124 lines
3.2 KiB
Text
124 lines
3.2 KiB
Text
/-
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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.eq
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open eq eq.ops decidable
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namespace bool
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local attribute bor [reducible]
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local attribute band [reducible]
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theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
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bool.cases_on b (or.inl rfl) (or.inr rfl)
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theorem cond_ff {A : Type} (t e : A) : cond ff t e = e :=
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rfl
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theorem cond_tt {A : Type} (t e : A) : cond tt t e = t :=
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rfl
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theorem eq_tt_of_ne_ff : ∀ {a : bool}, a ≠ ff → a = tt
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| @eq_tt_of_ne_ff tt H := rfl
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| @eq_tt_of_ne_ff ff H := absurd rfl H
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theorem eq_ff_of_ne_tt : ∀ {a : bool}, a ≠ tt → a = ff
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| @eq_ff_of_ne_tt tt H := absurd rfl H
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| @eq_ff_of_ne_tt ff H := rfl
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theorem absurd_of_eq_ff_of_eq_tt {B : Prop} {a : bool} (H₁ : a = ff) (H₂ : a = tt) : B :=
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absurd (H₁⁻¹ ⬝ H₂) ff_ne_tt
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theorem tt_bor (a : bool) : bor tt a = tt :=
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rfl
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notation a || b := bor a b
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theorem bor_tt (a : bool) : a || tt = tt :=
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bool.cases_on a rfl rfl
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theorem ff_bor (a : bool) : ff || a = a :=
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bool.cases_on a rfl rfl
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theorem bor_ff (a : bool) : a || ff = a :=
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bool.cases_on a rfl rfl
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theorem bor_self (a : bool) : a || a = a :=
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bool.cases_on a rfl rfl
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theorem bor.comm (a b : bool) : a || b = b || a :=
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by cases a; repeat (cases b | reflexivity)
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theorem bor.assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
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match a with
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| ff := by rewrite *ff_bor
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| tt := by rewrite *tt_bor
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end
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theorem or_of_bor_eq {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
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bool.rec_on a
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(assume H : ff || b = tt,
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have Hb : b = tt, from !ff_bor ▸ H,
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or.inr Hb)
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(assume H, or.inl rfl)
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theorem bor_inl {a b : bool} (H : a = tt) : a || b = tt :=
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by rewrite H
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theorem bor_inr {a b : bool} (H : b = tt) : a || b = tt :=
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bool.rec_on a (by rewrite H) (by rewrite H)
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theorem ff_band (a : bool) : ff && a = ff :=
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rfl
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theorem tt_band (a : bool) : tt && a = a :=
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bool.cases_on a rfl rfl
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theorem band_ff (a : bool) : a && ff = ff :=
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bool.cases_on a rfl rfl
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theorem band_tt (a : bool) : a && tt = a :=
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bool.cases_on a rfl rfl
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theorem band_self (a : bool) : a && a = a :=
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bool.cases_on a rfl rfl
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theorem band.comm (a b : bool) : a && b = b && a :=
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bool.cases_on a
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(bool.cases_on b rfl rfl)
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(bool.cases_on b rfl rfl)
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theorem band.assoc (a b c : bool) : (a && b) && c = a && (b && c) :=
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match a with
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| ff := by rewrite *ff_band
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| tt := by rewrite *tt_band
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end
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theorem band_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
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(calc ff = ff && b : ff_band
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... = a && b : H0
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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_intro {a b : bool} (H₁ : a = tt) (H₂ : b = tt) : a && b = tt :=
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by rewrite [H₁, H₂]
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theorem band_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
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band_elim_left (!band.comm ⬝ H)
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theorem bnot_bnot (a : bool) : bnot (bnot a) = a :=
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bool.cases_on a rfl rfl
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theorem bnot_false : bnot ff = tt :=
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rfl
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theorem bnot_true : bnot tt = ff :=
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rfl
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end bool
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