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
-/
2014-12-01 04:34:12 +00:00
import logic.eq
namespace bool
local attribute bor [reducible]
local attribute band [reducible]
theorem dichotomy (b : bool) : b = ff b = tt :=
by rec_simp
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 :=
by rec_simp
theorem eq_ff_of_ne_tt : ∀ {a : bool}, a ≠ tt → a = ff :=
by rec_simp
theorem absurd_of_eq_ff_of_eq_tt {B : Prop} {a : bool} (H₁ : a = ff) (H₂ : a = tt) : B :=
by rec_simp
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 :=
by rec_simp
theorem ff_bor [simp] (a : bool) : ff || a = a :=
by rec_simp
theorem bor_ff [simp] (a : bool) : a || ff = a :=
by rec_simp
theorem bor_self [simp] (a : bool) : a || a = a :=
by rec_simp
theorem bor_comm [simp] (a b : bool) : a || b = b || a :=
by rec_simp
theorem bor_assoc [simp] (a b c : bool) : (a || b) || c = a || (b || c) :=
by rec_simp
theorem bor_left_comm [simp] (a b c : bool) : a || (b || c) = b || (a || c) :=
by rec_simp
theorem or_of_bor_eq {a b : bool} : a || b = tt → a = tt b = tt :=
by rec_simp
theorem bor_inl {a b : bool} (H : a = tt) : a || b = tt :=
by rec_simp
theorem bor_inr {a b : bool} (H : b = tt) : a || b = tt :=
by rec_simp
theorem ff_band [simp] (a : bool) : ff && a = ff :=
rfl
theorem tt_band [simp] (a : bool) : tt && a = a :=
by rec_simp
theorem band_ff [simp] (a : bool) : a && ff = ff :=
by rec_simp
theorem band_tt [simp] (a : bool) : a && tt = a :=
by rec_simp
theorem band_self [simp] (a : bool) : a && a = a :=
by rec_simp
theorem band_comm [simp] (a b : bool) : a && b = b && a :=
by rec_simp
theorem band_assoc [simp] (a b c : bool) : (a && b) && c = a && (b && c) :=
by rec_simp
theorem band_left_comm [simp] (a b c : bool) : a && (b && c) = b && (a && c) :=
by rec_simp
theorem band_elim_left {a b : bool} (H : a && b = tt) : a = tt :=
by rec_simp
theorem band_intro {a b : bool} (H₁ : a = tt) (H₂ : b = tt) : a && b = tt :=
by rec_simp
theorem band_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
by rec_simp
theorem bnot_false [simp] : bnot ff = tt :=
rfl
theorem bnot_true [simp] : bnot tt = ff :=
rfl
theorem bnot_bnot [simp] (a : bool) : bnot (bnot a) = a :=
by rec_simp
theorem eq_tt_of_bnot_eq_ff {a : bool} : bnot a = ff → a = tt :=
by rec_simp
theorem eq_ff_of_bnot_eq_tt {a : bool} : bnot a = tt → a = ff :=
by rec_simp
definition bxor : bool → bool → bool
| ff ff := ff
| ff tt := tt
| tt ff := tt
| tt tt := ff
lemma ff_bxor_ff [simp] : bxor ff ff = ff := rfl
lemma ff_bxor_tt [simp] : bxor ff tt = tt := rfl
lemma tt_bxor_ff [simp] : bxor tt ff = tt := rfl
lemma tt_bxor_tt [simp] : bxor tt tt = ff := rfl
lemma bxor_self [simp] (a : bool) : bxor a a = ff :=
by rec_simp
lemma bxor_ff [simp] (a : bool) : bxor a ff = a :=
by rec_simp
lemma bxor_tt [simp] (a : bool) : bxor a tt = bnot a :=
by rec_simp
lemma ff_bxor [simp] (a : bool) : bxor ff a = a :=
by rec_simp
lemma tt_bxor [simp] (a : bool) : bxor tt a = bnot a :=
by rec_simp
lemma bxor_comm [simp] (a b : bool) : bxor a b = bxor b a :=
by rec_simp
lemma bxor_assoc [simp] (a b c : bool) : bxor (bxor a b) c = bxor a (bxor b c) :=
by rec_simp
lemma bxor_left_comm [simp] (a b c : bool) : bxor a (bxor b c) = bxor b (bxor a c) :=
by rec_simp
end bool