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Require Export base decidable orders.
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.
Lemma Some_inj {A} (a b : A) : Some a = Some b → a = b.
Proof. congruence. Qed.
Definition option_case {A B} (f : A → B) (b : B) (x : option A) :=
match x with
| None => b
| Some a => f a
end.
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.
Definition is_Some `(x : option A) := ∃ a, x = Some a.
Hint Extern 10 (is_Some _) => solve [eexists; eauto].
Ltac simplify_is_Some := repeat intro; repeat (
match goal with
| H : is_Some _ |- _ => destruct H as [??]
| |- is_Some _ => eauto
end || simplify_eqs).
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_1 `(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 is_Some_2 `(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.
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 ∘ Some_inj _ _)
end
| None => right (Some_ne_None _)
end
| None =>
match y with
| Some _ => right (None_ne_Some _)
| None => left eq_refl
end
end.
Inductive option_lift `(P : A → Prop) : option A → Prop :=
| option_lift_some x : P x → option_lift P (Some x)
| option_lift_None : option_lift P None.
Ltac option_lift_inv := repeat
match goal with
| H : option_lift _ (Some _) |- _ => inversion H; clear H; subst
| H : option_lift _ None |- _ => inversion H
end.
Lemma option_lift_inv_Some `(P : A → Prop) x : option_lift P (Some x) → P x.
Proof. intros. now option_lift_inv. Qed.
Definition option_lift_sig `(P : A → Prop) (x : option A) : option_lift P x → option (sig P) :=
match x with
| Some a => λ p, Some (exist _ a (option_lift_inv_Some P a p))
| None => λ _, None
end.
Definition option_lift_dsig `(P : A → Prop) `{∀ x : A, Decision (P x)}
(x : option A) : option_lift P x → option (dsig P) :=
match x with
| Some a => λ p, Some (dexist a (option_lift_inv_Some P a p))
| None => λ _, None
end.
Lemma option_lift_dsig_Some `(P : A → Prop) `{∀ x : A, Decision (P x)} x y px py :
option_lift_dsig P x px = Some (y↾py) ↔ x = Some y.
Proof.
split.
destruct x; simpl; intros; now simplify_eqs.
intros. subst. simpl. f_equal. now apply dsig_eq.
Qed.
Lemma option_lift_dsig_is_Some `(P : A → Prop) `{∀ x : A, Decision (P x)} x px :
is_Some (option_lift_dsig P x px) ↔ is_Some x.
Proof.
split.
intros [[??] ?]. eapply is_Some_2, option_lift_dsig_Some; eauto.
intros [??]. subst. eapply is_Some_2. reflexivity.
Qed.
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.
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_intersect: IntersectWith option := λ A f x y,
match x, y with
| Some a, Some b => Some (f a b)
| _, _ => None
end.
Section option_union_intersect.
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.
End option_union_intersect.