feat(library/data/bool): do not use ! as notation for bnot, rename band/bor -> and/or
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Leonardo de Moura
parent
72beb438e6
commit
bc6ebf34be
4 changed files with 66 additions and 69 deletions
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@ -1,7 +1,7 @@
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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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import general_notation
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import logic.core.connectives logic.core.decidable logic.core.inhabited
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open eq_ops eq decidable
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@ -35,104 +35,102 @@ namespace bool
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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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definition or (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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theorem or_tt_left (a : bool) : or tt a = tt :=
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rfl
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infixl `||` := bor
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infixl `||` := or
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theorem bor_tt_right (a : bool) : a || tt = tt :=
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theorem or_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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theorem or_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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theorem or_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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theorem or_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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theorem or_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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theorem or_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 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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(calc (ff || b) || c = b || c : {or_ff_left b}
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... = ff || (b || c) : or_ff_left (b || c)⁻¹)
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(calc (tt || b) || c = tt || c : {or_tt_left b}
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... = tt : or_tt_left c
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... = tt || (b || c) : or_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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theorem or_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 b) ▸ H,
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have Hb : b = tt, from (or_ff_left b) ▸ 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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definition and (a b : bool) :=
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rec_on a ff (rec_on b ff tt)
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infixl `&&` := band
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infixl `&&` := and
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theorem band_ff_left (a : bool) : ff && a = ff :=
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theorem and_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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theorem and_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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theorem and_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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theorem and_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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theorem and_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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theorem and_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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theorem and_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 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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(calc (ff && b) && c = ff && c : {and_ff_left b}
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... = ff : and_ff_left c
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... = ff && (b && c) : and_ff_left (b && c)⁻¹)
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(calc (tt && b) && c = b && c : {and_tt_left b}
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... = tt && (b && c) : and_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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theorem and_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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(calc ff = ff && b : (and_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 b a ⬝ H)
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theorem and_eq_tt_elim_right {a b : bool} (H : a && b = tt) : b = tt :=
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and_eq_tt_elim_left (and_comm b a ⬝ H)
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definition bnot (a : bool) :=
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definition not (a : bool) :=
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rec_on a tt ff
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notation `!` x:max := bnot x
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theorem bnot_bnot (a : bool) : !!a = a :=
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theorem bnot_bnot (a : bool) : not (not a) = a :=
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cases_on a rfl rfl
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theorem bnot_false : !ff = tt :=
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theorem bnot_false : not ff = tt :=
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rfl
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theorem bnot_true : !tt = ff :=
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theorem bnot_true : not tt = ff :=
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rfl
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protected theorem is_inhabited [instance] : inhabited bool :=
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@ -45,26 +45,26 @@ infixl `∩` := inter
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theorem mem_inter {T : Type} (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B) :=
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iff.intro
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(assume H, and.intro (band_eq_tt_elim_left H) (band_eq_tt_elim_right H))
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(assume H, and.intro (and_eq_tt_elim_left H) (and_eq_tt_elim_right H))
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(assume H,
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have e1 : A x = tt, from and.elim_left H,
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have e2 : B x = tt, from and.elim_right H,
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show A x && B x = tt, from e1⁻¹ ▸ e2⁻¹ ▸ band_tt_left tt)
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show A x && B x = tt, from e1⁻¹ ▸ e2⁻¹ ▸ and_tt_left tt)
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theorem inter_id {T : Type} (A : set T) : A ∩ A ∼ A :=
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take x, band_id (A x) ▸ iff.rfl
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take x, and_id (A x) ▸ iff.rfl
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theorem inter_empty_right {T : Type} (A : set T) : A ∩ ∅ ∼ ∅ :=
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take x, band_ff_right (A x) ▸ iff.rfl
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take x, and_ff_right (A x) ▸ iff.rfl
