-- Copyright (c) 2014 Microsoft Corporation. All rights reserved. -- Released under Apache 2.0 license as described in the file LICENSE. -- Authors: Leonardo de Moura, Jeremy Avigad import general_notation definition Prop := Type.{0} -- implication -- ----------- definition imp (a b : Prop) : Prop := a → b -- true and false -- -------------- inductive false : Prop theorem false_elim {c : Prop} (H : false) : c := false.rec c H inductive true : Prop := intro : true definition trivial := true.intro definition not (a : Prop) := a → false prefix `¬` := not -- not -- --- theorem not_intro {a : Prop} (H : a → false) : ¬a := H theorem not_elim {a : Prop} (H1 : ¬a) (H2 : a) : false := H1 H2 theorem absurd {a : Prop} {b : Prop} (H1 : a) (H2 : ¬a) : b := false_elim (H2 H1) theorem not_not_intro {a : Prop} (Ha : a) : ¬¬a := assume Hna : ¬a, absurd Ha Hna theorem mt {a b : Prop} (H1 : a → b) (H2 : ¬b) : ¬a := assume Ha : a, absurd (H1 Ha) H2 theorem not_false_trivial : ¬false := assume H : false, H theorem not_implies_left {a b : Prop} (H : ¬(a → b)) : ¬¬a := assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H theorem not_implies_right {a b : Prop} (H : ¬(a → b)) : ¬b := assume Hb : b, absurd (assume Ha : a, Hb) H