lean2/library/data/bool.lean
Leonardo de Moura 559dd586f2 feat(library): add 'decidable_eq' class
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2014-09-07 22:23:36 -07:00

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
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] : decidable_eq bool :=
decidable_eq.intro (λ (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