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theorem inter_empty_left {T : Type} (A : set T) : ∅ ∩ A ∼ ∅ :=
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take x, band_ff_left (A x) ▸ iff.rfl
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take x, and_ff_left (A x) ▸ iff.rfl
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theorem inter_comm {T : Type} (A B : set T) : A ∩ B ∼ B ∩ A :=
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take x, band_comm (A x) (B x) ▸ iff.rfl
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take x, and_comm (A x) (B x) ▸ iff.rfl
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theorem inter_assoc {T : Type} (A B C : set T) : (A ∩ B) ∩ C ∼ A ∩ (B ∩ C) :=
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take x, band_assoc (A x) (B x) (C x) ▸ iff.rfl
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take x, and_assoc (A x) (B x) (C x) ▸ iff.rfl
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definition union {T : Type} (A B : set T) : set T :=
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λx, A x || B x
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@ -72,26 +72,26 @@ infixl `∪` := union
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theorem mem_union {T : Type} (x : T) (A B : set T) : x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B) :=
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iff.intro
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(assume H, bor_to_or H)
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(assume H, or_to_or H)
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(assume H, or.elim H
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(assume Ha : A x = tt,
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show A x || B x = tt, from Ha⁻¹ ▸ bor_tt_left (B x))
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show A x || B x = tt, from Ha⁻¹ ▸ or_tt_left (B x))
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(assume Hb : B x = tt,
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show A x || B x = tt, from Hb⁻¹ ▸ bor_tt_right (A x)))
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show A x || B x = tt, from Hb⁻¹ ▸ or_tt_right (A x)))
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theorem union_id {T : Type} (A : set T) : A ∪ A ∼ A :=
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take x, bor_id (A x) ▸ iff.rfl
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take x, or_id (A x) ▸ iff.rfl
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theorem union_empty_right {T : Type} (A : set T) : A ∪ ∅ ∼ A :=
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take x, bor_ff_right (A x) ▸ iff.rfl
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take x, or_ff_right (A x) ▸ iff.rfl
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theorem union_empty_left {T : Type} (A : set T) : ∅ ∪ A ∼ A :=
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take x, bor_ff_left (A x) ▸ iff.rfl
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take x, or_ff_left (A x) ▸ iff.rfl
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theorem union_comm {T : Type} (A B : set T) : A ∪ B ∼ B ∪ A :=
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take x, bor_comm (A x) (B x) ▸ iff.rfl
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take x, or_comm (A x) (B x) ▸ iff.rfl
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theorem union_assoc {T : Type} (A B C : set T) : (A ∪ B) ∪ C ∼ A ∪ (B ∪ C) :=
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take x, bor_assoc (A x) (B x) (C x) ▸ iff.rfl
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take x, or_assoc (A x) (B x) (C x) ▸ iff.rfl
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end set
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@ -65,7 +65,6 @@ precedence `>`:50
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precedence `&&`:70 -- infixl
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precedence `||`:65 -- infixl
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precedence `!`:85 -- boolean negation, follow by rbp 100
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-- ### set operations
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@ -3,26 +3,26 @@
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-- BEGINWAIT
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-- ENDWAIT
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-- BEGINFINDP
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bool.bor_tt_left|∀ (a : bool), eq (bool.bor bool.tt a) bool.tt
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bool.and_tt_left|∀ (a : bool), eq (bool.and bool.tt a) a
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bool.and_tt_right|∀ (a : bool), eq (bool.and a bool.tt) a
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bool.tt|bool
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bool.bor_tt_right|∀ (a : bool), eq (bool.bor a bool.tt) bool.tt
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bool.band_tt_left|∀ (a : bool), eq (bool.band bool.tt a) a
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bool.band_tt_right|∀ (a : bool), eq (bool.band a bool.tt) a
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bool.or_tt_left|∀ (a : bool), eq (bool.or bool.tt a) bool.tt
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bool.and_eq_tt_elim_left|eq (bool.and ?a ?b) bool.tt → eq ?a bool.tt
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bool.and_eq_tt_elim_right|eq (bool.and ?a ?b) bool.tt → eq ?b bool.tt
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bool.cond_tt|∀ (t e : ?A), eq (bool.cond bool.tt t e) t
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bool.or_tt_right|∀ (a : bool), eq (bool.or a bool.tt) bool.tt
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bool.ff_ne_tt|not (eq bool.ff bool.tt)
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bool.band_eq_tt_elim_left|eq (bool.band ?a ?b) bool.tt → eq ?a bool.tt
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bool.band_eq_tt_elim_right|eq (bool.band ?a ?b) bool.tt → eq ?b bool.tt
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-- ENDFINDP
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-- BEGINWAIT
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-- ENDWAIT
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-- BEGINFINDP
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tt|bool
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bor_tt_left|∀ (a : bool), eq (bor tt a) tt
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bor_tt_right|∀ (a : bool), eq (bor a tt) tt
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band_tt_left|∀ (a : bool), eq (band tt a) a
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band_tt_right|∀ (a : bool), eq (band a tt) a
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and_tt_left|∀ (a : bool), eq (and tt a) a
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and_tt_right|∀ (a : bool), eq (and a tt) a
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or_tt_left|∀ (a : bool), eq (or tt a) tt
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and_eq_tt_elim_left|eq (and ?a ?b) tt → eq ?a tt
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and_eq_tt_elim_right|eq (and ?a ?b) tt → eq ?b tt
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cond_tt|∀ (t e : ?A), eq (cond tt t e) t
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or_tt_right|∀ (a : bool), eq (or a tt) tt
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ff_ne_tt|not (eq ff tt)
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band_eq_tt_elim_left|eq (band ?a ?b) tt → eq ?a tt
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band_eq_tt_elim_right|eq (band ?a ?b) tt → eq ?b tt
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-- ENDFINDP
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