A B c2} : Maybe (c1 ∘ c2) := λ x,
maybe c1 x ≫= maybe c2.
Arguments maybe_comp _ _ _ _ _ _ _ !_ /.
Instance maybe_inl {A B} : Maybe (@inl A B) := λ xy,
match xy with inl x => Some x | _ => None end.
Instance maybe_inr {A B} : Maybe (@inr A B) := λ xy,
match xy with inr y => Some y | _ => None end.
Instance maybe_Some {A} : Maybe (@Some A) := id.
Arguments maybe_Some _ !_ /.
(** * 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: Comm (=) f → Comm (=) (union_with f).
Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm 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: Comm (=) f → Comm (=) (intersection_with f).
Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm 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 : appcontext C [@mguard option _ ?P ?dec] |- _ =>
change (@mguard option _ P dec) with (λ A (x : P → option A),
match @decide P dec with left H' => x H' | _ => None end) in *;
destruct_decide (@decide P dec) as Hx
| |- appcontext C [@mguard option _ ?P ?dec] =>
change (@mguard option _ P dec) with (λ A (x : P → option A),
match @decide P dec with left H' => x H' | _ => None end) in *;
destruct_decide (@decide P dec) as Hx
end.
Tactic Notation "case_option_guard" :=
let H := fresh in case_option_guard as H.
Lemma option_guard_True {A} P `{Decision P} (x : option A) :
P → guard P; x = x.
Proof. intros. by case_option_guard. Qed.
Lemma option_guard_False {A} P `{Decision P} (x : option A) :
¬P → guard P; x = None.
Proof. intros. by case_option_guard. Qed.
Lemma option_guard_iff {A} P Q `{Decision P, Decision Q} (x : option A) :
(P ↔ Q) → guard P; x = guard Q; x.
Proof. intros [??]. repeat case_option_guard; intuition. Qed.
Tactic Notation "simpl_option" "by" tactic3(tac) :=
let assert_Some_None A o H := first
[ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x;
assert (o = Some x') as H by tac
| assert (o = None) as H by tac ]
in repeat match goal with
| H : appcontext [@mret _ _ ?A] |- _ =>
change (@mret _ _ A) with (@Some A) in H
| |- appcontext [@mret _ _ ?A] => change (@mret _ _ A) with (@Some A)
| H : context [mbind (M:=option) (A:=?A) ?f ?o] |- _ =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx in H; clear Hx
| H : context [fmap (M:=option) (A:=?A) ?f ?o] |- _ =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx in H; clear Hx
| H : context [default (A:=?A) _ ?o _] |- _ =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx in H; clear Hx
| H : context [from_option (A:=?A) _ ?o] |- _ =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx in H; clear Hx
| H : context [ match ?o with _ => _ end ] |- _ =>
match type of o with
| option ?A =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx in H; clear Hx
end
| |- context [mbind (M:=option) (A:=?A) ?f ?o] =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx; clear Hx
| |- context [fmap (M:=option) (A:=?A) ?f ?o] =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx; clear Hx
| |- context [default (A:=?A) _ ?o _] =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx; clear Hx
| |- context [from_option (A:=?A) _ ?o] =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx; clear Hx
| |- context [ match ?o with _ => _ end ] =>
match type of o with
| option ?A =>
let Hx := fresh in assert_Some_None A o Hx; rewrite Hx; clear Hx
end
| H : context [decide _] |- _ => rewrite decide_True in H by tac
| H : context [decide _] |- _ => rewrite decide_False in H by tac
| H : context [mguard _ _] |- _ => rewrite option_guard_False in H by tac
| H : context [mguard _ _] |- _ => rewrite option_guard_True in H by tac
| _ => rewrite decide_True by tac
| _ => rewrite decide_False by tac
| _ => rewrite option_guard_True by tac
| _ => rewrite option_guard_False by tac
| H : context [None ∪ _] |- _ => rewrite (left_id_L None (∪)) in H
| H : context [_ ∪ None] |- _ => rewrite (right_id_L None (∪)) in H
| |- context [None ∪ _] => rewrite (left_id_L None (∪))
| |- context [_ ∪ None] => rewrite (right_id_L None (∪))
end.
Tactic Notation "simplify_option_equality" "by" tactic3(tac) :=
repeat match goal with
| _ => progress simplify_equality'
| _ => progress simpl_option by tac
| _ : maybe _ ?x = Some _ |- _ => is_var x; destruct x
| _ : maybe2 _ ?x = Some _ |- _ => is_var x; destruct x
| _ : maybe3 _ ?x = Some _ |- _ => is_var x; destruct x
| _ : maybe4 _ ?x = Some _ |- _ => is_var x; destruct x
| 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;
let y := fresh in destruct o as [y|] eqn:?;
[change (f y = x) in H|change (None = x) in H]
| 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;
let y := fresh in destruct o as [y|] eqn:?;
[change (x = f y) in H|change (x = None) in H]
| 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;
let y := fresh in destruct o as [y|] eqn:?;
[change (Some (f y) = x) in H|change (None = x) in H]
| 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;
let y := fresh in destruct o as [y|] eqn:?;
[change (x = Some (f y)) in H|change (x = None) in H]
| _ => progress case_decide
| _ => progress case_option_guard
end.
Tactic Notation "simplify_option_equality" := simplify_option_equality by eauto.