lean2/library/logic/axioms/prop_decidable.lean
2014-11-30 20:34:12 -08:00

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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
-- logic.axioms.prop_decidable
-- ===========================
import logic.axioms.classical logic.axioms.hilbert
open decidable inhabited nonempty
-- Excluded middle + Hilbert implies every proposition is decidable
-- First, we show that (decidable a) is inhabited for any 'a' using the excluded middle
theorem decidable_inhabited [instance] (a : Prop) : inhabited (decidable a) :=
nonempty_imp_inhabited
(or.elim (em a)
(assume Ha, nonempty.intro (inl Ha))
(assume Hna, nonempty.intro (inr Hna)))
-- Note that decidable_inhabited is marked as an instance, and it is silently used
-- for synthesizing the implicit argument in the following 'epsilon'
theorem prop_decidable [instance] (a : Prop) : decidable a :=
epsilon (λd, true)