feat(library/data/set): use Prop instead of bool when defining set
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Leonardo de Moura
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1 changed files with 73 additions and 56 deletions
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@ -5,96 +5,113 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.set
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Author: Jeremy Avigad, Leonardo de Moura
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-/
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import data.bool
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open eq.ops bool
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import logic
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open eq.ops
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namespace set
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definition set (T : Type) :=
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T → bool
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T → Prop
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definition mem [reducible] {T : Type} (x : T) (s : set T) :=
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(s x) = tt
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s x
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notation e ∈ s := mem e s
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definition eqv {T : Type} (A B : set T) : Prop :=
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variable {T : Type}
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definition eqv (A B : set T) : Prop :=
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∀x, x ∈ A ↔ x ∈ B
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notation a ∼ b := eqv a b
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theorem eqv_refl {T : Type} (A : set T) : A ∼ A :=
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theorem eqv_refl (A : set T) : A ∼ A :=
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take x, iff.rfl
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theorem eqv_symm {T : Type} {A B : set T} (H : A ∼ B) : B ∼ A :=
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theorem eqv_symm {A B : set T} (H : A ∼ B) : B ∼ A :=
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take x, iff.symm (H x)
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theorem eqv_trans {T : Type} {A B C : set T} (H1 : A ∼ B) (H2 : B ∼ C) : A ∼ C :=
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theorem eqv_trans {A B C : set T} (H1 : A ∼ B) (H2 : B ∼ C) : A ∼ C :=
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take x, iff.trans (H1 x) (H2 x)
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definition empty [reducible] {T : Type} : set T :=
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λx, ff
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definition empty [reducible] : set T :=
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λx, false
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notation `∅` := empty
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theorem mem_empty {T : Type} (x : T) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, absurd H ff_ne_tt
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theorem mem_empty (x : T) : ¬ (x ∈ ∅) :=
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assume H : x ∈ ∅, H
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definition univ {T : Type} : set T :=
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λx, tt
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definition univ : set T :=
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λx, true
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theorem mem_univ {T : Type} (x : T) : x ∈ univ :=
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rfl
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theorem mem_univ (x : T) : x ∈ univ :=
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trivial
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definition inter [reducible] {T : Type} (A B : set T) : set T :=
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λx, A x && B x
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definition inter [reducible] (A B : set T) : set T :=
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λx, x ∈ A ∧ x ∈ B
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notation a ∩ b := inter a b
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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,
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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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theorem mem_inter (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B) :=
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!iff.refl
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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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theorem inter_id (A : set T) : A ∩ A ∼ A :=
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take x, iff.intro
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(assume H, and.elim_left H)
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(assume H, and.intro H H)
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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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theorem inter_empty_right (A : set T) : A ∩ ∅ ∼ ∅ :=
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take x, iff.intro
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(assume H, and.elim_right H)
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(assume H, false.elim H)
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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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theorem inter_empty_left (A : set T) : ∅ ∩ A ∼ ∅ :=
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take x, iff.intro
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(assume H, and.elim_left H)
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(assume H, false.elim H)
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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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theorem inter_comm (A B : set T) : A ∩ B ∼ B ∩ A :=
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take x, !and.comm
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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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theorem inter_assoc (A B C : set T) : (A ∩ B) ∩ C ∼ A ∩ (B ∩ C) :=
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take x, !and.assoc
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definition union [reducible] {T : Type} (A B : set T) : set T :=
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λx, A x || B x
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definition union [reducible] (A B : set T) : set T :=
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λx, x ∈ A ∨ x ∈ B
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notation a ∪ b := union a b
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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.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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(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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theorem mem_union (x : T) (A B : set T) : x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B) :=
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!iff.refl
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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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theorem union_id (A : set T) : A ∪ A ∼ A :=
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take x, iff.intro
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(assume H,
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match H with
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| or.inl H₁ := H₁
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| or.inr H₂ := H₂
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end)
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(assume H, or.inl H)
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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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theorem union_empty_right (A : set T) : A ∪ ∅ ∼ A :=
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take x, iff.intro
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(assume H, match H with
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| or.inl H₁ := H₁
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| or.inr H₂ := false.elim H₂
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end)
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(assume H, or.inl H)
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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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theorem union_empty_left (A : set T) : ∅ ∪ A ∼ A :=
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take x, iff.intro
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(assume H, match H with
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| or.inl H₁ := false.elim H₁
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| or.inr H₂ := H₂
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end)
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(assume H, or.inr H)
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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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theorem union_comm (A B : set T) : A ∪ B ∼ B ∪ A :=
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take x, or.comm
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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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theorem union_assoc (A B C : set T) : (A ∪ B) ∪ C ∼ A ∪ (B ∪ C) :=
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take x, or.assoc
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definition subset (A B : set T) := ∀ x, x ∈ A → x ∈ B
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infix `⊆`:50 := subset
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definition eqv_of_subset (A B : set T) : A ⊆ B → B ⊆ A → A ∼ B :=
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assume H₁ H₂, take x, iff.intro (H₁ x) (H₂ x)
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end set
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