144 lines
4 KiB
Text
144 lines
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 general_notation
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import logic.connectives logic.decidable logic.inhabited
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open eq eq.ops decidable
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inductive bool : Type :=
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ff : bool,
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tt : bool
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namespace bool
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protected definition rec_on {C : bool → Type} (b : bool) (H₁ : C ff) (H₂ : C tt) : C b :=
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rec H₁ H₂ b
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protected definition cases_on {p : bool → Prop} (b : bool) (H₁ : p ff) (H₂ : p tt) : p b :=
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rec H₁ H₂ b
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definition cond {A : Type} (b : bool) (t e : A) :=
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rec_on b e t
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theorem dichotomy (b : bool) : b = ff ∨ b = tt :=
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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 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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definition bor (a b : bool) :=
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rec_on a (rec_on b ff tt) tt
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theorem bor.tt_left (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_right (a : bool) : a || tt = tt :=
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cases_on a rfl rfl
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theorem bor.ff_left (a : bool) : ff || a = a :=
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cases_on a rfl rfl
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theorem bor.ff_right (a : bool) : a || ff = a :=
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cases_on a rfl rfl
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theorem bor.id (a : bool) : a || a = a :=
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cases_on a rfl rfl
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theorem bor.comm (a b : bool) : a || b = b || a :=
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cases_on a
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(cases_on b rfl rfl)
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(cases_on b rfl rfl)
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theorem bor.assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
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cases_on a
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(calc (ff || b) || c = b || c : {!bor.ff_left}
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... = ff || (b || c) : !bor.ff_left⁻¹)
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(calc (tt || b) || c = tt || c : {!bor.tt_left}
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... = tt : !bor.tt_left
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... = tt || (b || c) : !bor.tt_left⁻¹)
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theorem bor.to_or {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
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rec_on a
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(assume H : ff || b = tt,
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have Hb : b = tt, from !bor.ff_left ▸ H,
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or.inr Hb)
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(assume H, or.inl rfl)
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definition band (a b : bool) :=
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rec_on a ff (rec_on b ff tt)
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notation a && b := band a b
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theorem band.ff_left (a : bool) : ff && a = ff :=
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rfl
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theorem band.tt_left (a : bool) : tt && a = a :=
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cases_on a rfl rfl
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theorem band.ff_right (a : bool) : a && ff = ff :=
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cases_on a rfl rfl
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theorem band.tt_right (a : bool) : a && tt = a :=
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cases_on a rfl rfl
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theorem band.id (a : bool) : a && a = a :=
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cases_on a rfl rfl
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theorem band.comm (a b : bool) : a && b = b && a :=
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cases_on a
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(cases_on b rfl rfl)
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(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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cases_on a
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(calc (ff && b) && c = ff && c : {!band.ff_left}
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... = ff : !band.ff_left
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... = ff && (b && c) : !band.ff_left⁻¹)
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(calc (tt && b) && c = b && c : {!band.tt_left}
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... = tt && (b && c) : !band.tt_left⁻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
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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 (!band.comm ⬝ H)
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definition bnot (a : bool) :=
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rec_on a tt ff
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theorem bnot.bnot (a : bool) : bnot (bnot a) = a :=
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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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protected definition is_inhabited [instance] : inhabited bool :=
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inhabited.mk ff
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protected definition has_decidable_eq [instance] : decidable_eq bool :=
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take a b : bool,
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rec_on a
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(rec_on b (inl rfl) (inr ff_ne_tt))
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(rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl))
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end bool
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