52 lines
1.8 KiB
Text
52 lines
1.8 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.connectives.basic logic.connectives.eq logic.classes.inhabited logic.classes.decidable
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using eq_ops decidable
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namespace option
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inductive option (A : Type) : Type :=
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none {} : option A,
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some : A → option A
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theorem induction_on {A : Type} {p : option A → Prop} (o : option A)
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(H1 : p none) (H2 : ∀a, p (some a)) : p o :=
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option_rec H1 H2 o
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definition rec_on {A : Type} {C : option A → Type} (o : option A)
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(H1 : C none) (H2 : ∀a, C (some a)) : C o :=
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option_rec H1 H2 o
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definition is_none {A : Type} (o : option A) : Prop :=
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option_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_trivial
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theorem none_ne_some {A : Type} (a : A) : none ≠ some a :=
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assume H : none = some a, absurd
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(H ▸ is_none_none)
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(not_is_none_some a)
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theorem some_inj {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
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congr_arg (option_rec a₁ (λ a, a)) H
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theorem option_inhabited [instance] (A : Type) : inhabited (option A) :=
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inhabited_mk none
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theorem decidable_eq [instance] {A : Type} {H : ∀a₁ a₂ : A, decidable (a₁ = a₂)}
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(o₁ o₂ : option A) : decidable (o₁ = o₂) :=
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rec_on o₁
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(rec_on o₂ (inl (refl _)) (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 ▸ refl _))
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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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