128 lines
3.5 KiB
Text
128 lines
3.5 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.eq
|
||
open eq eq.ops decidable
|
||
|
||
namespace bool
|
||
local attribute bor [reducible]
|
||
local attribute band [reducible]
|
||
|
||
theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
|
||
bool.cases_on b (or.inl rfl) (or.inr rfl)
|
||
|
||
theorem cond_ff [simp] {A : Type} (t e : A) : cond ff t e = e :=
|
||
rfl
|
||
|
||
theorem cond_tt [simp] {A : Type} (t e : A) : cond tt t e = t :=
|
||
rfl
|
||
|
||
theorem eq_tt_of_ne_ff : ∀ {a : bool}, a ≠ ff → a = tt
|
||
| @eq_tt_of_ne_ff tt H := rfl
|
||
| @eq_tt_of_ne_ff ff H := absurd rfl H
|
||
|
||
theorem eq_ff_of_ne_tt : ∀ {a : bool}, a ≠ tt → a = ff
|
||
| @eq_ff_of_ne_tt tt H := absurd rfl H
|
||
| @eq_ff_of_ne_tt ff H := rfl
|
||
|
||
theorem absurd_of_eq_ff_of_eq_tt {B : Prop} {a : bool} (H₁ : a = ff) (H₂ : a = tt) : B :=
|
||
absurd (H₁⁻¹ ⬝ H₂) ff_ne_tt
|
||
|
||
theorem tt_bor [simp] (a : bool) : bor tt a = tt :=
|
||
rfl
|
||
|
||
notation a || b := bor a b
|
||
|
||
theorem bor_tt [simp] (a : bool) : a || tt = tt :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem ff_bor [simp] (a : bool) : ff || a = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem bor_ff [simp] (a : bool) : a || ff = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem bor_self [simp] (a : bool) : a || a = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem bor.comm [simp] (a b : bool) : a || b = b || a :=
|
||
by cases a; repeat (cases b | reflexivity)
|
||
|
||
theorem bor.assoc [simp] (a b c : bool) : (a || b) || c = a || (b || c) :=
|
||
match a with
|
||
| ff := by rewrite *ff_bor
|
||
| tt := by rewrite *tt_bor
|
||
end
|
||
|
||
theorem or_of_bor_eq {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
|
||
bool.rec_on a
|
||
(suppose ff || b = tt,
|
||
have b = tt, from !ff_bor ▸ this,
|
||
or.inr this)
|
||
(suppose tt || b = tt,
|
||
or.inl rfl)
|
||
|
||
theorem bor_inl {a b : bool} (H : a = tt) : a || b = tt :=
|
||
by rewrite H
|
||
|
||
theorem bor_inr {a b : bool} (H : b = tt) : a || b = tt :=
|
||
bool.rec_on a (by rewrite H) (by rewrite H)
|
||
|
||
theorem ff_band [simp] (a : bool) : ff && a = ff :=
|
||
rfl
|
||
|
||
theorem tt_band [simp] (a : bool) : tt && a = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem band_ff [simp] (a : bool) : a && ff = ff :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem band_tt [simp] (a : bool) : a && tt = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem band_self [simp] (a : bool) : a && a = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem band.comm [simp] (a b : bool) : a && b = b && a :=
|
||
bool.cases_on a
|
||
(bool.cases_on b rfl rfl)
|
||
(bool.cases_on b rfl rfl)
|
||
|
||
theorem band.assoc [simp] (a b c : bool) : (a && b) && c = a && (b && c) :=
|
||
match a with
|
||
| ff := by rewrite *ff_band
|
||
| tt := by rewrite *tt_band
|
||
end
|
||
|
||
theorem band_elim_left {a b : bool} (H : a && b = tt) : a = tt :=
|
||
or.elim (dichotomy a)
|
||
(suppose a = ff,
|
||
absurd (by msimp) ff_ne_tt)
|
||
(suppose a = tt, this)
|
||
|
||
theorem band_intro {a b : bool} (H₁ : a = tt) (H₂ : b = tt) : a && b = tt :=
|
||
by rewrite [H₁, H₂]
|
||
|
||
theorem band_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
|
||
band_elim_left (!band.comm ⬝ H)
|
||
|
||
theorem bnot_bnot [simp] (a : bool) : bnot (bnot a) = a :=
|
||
bool.cases_on a rfl rfl
|
||
|
||
theorem bnot_false [simp] : bnot ff = tt :=
|
||
rfl
|
||
|
||
theorem bnot_true [simp] : bnot tt = ff :=
|
||
rfl
|
||
|
||
theorem eq_tt_of_bnot_eq_ff {a : bool} : bnot a = ff → a = tt :=
|
||
bool.cases_on a (by contradiction) (λ h, rfl)
|
||
|
||
theorem eq_ff_of_bnot_eq_tt {a : bool} : bnot a = tt → a = ff :=
|
||
bool.cases_on a (λ h, rfl) (by contradiction)
|
||
|
||
definition bxor (x:bool) (y:bool) := cond x (bnot y) y
|
||
end bool
|