53 lines
1.2 KiB
Text
53 lines
1.2 KiB
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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-- Authors: Leonardo de Moura, Jeremy Avigad
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import general_notation type
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-- implication
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-- -----------
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definition imp (a b : Prop) : Prop := a → b
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-- true and false
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-- --------------
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inductive false : Prop
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theorem false_elim {c : Prop} (H : false) : c :=
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false.rec c H
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inductive true : Prop :=
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intro : true
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definition trivial := true.intro
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definition not (a : Prop) := a → false
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prefix `¬` := not
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-- not
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-- ---
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theorem not_intro {a : Prop} (H : a → false) : ¬a := H
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theorem not_elim {a : Prop} (H1 : ¬a) (H2 : a) : false := H1 H2
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theorem absurd {a : Prop} {b : Prop} (H1 : a) (H2 : ¬a) : b :=
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false_elim (H2 H1)
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theorem not_not_intro {a : Prop} (Ha : a) : ¬¬a :=
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assume Hna : ¬a, absurd Ha Hna
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theorem mt {a b : Prop} (H1 : a → b) (H2 : ¬b) : ¬a :=
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assume Ha : a, absurd (H1 Ha) H2
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theorem not_false_trivial : ¬false :=
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assume H : false, H
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theorem not_implies_left {a b : Prop} (H : ¬(a → b)) : ¬¬a :=
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assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
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theorem not_implies_right {a b : Prop} (H : ¬(a → b)) : ¬b :=
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assume Hb : b, absurd (assume Ha : a, Hb) H
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