56 lines
1.8 KiB
Text
56 lines
1.8 KiB
Text
/-
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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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-/
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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} : option A → Prop
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| none := true
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| (some v) := false
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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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sorry -- by contradiction
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theorem some.inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
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sorry -- by injection H; assumption
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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] : ∀ o₁ o₂ : option A, decidable (o₁ = o₂)
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| none none := tt sorry -- by left; reflexivity
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| none (some v₂) := ff sorry -- by right; contradiction
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| (some v₁) none := ff sorry -- by right; contradiction
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| (some v₁) (some v₂) :=
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match H v₁ v₂ with
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| tt e := tt sorry -- by left; congruence; assumption
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| ff n := ff sorry -- by right; intro h; injection h; contradiction
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end
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inline protected definition fmap {A B : Type} (f : A → B) (e : option A) : option B :=
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option.cases_on e
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none
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(λ a, some (f a))
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inline protected definition bind {A B : Type} (a : option A) (b : A → option B) : option B :=
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option.cases_on a
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none
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(λ a, b a)
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inline protected definition return {A : Type} (a : A) : option A :=
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some a
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end option
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definition option.is_monad [instance] : monad option :=
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monad.mk @option.fmap @option.return @option.bind
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