lean4game/server/adam/Adam/Levels/SetTheory/T04_xx.lean

import Adam.Metadata

import Mathlib
import Duper.Tactic


import Mathlib.Order.Disjoint

set_option tactic.hygienic false

Game "Adam"
World "SetTheory"
Level 4

Title "Mengen"

Introduction
"
`∈ ∉ ⊆ ⊂ ⋃ ⋂`
"

open Set

-- open_locale cardinal

variables {X : Type _} (x : X) (A B : Set X)

#check A.Nonempty
#check Nonempty A
#check insert x A -- {x} ∪ A
-- #check disjoint A B -- needs Mathlib.Order.Disjoint
#check A ∆ B
#check Nontrivial A

#check ({2} : Set ℕ)

example : ({2} : Set ℕ) ⊆ {4, 2, 3} := by
  simp only [mem_singleton_iff, mem_insert_iff, or_self, singleton_subset_iff, or_false, or_true]

example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
  intro n hn
  simp only [mem_insert_iff, mem_singleton_iff] at *
  tauto

example : {2, 3, 5} ⊆ (univ : Set ℕ) := by
  tauto

example : 3 ∈ ({4, 2, 5, 7, 3} : Set ℕ) := by
  tauto

example : ({4, 9} : Set ℕ) = Set.insert 4 {9} := by
  rfl




#check Finset.card

variable (A : Finset ℕ)

#check A.card

-- card_union_eq
example (A B : Finset ℕ) (h : Disjoint A B) : (A ∪ B).card = A.card + B.card := by
  -- Not a suitable proof for the course.
  rw [← Finset.disjUnion_eq_union A B h, Finset.card_disjUnion _ _ _]

example : ¬Disjoint {n : ℤ | ∃ k, n = 2 * k} {3, 5, 6, 9, 11} := by
  rw [not_disjoint_iff]
  use 6
  constructor
  rw [mem_setOf_eq]
  use 3
  tauto

example : {n : ℕ | Even n ∨ Odd n} = univ := by
  rw [setOf_or]




#check Set.eq_of_mem_singleton
#check {n : ℤ | ∃ k, n = 2 * k}
#check {n : ℤ // ∃ k, n = 2 * k}
#check ℤ
variables (C : Set ℤ)
#check {n ∈ C | ∃ (k : ℤ), n = 2 * k}
#check {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k}
#check {x ∈ A | x = x}
#check {y | y ∈ A}
#check setOf_and
#check setOf_or
#check Disjoint C univ

example : {n ∈ C | ∃ (k : ℤ), n = 2 * k} = {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k} := by
  rfl