2014-12-22 19:54:01 +00:00
|
|
|
/-
|
|
|
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
|
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
2014-08-30 03:45:57 +00:00
|
|
|
|
2014-12-22 19:54:01 +00:00
|
|
|
Module: logic.axioms.prop_decidable
|
|
|
|
|
Author: Leonardo de Moura
|
2015-04-08 23:28:13 +00:00
|
|
|
|
|
|
|
|
Excluded middle + Hilbert implies every proposition is decidable.
|
2014-12-22 19:54:01 +00:00
|
|
|
-/
|
2015-02-01 15:58:58 +00:00
|
|
|
import logic.axioms.prop_complete logic.axioms.hilbert
|
2014-09-03 23:00:38 +00:00
|
|
|
open decidable inhabited nonempty
|
2014-07-19 19:09:47 +00:00
|
|
|
|
2014-08-20 22:49:44 +00:00
|
|
|
theorem decidable_inhabited [instance] (a : Prop) : inhabited (decidable a) :=
|
2014-12-22 19:54:01 +00:00
|
|
|
inhabited_of_nonempty
|
2014-09-05 04:25:21 +00:00
|
|
|
(or.elim (em a)
|
2014-09-04 23:36:06 +00:00
|
|
|
(assume Ha, nonempty.intro (inl Ha))
|
|
|
|
|
(assume Hna, nonempty.intro (inr Hna)))
|
2014-07-19 19:09:47 +00:00
|
|
|
|
2014-07-29 02:58:57 +00:00
|
|
|
theorem prop_decidable [instance] (a : Prop) : decidable a :=
|
2015-04-08 23:28:13 +00:00
|
|
|
arbitrary (decidable a)
|
