2014-08-12 00:35:25 +00:00
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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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2014-09-05 16:31:27 +00:00
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-- Author: Jeremy Avigad, Floris van Doorn
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2014-08-12 00:35:25 +00:00
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-- Empty type
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-- ----------
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2014-12-01 04:34:12 +00:00
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import logic.cast logic.subsingleton
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2014-08-12 00:35:25 +00:00
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2014-09-05 05:31:52 +00:00
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namespace empty
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2014-09-19 22:04:52 +00:00
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protected theorem elim (A : Type) (H : empty) : A :=
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2014-09-05 05:31:52 +00:00
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rec (λe, A) H
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2014-10-09 01:41:18 +00:00
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protected theorem subsingleton [instance] : subsingleton empty :=
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subsingleton.intro (λ a b, !elim a)
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2014-09-05 05:31:52 +00:00
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end empty
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2014-11-09 01:55:42 +00:00
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2014-12-01 04:34:12 +00:00
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protected definition empty.has_decidable_eq [instance] : decidable_eq empty :=
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take (a b : empty), decidable.inl (!empty.elim a)
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2014-11-09 01:55:42 +00:00
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definition tneg.tneg (A : Type) := A → empty
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prefix `~` := tneg.tneg
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namespace tneg
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variables {A B : Type}
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protected definition intro (H : A → empty) : ~A := H
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protected definition elim (H1 : ~A) (H2 : A) : empty := H1 H2
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protected definition empty : ~empty := λH : empty, H
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definition tabsurd (H1 : A) (H2 : ~A) : B := !empty.elim (H2 H1)
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definition tneg_tneg_intro (H : A) : ~~A := λH2 : ~A, tneg.elim H2 H
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definition tmt (H1 : A → B) (H2 : ~B) : ~A := λHA : A, tabsurd (H1 HA) H2
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definition tneg_pi_left {B : A → Type} (H : ~Πa, B a) : ~~A :=
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λHnA : ~A, tneg.elim H (λHA : A, tabsurd HA HnA)
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definition tneg_function_right (H : ~(A → B)) : ~B :=
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λHB : B, tneg.elim H (λHA : A, HB)
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end tneg
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