lean2/library/standard/prop_decidable.lean

19 lines
792 B
Text
Raw Normal View History

-- 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 classical hilbert decidable
using decidable
-- Excluded middle + Hilbert implies every proposition is decidable
-- First, we show that (decidable a) is inhabited for any 'a' using the excluded middle
theorem inhabited_decidable [instance] (a : Prop) : inhabited (decidable a)
:= or_elim (em a)
(assume Ha, inhabited_intro (inl Ha))
(assume Hna, inhabited_intro (inr Hna))
-- Note that inhabited_decidable 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 (λ d, true)