121 lines
3.4 KiB
Text
121 lines
3.4 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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import logic decidable
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using eq_proofs
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namespace bool
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inductive bool : Type :=
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| b0 : bool
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| b1 : bool
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notation `'0`:max := b0
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notation `'1`:max := b1
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theorem induction_on {p : bool → Prop} (b : bool) (H0 : p '0) (H1 : p '1) : 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 b0
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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 = '0 ∨ b = '1
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:= induction_on b (or_intro_left _ (refl '0)) (or_intro_right _ (refl '1))
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theorem cond_b0 {A : Type} (t e : A) : cond '0 t e = e
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:= refl (cond '0 t e)
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theorem cond_b1 {A : Type} (t e : A) : cond '1 t e = t
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:= refl (cond '1 t e)
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theorem b0_ne_b1 : ¬ '0 = '1
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:= assume H : '0 = '1, absurd
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(calc true = cond '1 true false : (cond_b1 _ _)⁻¹
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... = cond '0 true false : {H⁻¹}
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... = false : cond_b0 _ _)
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true_ne_false
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definition bor (a b : bool)
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:= bool_rec (bool_rec '0 '1 b) '1 a
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theorem bor_b1_left (a : bool) : bor '1 a = '1
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:= refl (bor '1 a)
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infixl `||`:65 := bor
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theorem bor_b1_right (a : bool) : a || '1 = '1
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:= induction_on a (refl ('0 || '1)) (refl ('1 || '1))
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theorem bor_b0_left (a : bool) : '0 || a = a
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:= induction_on a (refl ('0 || '0)) (refl ('0 || '1))
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theorem bor_b0_right (a : bool) : a || '0 = a
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:= induction_on a (refl ('0 || '0)) (refl ('1 || '0))
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theorem bor_id (a : bool) : a || a = a
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:= induction_on a (refl ('0 || '0)) (refl ('1 || '1))
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theorem bor_swap (a b : bool) : a || b = b || a
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:= induction_on a
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(induction_on b (refl ('0 || '0)) (refl ('0 || '1)))
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(induction_on b (refl ('1 || '0)) (refl ('1 || '1)))
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definition band (a b : bool)
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:= bool_rec '0 (bool_rec '0 '1 b) a
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infixl `&&`:75 := band
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theorem band_b0_left (a : bool) : '0 && a = '0
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:= refl ('0 && a)
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theorem band_b1_left (a : bool) : '1 && a = a
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:= induction_on a (refl ('1 && '0)) (refl ('1 && '1))
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theorem band_b0_right (a : bool) : a && '0 = '0
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:= induction_on a (refl ('0 && '0)) (refl ('1 && '0))
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theorem band_b1_right (a : bool) : a && '1 = a
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:= induction_on a (refl ('0 && '1)) (refl ('1 && '1))
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theorem band_id (a : bool) : a && a = a
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:= induction_on a (refl ('0 && '0)) (refl ('1 && '1))
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theorem band_swap (a b : bool) : a && b = b && a
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:= induction_on a
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(induction_on b (refl ('0 && '0)) (refl ('0 && '1)))
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(induction_on b (refl ('1 && '0)) (refl ('1 && '1)))
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theorem band_eq_b1_elim_left {a b : bool} (H : a && b = '1) : a = '1
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:= or_elim (dichotomy a)
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(assume H0 : a = '0,
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absurd_elim (a = '1)
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(calc '0 = '0 && b : (band_b0_left _)⁻¹
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... = a && b : {H0⁻¹}
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... = '1 : H)
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b0_ne_b1)
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(assume H1 : a = '1, H1)
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theorem band_eq_b1_elim_right {a b : bool} (H : a && b = '1) : b = '1
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:= band_eq_b1_elim_left (trans (band_swap b a) H)
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definition bnot (a : bool) := bool_rec '1 '0 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 (!!'0)) (refl (!!'1))
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theorem bnot_false : !'0 = '1
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:= refl _
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theorem bnot_true : !'1 = '0
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:= refl _
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using decidable
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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 '0)) (inr b0_ne_b1) b)
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(bool_rec (inr (not_eq_symm b0_ne_b1)) (inl (refl '1)) b)
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a
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end
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