lean2/library/data/bool.lean

146 lines
4.1 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.core.connectives logic.classes.decidable logic.classes.inhabited
using eq_ops decidable
inductive bool : Type :=
ff : bool,
tt : bool
namespace bool
theorem cases_on {p : bool → Prop} (b : bool) (H0 : p ff) (H1 : p tt) : p b :=
bool_rec H0 H1 b
theorem bool_inhabited [instance] : inhabited bool :=
inhabited_mk ff
definition cond {A : Type} (b : bool) (t e : A) :=
bool_rec e t b
theorem dichotomy (b : bool) : b = ff b = tt :=
cases_on b (or_inl (refl ff)) (or_inr (refl tt))
theorem cond_ff {A : Type} (t e : A) : cond ff t e = e :=
rfl
theorem cond_tt {A : Type} (t e : A) : cond tt t e = t :=
rfl
theorem ff_ne_tt : ¬ ff = tt :=
assume H : ff = tt, absurd
(calc true = cond tt true false : (cond_tt _ _)⁻¹
... = cond ff true false : {H⁻¹}
... = false : cond_ff _ _)
true_ne_false
theorem decidable_eq [instance] (a b : bool) : decidable (a = b) :=
bool_rec
(bool_rec (inl (refl ff)) (inr ff_ne_tt) b)
(bool_rec (inr (ne_symm ff_ne_tt)) (inl (refl tt)) b)
a
definition bor (a b : bool) :=
bool_rec (bool_rec ff tt b) tt a
theorem bor_tt_left (a : bool) : bor tt a = tt :=
rfl
infixl `||` := bor
theorem bor_tt_right (a : bool) : a || tt = tt :=
cases_on a (refl (ff || tt)) (refl (tt || tt))
theorem bor_ff_left (a : bool) : ff || a = a :=
cases_on a (refl (ff || ff)) (refl (ff || tt))
theorem bor_ff_right (a : bool) : a || ff = a :=
cases_on a (refl (ff || ff)) (refl (tt || ff))
theorem bor_id (a : bool) : a || a = a :=
cases_on a (refl (ff || ff)) (refl (tt || tt))
theorem bor_comm (a b : bool) : a || b = b || a :=
cases_on a
(cases_on b (refl (ff || ff)) (refl (ff || tt)))
(cases_on b (refl (tt || ff)) (refl (tt || tt)))
theorem bor_assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
cases_on a
(calc (ff || b) || c = b || c : {bor_ff_left b}
... = ff || (b || c) : bor_ff_left (b || c)⁻¹)
(calc (tt || b) || c = tt || c : {bor_tt_left b}
... = tt : bor_tt_left c
... = tt || (b || c) : bor_tt_left (b || c)⁻¹)
theorem bor_to_or {a b : bool} : a || b = tt → a = tt b = tt :=
bool_rec
(assume H : ff || b = tt,
have Hb : b = tt, from (bor_ff_left b) ▸ H,
or_inr Hb)
(assume H, or_inl (refl tt))
a
definition band (a b : bool) :=
bool_rec ff (bool_rec ff tt b) a
infixl `&&` := band
theorem band_ff_left (a : bool) : ff && a = ff :=
rfl
theorem band_tt_left (a : bool) : tt && a = a :=
cases_on a (refl (tt && ff)) (refl (tt && tt))
theorem band_ff_right (a : bool) : a && ff = ff :=
cases_on a (refl (ff && ff)) (refl (tt && ff))
theorem band_tt_right (a : bool) : a && tt = a :=
cases_on a (refl (ff && tt)) (refl (tt && tt))
theorem band_id (a : bool) : a && a = a :=
cases_on a (refl (ff && ff)) (refl (tt && tt))
theorem band_comm (a b : bool) : a && b = b && a :=
cases_on a
(cases_on b (refl (ff && ff)) (refl (ff && tt)))
(cases_on b (refl (tt && ff)) (refl (tt && tt)))
theorem band_assoc (a b c : bool) : (a && b) && c = a && (b && c) :=
cases_on a
(calc (ff && b) && c = ff && c : {band_ff_left b}
... = ff : band_ff_left c
... = ff && (b && c) : band_ff_left (b && c)⁻¹)
(calc (tt && b) && c = b && c : {band_tt_left b}
... = tt && (b && c) : band_tt_left (b && c)⁻¹)
theorem band_eq_tt_elim_left {a b : bool} (H : a && b = tt) : a = tt :=
or_elim (dichotomy a)
(assume H0 : a = ff,
absurd
(calc ff = ff && b : (band_ff_left _)⁻¹
... = a && b : {H0⁻¹}
... = tt : H)
ff_ne_tt)
(assume H1 : a = tt, H1)
theorem band_eq_tt_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
band_eq_tt_elim_left (trans (band_comm b a) H)
definition bnot (a : bool) := bool_rec tt ff a
notation `!` x:max := bnot x
theorem bnot_bnot (a : bool) : !!a = a :=
cases_on a (refl (!!ff)) (refl (!!tt))
theorem bnot_false : !ff = tt :=
rfl
theorem bnot_true : !tt = ff :=
rfl
end bool