146 lines
4.1 KiB
Text
146 lines
4.1 KiB
Text
-- 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.connectives.basic 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_elim
|
||
(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
|