lean2/hott/types/bool.hlean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Floris van Doorn

Partially ported from the standard library
-/

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 (sum.inl rfl) (sum.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 eq_tt_of_ne_ff : Π {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 : Π {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 : Type} {a : bool} (H₁ : a = ff) (H₂ : a = tt) : B :=
  absurd (H₁⁻¹ ⬝ H₂) ff_ne_tt

  theorem tt_bor (a : bool) : bor tt a = tt :=
  rfl

  notation a || b := bor a b

  theorem bor_tt (a : bool) : a || tt = tt :=
  bool.cases_on a rfl rfl

  theorem ff_bor (a : bool) : ff || a = a :=
  bool.cases_on a rfl rfl

  theorem bor_ff (a : bool) : a || ff = a :=
  bool.cases_on a rfl rfl

  theorem bor_self (a : bool) : a || a = a :=
  bool.cases_on a rfl rfl

  theorem bor.comm (a b : bool) : a || b = b || a :=
  by cases a; repeat (cases b | reflexivity)

  theorem bor.assoc (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,
      sum.inr this)
    (suppose tt || b = tt,
      sum.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 (a : bool) : ff && a = ff :=
  rfl

  theorem tt_band (a : bool) : tt && a = a :=
  bool.cases_on a rfl rfl

  theorem band_ff (a : bool) : a && ff = ff :=
  bool.cases_on a rfl rfl

  theorem band_tt (a : bool) : a && tt = a :=
  bool.cases_on a rfl rfl

  theorem band_self (a : bool) : a && a = a :=
  bool.cases_on a rfl rfl

  theorem band.comm (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 (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 :=
  sum.elim (dichotomy a)
    (suppose a = ff,
      absurd
        (calc ff = ff && b : ff_band
             ... = a && b  : this
             ... = tt      : H)
        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 (a : bool) : bnot (bnot a) = a :=
  bool.cases_on a rfl rfl

  theorem bnot_empty : bnot ff = tt :=
  rfl

  theorem bnot_unit  : 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

  /- HoTT-related stuff -/
  open is_equiv equiv function is_trunc option unit decidable


  definition is_equiv_bnot [constructor] [instance] [priority 500] : is_equiv bnot :=
  begin
    fapply is_equiv.mk,
      exact bnot,
      all_goals (intro b;cases b), do 6 reflexivity
  --    all_goals (focus (intro b;cases b;all_goals reflexivity)),
  end

  definition bnot_ne : Π(b : bool), bnot b ≠ b
  | bnot_ne tt := ff_ne_tt
  | bnot_ne ff := ne.symm ff_ne_tt

  definition equiv_bnot [constructor] : bool ≃ bool := equiv.mk bnot _
  definition eq_bnot    : bool = bool := ua equiv_bnot

  definition eq_bnot_ne_idp : eq_bnot ≠ idp :=
  assume H : eq_bnot = idp,
  have H2 : bnot = id, from !cast_ua_fn⁻¹ ⬝ ap cast H,
  absurd (ap10 H2 tt) ff_ne_tt

  theorem is_set_bool : is_set bool := _
  theorem not_is_prop_bool_eq_bool : ¬ is_prop (bool = bool) :=
  λ H, eq_bnot_ne_idp !is_prop.elim

  definition bool_equiv_option_unit [constructor] : bool ≃ option unit :=
  begin
    fapply equiv.MK,
    { intro b, cases b, exact none, exact some star},
    { intro u, cases u, exact ff, exact tt},
    { intro u, cases u with u, reflexivity, cases u, reflexivity},
    { intro b, cases b, reflexivity, reflexivity},
  end

  /- pointed and truncated bool -/
  open pointed
  definition pointed_bool [instance] [constructor] : pointed bool :=
  pointed.mk ff

  definition pbool [constructor] : Set* :=
  pSet.mk' bool

  definition tbool [constructor] : Set := trunctype.mk bool _

  notation `bool*` := pbool


end bool