refactor(library/data/bool): use new style
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Leonardo de Moura
parent
60d8369688
commit
317e910054
4 changed files with 72 additions and 72 deletions
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@ -22,115 +22,115 @@ namespace bool
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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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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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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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(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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... = false : !cond.ff)
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true_ne_false
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definition or (a b : bool) :=
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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 or_tt_left (a : bool) : or tt a = tt :=
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theorem bor.tt_left (a : bool) : bor tt a = tt :=
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rfl
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infixl `||` := or
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infixl `||` := bor
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theorem or_tt_right (a : bool) : a || tt = tt :=
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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 or_ff_left (a : bool) : ff || a = a :=
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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 or_ff_right (a : bool) : a || ff = a :=
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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 or_id (a : bool) : a || a = a :=
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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 or_comm (a b : bool) : a || b = b || a :=
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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 or_assoc (a b c : bool) : (a || b) || c = a || (b || c) :=
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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 : {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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(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 or_to_or {a b : bool} : a || b = tt → a = tt ∨ b = tt :=
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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 (or_ff_left b) ▸ H,
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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 and (a b : bool) :=
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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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infixl `&&` := and
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infixl `&&` := band
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theorem and_ff_left (a : bool) : ff && a = ff :=
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theorem band.ff_left (a : bool) : ff && a = ff :=
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rfl
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theorem and_tt_left (a : bool) : tt && a = a :=
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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 and_ff_right (a : bool) : a && ff = ff :=
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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 and_tt_right (a : bool) : a && tt = a :=
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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 and_id (a : bool) : a && a = a :=
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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 and_comm (a b : bool) : a && b = b && a :=
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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 and_assoc (a b c : bool) : (a && b) && c = a && (b && c) :=
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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 : {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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(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 and_eq_tt_elim_left {a b : bool} (H : a && b = tt) : a = tt :=
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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 : (and_ff_left _)⁻¹
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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 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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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 not (a : bool) :=
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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) : not (not a) = a :=
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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 : not ff = tt :=
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theorem bnot.false : bnot ff = tt :=
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rfl
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theorem bnot_true : not tt = ff :=
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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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@ -148,7 +148,7 @@ namespace num
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(λp, calc
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pred (succ (pos p)) = pred (pos (pos_num.succ p)) : rfl
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... = cond ff zero (pos (pos_num.pred (pos_num.succ p))) : {succ_not_is_one}
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... = pos (pos_num.pred (pos_num.succ p)) : cond_ff _ _
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... = pos (pos_num.pred (pos_num.succ p)) : !cond.ff
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... = pos p : {pos_num.pred_succ})
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definition add (a b : num) : num :=
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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 (and_eq_tt_elim_left H) (and_eq_tt_elim_right H))
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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,
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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 ▸ and_tt_left tt)
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show A x && B x = tt, from e1⁻¹ ▸ e2⁻¹ ▸ band.tt_left tt)
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theorem inter_id {T : Type} (A : set T) : A ∩ A ∼ A :=
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take x, and_id (A x) ▸ iff.rfl
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take x, band.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, and_ff_right (A x) ▸ iff.rfl
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take x, band.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, and_ff_left (A x) ▸ iff.rfl
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take x, band.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, and_comm (A x) (B x) ▸ iff.rfl
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take x, band.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, and_assoc (A x) (B x) (C x) ▸ iff.rfl
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take x, band.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, or_to_or H)
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(assume H, bor.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 ▸ or_tt_left (B x))
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show A x || B x = tt, from Ha⁻¹ ▸ bor.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 ▸ or_tt_right (A x)))
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show A x || B x = tt, from Hb⁻¹ ▸ bor.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, or_id (A x) ▸ iff.rfl
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take x, bor.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, or_ff_right (A x) ▸ iff.rfl
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take x, bor.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, or_ff_left (A x) ▸ iff.rfl
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take x, bor.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, or_comm (A x) (B x) ▸ iff.rfl
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take x, bor.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, or_assoc (A x) (B x) (C x) ▸ iff.rfl
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take x, bor.assoc (A x) (B x) (C x) ▸ iff.rfl
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end set
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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.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.band.tt_left|∀ (a : bool), eq (bool.band bool.tt a) a
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bool.tt|bool
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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.band.eq_tt_elim_right|eq (bool.band ?a ?b) bool.tt → eq ?b 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.tt_right|∀ (a : bool), eq (bool.band a bool.tt) a
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bool.bor.tt_right|∀ (a : bool), eq (bool.bor a bool.tt) bool.tt
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bool.bor.tt_left|∀ (a : bool), eq (bool.bor bool.tt a) bool.tt
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bool.ff_ne_tt|not (eq bool.ff bool.tt)
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bool.cond.tt|∀ (t e : ?A), eq (bool.cond bool.tt t e) t
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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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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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band.tt_left|∀ (a : bool), eq (band tt a) a
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band.eq_tt_elim_right|eq (band ?a ?b) tt → eq ?b tt
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band.eq_tt_elim_left|eq (band ?a ?b) tt → eq ?a tt
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band.tt_right|∀ (a : bool), eq (band a tt) a
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bor.tt_right|∀ (a : bool), eq (bor a tt) tt
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bor.tt_left|∀ (a : bool), eq (bor tt a) tt
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ff_ne_tt|not (eq ff tt)
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cond.tt|∀ (t e : ?A), eq (cond tt t e) t
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-- ENDFINDP
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