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.
-- Author: Leonardo de Moura
import logic.core.eq logic.classes.inhabited logic.classes.decidable
open eq_ops decidable
inductive option (A : Type) : Type :=
none {} : option A,
some : A → option A
namespace option
theorem induction_on [protected] {A : Type} {p : option A → Prop} (o : option A)
(H1 : p none) (H2 : ∀a, p (some a)) : p o :=
rec H1 H2 o
definition rec_on [protected] {A : Type} {C : option A → Type} (o : option A)
(H1 : C none) (H2 : ∀a, C (some a)) : C o :=
rec H1 H2 o
definition is_none {A : Type} (o : option A) : Prop :=
rec true (λ a, false) o
theorem is_none_none {A : Type} : is_none (@none A) :=
trivial
theorem not_is_none_some {A : Type} (a : A) : ¬ is_none (some a) :=
not_false_trivial
theorem none_ne_some {A : Type} (a : A) : none ≠ some a :=
assume H : none = some a, absurd
(H ▸ is_none_none)
(not_is_none_some a)
theorem equal [protected] {A : Type} {a₁ a₂ : A} (H : some a₁ = some a₂) : a₁ = a₂ :=
congr_arg (option.rec a₁ (λ a, a)) H
theorem is_inhabited [protected] [instance] (A : Type) : inhabited (option A) :=
inhabited.mk none
theorem has_decidable_eq [protected] [instance] {A : Type} {H : ∀a₁ a₂ : A, decidable (a₁ = a₂)}
(o₁ o₂ : option A) : decidable (o₁ = o₂) :=
rec_on o₁
(rec_on o₂ (inl rfl) (take a₂, (inr (none_ne_some a₂))))
(take a₁ : A, rec_on o₂
(inr (ne.symm (none_ne_some a₁)))
(take a₂ : A, decidable.rec_on (H a₁ a₂)
(assume Heq : a₁ = a₂, inl (Heq ▸ rfl))
(assume Hne : a₁ ≠ a₂, inr (assume Hn : some a₁ = some a₂, absurd (equal Hn) Hne))))
end option