lean2/library/standard/set.lean

--- Copyright (c) 2014 Jeremy Avigad. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Jeremy Avigad, Leonardo de Moura
import logic funext bool
using eq_proofs bool

namespace set
definition set (T : Type) := T → bool
definition mem {T : Type} (x : T) (s : set T) := (s x) = '1
infix `∈`:50 := mem

section
parameter {T : Type}

definition empty : set T := λx, '0
notation `∅`:max := empty

theorem mem_empty (x : T) : ¬ (x ∈ ∅)
:= assume H : x ∈ ∅, absurd H b0_ne_b1

definition univ : set T := λx, '1

theorem mem_univ (x : T) : x ∈ univ
:= refl _

definition inter (A B : set T) : set T := λx, A x && B x
infixl `∩`:70 := inter

theorem mem_inter (x : T) (A B : set T) : x ∈ A ∩ B ↔ (x ∈ A ∧ x ∈ B)
:= iff_intro
     (assume H, and_intro (band_eq_b1_elim_left H) (band_eq_b1_elim_right H))
     (assume H,
       have e1 : A x = '1, from and_elim_left H,
       have e2 : B x = '1, from and_elim_right H,
       show A x && B x = '1, from e1⁻¹ ▸ e2⁻¹ ▸ band_b1_left '1)

theorem inter_comm (A B : set T) : A ∩ B = B ∩ A
:= funext (λx, band_comm (A x) (B x))

theorem inter_assoc (A B C : set T) : (A ∩ B) ∩ C = A ∩ (B ∩ C)
:= funext (λx, band_assoc (A x) (B x) (C x))

definition union (A B : set T) : set T := λx, A x || B x
infixl `∪`:65 := union

theorem mem_union (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 = '1,
        show A x || B x = '1, from Ha⁻¹ ▸ bor_b1_left (B x))
       (assume Hb : B x = '1,
        show A x || B x = '1, from Hb⁻¹ ▸ bor_b1_right (A x)))

theorem union_comm (A B : set T) : A ∪ B = B ∪ A
:= funext (λx, bor_comm (A x) (B x))

theorem union_assoc (A B C : set T) : (A ∪ B) ∪ C = A ∪ (B ∪ C)
:= funext (λx, bor_assoc (A x) (B x) (C x))

end
end