lean2/library/data/bool.lean

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-- 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 general_notation
import logic.connectives logic.decidable logic.inhabited
open eq eq.ops decidable
inductive bool : Type :=
ff : bool,
tt : bool
namespace bool
definition cond {A : Type} (b : bool) (t e : A) :=
rec_on b e t
theorem dichotomy (b : bool) : b = ff b = tt :=
cases_on b (or.inl rfl) (or.inr rfl)
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
definition bor (a b : bool) :=
rec_on a (rec_on b ff tt) tt
theorem bor.tt_left (a : bool) : bor tt a = tt :=
rfl
notation a || b := bor a b
theorem bor.tt_right (a : bool) : a || tt = tt :=
cases_on a rfl rfl
theorem bor.ff_left (a : bool) : ff || a = a :=
cases_on a rfl rfl
theorem bor.ff_right (a : bool) : a || ff = a :=
cases_on a rfl rfl
theorem bor.id (a : bool) : a || a = a :=
cases_on a rfl rfl
theorem bor.comm (a b : bool) : a || b = b || a :=
cases_on a
(cases_on b rfl rfl)
(cases_on b rfl rfl)
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}
... = ff || (b || c) : !bor.ff_left⁻¹)
(calc (tt || b) || c = tt || c : {!bor.tt_left}
... = tt : !bor.tt_left
... = tt || (b || c) : !bor.tt_left⁻¹)
theorem bor.to_or {a b : bool} : a || b = tt → a = tt b = tt :=
rec_on a
(assume H : ff || b = tt,
have Hb : b = tt, from !bor.ff_left ▸ H,
or.inr Hb)
(assume H, or.inl rfl)
definition band (a b : bool) :=
rec_on a ff (rec_on b ff tt)
notation a && b := band a b
theorem band.ff_left (a : bool) : ff && a = ff :=
rfl
theorem band.tt_left (a : bool) : tt && a = a :=
cases_on a rfl rfl
theorem band.ff_right (a : bool) : a && ff = ff :=
cases_on a rfl rfl
theorem band.tt_right (a : bool) : a && tt = a :=
cases_on a rfl rfl
theorem band.id (a : bool) : a && a = a :=
cases_on a rfl rfl
theorem band.comm (a b : bool) : a && b = b && a :=
cases_on a
(cases_on b rfl rfl)
(cases_on b rfl rfl)
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}
... = ff : !band.ff_left
... = ff && (b && c) : !band.ff_left⁻¹)
(calc (tt && b) && c = b && c : {!band.tt_left}
... = tt && (b && c) : !band.tt_left⁻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 (!band.comm ⬝ H)
definition bnot (a : bool) :=
rec_on a tt ff
theorem bnot.bnot (a : bool) : bnot (bnot a) = a :=
cases_on a rfl rfl
theorem bnot.false : bnot ff = tt :=
rfl
theorem bnot.true : bnot tt = ff :=
rfl
protected definition is_inhabited [instance] : inhabited bool :=
inhabited.mk ff
protected definition has_decidable_eq [instance] : decidable_eq bool :=
take a b : bool,
rec_on a
(rec_on b (inl rfl) (inr ff_ne_tt))
(rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl))
end bool