24 lines
873 B
Text
24 lines
873 B
Text
/-
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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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Excluded middle + Hilbert implies every proposition is decidable.
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-/
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import logic.axioms.prop_complete logic.axioms.hilbert data.sum
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open decidable inhabited nonempty sum
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theorem decidable_inhabited [instance] [priority 0] (a : Prop) : inhabited (decidable a) :=
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inhabited_of_nonempty
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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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theorem prop_decidable [instance] [priority 0] (a : Prop) : decidable a :=
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arbitrary (decidable a)
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theorem type_decidable (A : Type) : A + (A → false) :=
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match prop_decidable (nonempty A) with
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| inl Hp := sum.inl (inhabited.value (inhabited_of_nonempty Hp))
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| inr Hn := sum.inr (λ a, absurd (nonempty.intro a) Hn)
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end
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