<commit_before>/-
Copyright (c) 2014 <NAME>. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: data.set
Author: <NAME>, <NAME>
-/

import data.bool
open eq.ops bool

namespace set
definition set (T : Type) :=
T → bool
definition mem [reducible] {T : Type} (x : T) (s : set T) :=
(s x) = tt
notation e ∈ s := mem e s

definition eqv {T : Type} (A B : set T) : Prop :=
∀x, x ∈ A ↔ x ∈ B
notation a ∼ b := eqv a b

theorem eqv_refl {T : Type} (A : set T) : A ∼ A :=
take x, iff.rfl

theorem eqv_symm {T : Type} {A B : set T} (H : A ∼ B) : B ∼ A :=
take x, iff.symm (H x)

theorem eqv_trans {T : Type} {A B C : set T} (H1 : A ∼ B) (H2 : B ∼ C) : A ∼ C :=
take x, iff.trans (H1 x) (H2 x)

definition empty [reducible] {T : Type} : set T :=
λx, ff
notation `∅` := empty

theorem mem_empty {T : Type} (x : T) : ¬ (x ∈ ∅) :=
assume H : x ∈ ∅, absurd H ff_ne_tt

definition univ {T : Type} : set T :=
λx, tt

theorem mem_univ {T : Type} (x : T) : x ∈ univ :=
rfl

definition inter [reducible] {T : Type} (A B : set T) : set T :=
λx, A x && B x
notation a ∩ b := inter a b

theorem mem_inter {T : Type} (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B) :=
iff.intro
  (assume H, and.intro (band.eq_tt_elim_left H) (band.eq_tt_elim_right H))
  (assume H,
    have e1 : A x = tt, from and.elim_left H,
    have e2 : B x = tt, from and.elim_right H,
    show A x && B x = tt, from e1⁻¹ ▸ e2⁻¹ ▸ band.tt_left tt)

theorem inter_id {T : Type} (A : set T) : A ∩ A ∼ A :=
take x, band.id (A x) ▸ iff.rfl

theorem inter_empty_right {T : Type} (A : set T) : A ∩ ∅ ∼ ∅ :=
take x, band.ff_right (A x) ▸ iff.rfl

theorem inter_empty_left {T : Type} (A : set T) : ∅ ∩ A ∼ ∅ :=
take x, band.ff_left (A x) ▸ iff.rfl

theorem inter_comm {T : Type} (A B : set T) : A ∩ B ∼ B ∩ A :=
take x, band.comm (A x) (B x) ▸ iff.rfl

theorem inter_assoc {T : Type} (A B C : set T) : (A ∩ B) ∩ C ∼ A ∩ (B ∩ C) :=
take x, band.assoc (A x) (B x) (C x) ▸ iff.rfl

definition union [reducible] {T : Type} (A B : set T) : set T :=
λx, A x || B x
notation a ∪ b := union a b

theorem mem_union {T : Type} (x : T) (A B : set T) : x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B) :=
iff.intro
  (assume H, bor.to_or H)
  (assume H, or.elim H
    (assume Ha : A x = tt,
      show A x || B x = tt, from Ha⁻¹ ▸ bor.tt_left (B x))
    (assume Hb : B x = tt,
      show A x || B x = tt, from Hb⁻¹ ▸ bor.tt_right (A x)))

theorem union_id {T : Type} (A : set T) : A ∪ A ∼ A :=
take x, bor.id (A x) ▸ iff.rfl

theorem union_empty_right {T : Type} (A : set T) : A ∪ ∅ ∼ A :=
take x, bor.ff_right (A x) ▸ iff.rfl

theorem union_empty_left {T : Type} (A : set T) : ∅ ∪ A ∼ A :=
take x, bor.ff_left (A x) ▸ iff.rfl

theorem union_comm {T : Type} (A B : set T) : A ∪ B ∼ B ∪ A :=
take x, bor.comm (A x) (B x) ▸ iff.rfl

theorem union_assoc {T : Type} (A B C : set T) : (A ∪ B) ∪ C ∼ A ∪ (B ∪ C) :=
take x, bor.assoc (A x) (B x) (C x) ▸ iff.rfl

