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 logic.core.connectives logic.classes.decidable logic.classes.inhabited
open eq_ops eq decidable
inductive bool : Type :=
ff : bool,
tt : bool
namespace bool
definition rec_on [protected] {C : bool → Type} (b : bool) (H₁ : C ff) (H₂ : C tt) : C b :=
rec H₁ H₂ b
theorem cases_on [protected] {p : bool → Prop} (b : bool) (H₁ : p ff) (H₂ : p tt) : p b :=
rec H₁ H₂ b
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
infixl `||` := bor
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 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 :=
rec_on a
(assume H : ff || b = tt,
have Hb : b = tt, from (bor_ff_left b) ▸ H,
or.inr Hb)
(assume H, or.inl rfl)
definition band (a b : bool) :=
rec_on a ff (rec_on b ff tt)
infixl `&&` := band
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 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 (band_comm b a ⬝ H)
definition bnot (a : bool) :=
rec_on a tt ff
notation `!` x:max := bnot x
theorem bnot_bnot (a : bool) : !!a = a :=
cases_on a rfl rfl
theorem bnot_false : !ff = tt :=
rfl
theorem bnot_true : !tt = ff :=
rfl
theorem is_inhabited [protected] [instance] : inhabited bool :=
inhabited.mk ff
theorem has_decidable_eq [protected] [instance] (a b : bool) : decidable (a = b) :=
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