143 lines
No EOL
4.5 KiB
Text
143 lines
No EOL
4.5 KiB
Text
------------------------------------------------------------------------------------------------------ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic.connectives.basic logic.classes.decidable logic.classes.inhabited
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using eq_proofs decidable
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namespace bool
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inductive bool : Type :=
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| ff : bool
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| tt : bool
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theorem induction_on {p : bool → Prop} (b : bool) (H0 : p ff) (H1 : p tt) : p b :=
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bool_rec H0 H1 b
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theorem inhabited_bool [instance] : inhabited bool :=
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inhabited_intro ff
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definition cond {A : Type} (b : bool) (t e : A) :=
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bool_rec e t b
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theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
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induction_on b (or_inl (refl ff)) (or_inr (refl tt))
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theorem cond_ff {A : Type} (t e : A) : cond ff t e = e :=
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refl (cond ff t e)
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theorem cond_tt {A : Type} (t e : A) : cond tt t e = t :=
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refl (cond tt t e)
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theorem ff_ne_tt : ¬ ff = tt :=
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assume H : ff = tt, absurd
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(calc true = cond tt true false : (cond_tt _ _)⁻¹
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... = cond ff true false : {H⁻¹}
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... = false : cond_ff _ _)
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true_ne_false
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theorem decidable_eq [instance] (a b : bool) : decidable (a = b) :=
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bool_rec
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(bool_rec (inl (refl ff)) (inr ff_ne_tt) b)
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(bool_rec (inr (ne_symm ff_ne_tt)) (inl (refl tt)) b)
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a
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definition bor (a b : bool) :=
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bool_rec (bool_rec ff tt b) tt a
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theorem bor_tt_left (a : bool) : bor tt a = tt :=
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refl (bor tt a)
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infixl `||`:65 := bor
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theorem bor_tt_right (a : bool) : a || tt = tt :=
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induction_on a (refl (ff || tt)) (refl (tt || tt))
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theorem bor_ff_left (a : bool) : ff || a = a :=
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induction_on a (refl (ff || ff)) (refl (ff || tt))
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theorem bor_ff_right (a : bool) : a || ff = a :=
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induction_on a (refl (ff || ff)) (refl (tt || ff))
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theorem bor_id (a : bool) : a || a = a :=
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induction_on a (refl (ff || ff)) (refl (tt || tt))
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theorem bor_comm (a b : bool) : a || b = b || a :=
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induction_on a
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(induction_on b (refl (ff || ff)) (refl (ff || tt)))
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(induction_on b (refl (tt || ff)) (refl (tt || tt)))
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theorem bor_assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
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induction_on a
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(calc (ff || b) || c = b || c : {bor_ff_left b}
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... = ff || (b || c) : bor_ff_left (b || c)⁻¹)
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(calc (tt || b) || c = tt || c : {bor_tt_left b}
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... = tt : bor_tt_left c
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... = tt || (b || c) : bor_tt_left (b || c)⁻¹)
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theorem bor_to_or {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
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bool_rec
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(assume H : ff || b = tt,
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have Hb : b = tt, from (bor_ff_left b) ▸ H,
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or_inr Hb)
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(assume H, or_inl (refl tt))
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a
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definition band (a b : bool) :=
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bool_rec ff (bool_rec ff tt b) a
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infixl `&&`:75 := band
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theorem band_ff_left (a : bool) : ff && a = ff :=
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refl (ff && a)
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theorem band_tt_left (a : bool) : tt && a = a :=
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induction_on a (refl (tt && ff)) (refl (tt && tt))
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theorem band_ff_right (a : bool) : a && ff = ff :=
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induction_on a (refl (ff && ff)) (refl (tt && ff))
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theorem band_tt_right (a : bool) : a && tt = a :=
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induction_on a (refl (ff && tt)) (refl (tt && tt))
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theorem band_id (a : bool) : a && a = a :=
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induction_on a (refl (ff && ff)) (refl (tt && tt))
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theorem band_comm (a b : bool) : a && b = b && a :=
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induction_on a
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(induction_on b (refl (ff && ff)) (refl (ff && tt)))
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(induction_on b (refl (tt && ff)) (refl (tt && tt)))
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theorem band_assoc (a b c : bool) : (a && b) && c = a && (b && c) :=
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induction_on a
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(calc (ff && b) && c = ff && c : {band_ff_left b}
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... = ff : band_ff_left c
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... = ff && (b && c) : band_ff_left (b && c)⁻¹)
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(calc (tt && b) && c = b && c : {band_tt_left b}
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... = tt && (b && c) : band_tt_left (b && c)⁻¹)
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theorem band_eq_tt_elim_left {a b : bool} (H : a && b = tt) : a = tt :=
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or_elim (dichotomy a)
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(assume H0 : a = ff,
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absurd_elim (a = tt)
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(calc ff = ff && b : (band_ff_left _)⁻¹
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... = a && b : {H0⁻¹}
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... = tt : H)
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ff_ne_tt)
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(assume H1 : a = tt, H1)
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theorem band_eq_tt_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
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band_eq_tt_elim_left (trans (band_comm b a) H)
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definition bnot (a : bool) := bool_rec tt ff a
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prefix `!`:85 := bnot
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theorem bnot_bnot (a : bool) : !!a = a :=
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induction_on a (refl (!!ff)) (refl (!!tt))
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theorem bnot_false : !ff = tt := refl _
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theorem bnot_true : !tt = ff := refl _
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end |