end set<commit_msg>feat(library/data/set): use Prop instead of bool when defining set<commit_after>/-
Copyright (c) 2014 <NAME>. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: data.set
Author: <NAME>, <NAME>
-/
import logic
open eq.ops

namespace set
definition set (T : Type) :=
T → Prop
definition mem [reducible] {T : Type} (x : T) (s : set T) :=
s x
notation e ∈ s := mem e s

variable {T : Type}
definition eqv (A B : set T) : Prop :=
∀x, x ∈ A ↔ x ∈ B
notation a ∼ b := eqv a b

theorem eqv_refl (A : set T) : A ∼ A :=
take x, iff.rfl

theorem eqv_symm {A B : set T} (H : A ∼ B) : B ∼ A :=
take x, iff.symm (H x)

theorem eqv_trans {A B C : set T} (H1 : A ∼ B) (H2 : B ∼ C) : A ∼ C :=
take x, iff.trans (H1 x) (H2 x)

definition empty [reducible] : set T :=
λx, false
notation `∅` := empty

theorem mem_empty (x : T) : ¬ (x ∈ ∅) :=
assume H : x ∈ ∅, H

definition univ : set T :=
λx, true

theorem mem_univ (x : T) : x ∈ univ :=
trivial

definition inter [reducible] (A B : set T) : set T :=
λx, x ∈ A ∧ x ∈ B
notation a ∩ b := inter a b

theorem mem_inter (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B) :=
!iff.refl

theorem inter_id (A : set T) : A ∩ A ∼ A :=
take x, iff.intro
  (assume H, and.elim_left H)
  (assume H, and.intro H H)

theorem inter_empty_right (A : set T) : A ∩ ∅ ∼ ∅ :=
take x, iff.intro
  (assume H, and.elim_right H)
  (assume H, false.elim H)

theorem inter_empty_left (A : set T) : ∅ ∩ A ∼ ∅ :=
take x, iff.intro
  (assume H, and.elim_left H)
  (assume H, false.elim H)

theorem inter_comm (A B : set T) : A ∩ B ∼ B ∩ A :=
take x, !and.comm

theorem inter_assoc (A B C : set T) : (A ∩ B) ∩ C ∼ A ∩ (B ∩ C) :=
take x, !and.assoc

definition union [reducible] (A B : set T) : set T :=
λx, x ∈ A ∨ x ∈ B
notation a ∪ b := union a b

theorem mem_union (x : T) (A B : set T) : x ∈ A ∪ B ↔ (x ∈ A ∨ x ∈ B) :=
!iff.refl

theorem union_id (A : set T) : A ∪ A ∼ A :=
take x, iff.intro
  (assume H,
    match H with
    | or.inl H₁ := H₁
    | or.inr H₂ := H₂
    end)
  (assume H, or.inl H)

theorem union_empty_right (A : set T) : A ∪ ∅ ∼ A :=
take x, iff.intro
  (assume H, match H with
             | or.inl H₁ := H₁
             | or.inr H₂ := false.elim H₂
             end)
  (assume H, or.inl H)

theorem union_empty_left (A : set T) : ∅ ∪ A ∼ A :=
take x, iff.intro
  (assume H, match H with
             | or.inl H₁ := false.elim H₁
             | or.inr H₂ := H₂
             end)
  (assume H, or.inr H)

theorem union_comm (A B : set T) : A ∪ B ∼ B ∪ A :=
take x, or.comm

theorem union_assoc (A B C : set T) : (A ∪ B) ∪ C ∼ A ∪ (B ∪ C) :=
take x, or.assoc

definition subset (A B : set T) := ∀ x, x ∈ A → x ∈ B
infix `⊆`:50 := subset

definition eqv_of_subset (A B : set T) : A ⊆ B → B ⊆ A → A ∼ B :=
assume H₁ H₂, take x, iff.intro (H₁ x) (H₂ x)

end set
{'fraction_non_alphanumeric': 0.1674765706351961, 'fraction_numerical': 0.004512322110378341, 'mean_word_length': 2.471686746987952, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '4972250', 'n_tokens_mistral': 2786, 'n_tokens_neox': 2599, 'n_words': 1136}