lean2/library/data/option.lean

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/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: data.option
Author: Leonardo de Moura
-/
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import logic.eq
open eq.ops decidable
namespace option
definition is_none {A : Type} : option A → Prop
| none := true
| (some v) := false
theorem is_none_none {A : Type} : is_none (@none A) :=
trivial
theorem not_is_none_some {A : Type} (a : A) : ¬ is_none (some a) :=
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not_false
theorem none_ne_some {A : Type} (a : A) : none ≠ some a :=
by contradiction
theorem some.inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
by injection H; assumption
protected definition is_inhabited [instance] (A : Type) : inhabited (option A) :=
inhabited.mk none
protected definition has_decidable_eq [instance] {A : Type} [H : decidable_eq A] : ∀ o₁ o₂ : option A, decidable (o₁ = o₂)
| none none := by left; reflexivity
| none (some v₂) := by right; contradiction
| (some v₁) none := by right; contradiction
| (some v₁) (some v₂) :=
match H v₁ v₂ with
| inl e := by left; congruence; assumption
| inr n := by right; intro h; injection h; refine absurd _ n; assumption
end
end option