35 lines
1.3 KiB
Text
35 lines
1.3 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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import logic.eq
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open eq.ops decidable
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namespace option
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definition is_none {A : Type} (o : option A) : Prop :=
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rec true (λ a, false) o
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theorem is_none_none {A : Type} : is_none (@none A) :=
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trivial
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theorem not_is_none_some {A : Type} (a : A) : ¬ is_none (some a) :=
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not_false
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theorem none_ne_some {A : Type} (a : A) : none ≠ some a :=
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assume H, no_confusion H
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theorem some.inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
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no_confusion H (λe, e)
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protected definition is_inhabited [instance] (A : Type) : inhabited (option A) :=
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inhabited.mk none
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protected definition has_decidable_eq [instance] {A : Type} (H : decidable_eq A) : decidable_eq (option A) :=
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take o₁ o₂ : option A,
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rec_on o₁
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(rec_on o₂ (inl rfl) (take a₂, (inr (none_ne_some a₂))))
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(take a₁ : A, rec_on o₂
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(inr (ne.symm (none_ne_some a₁)))
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(take a₂ : A, decidable.rec_on (H a₁ a₂)
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(assume Heq : a₁ = a₂, inl (Heq ▸ rfl))
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(assume Hne : a₁ ≠ a₂, inr (assume Hn : some a₁ = some a₂, absurd (some.inj Hn) Hne))))
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end option
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