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