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import TestGame.Metadata
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import Mathlib
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import Duper.Tactic
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import Mathlib.Order.Disjoint
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set_option tactic.hygienic false
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Game "TestGame"
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World "SetTheory"
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Level 4
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Title "Mengen"
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Introduction
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"
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`∈ ∉ ⊆ ⊂ ⋃ ⋂`
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"
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open Set
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-- open_locale cardinal
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variables {X : Type _} (x : X) (A B : Set X)
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#check A.Nonempty
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#check Nonempty A
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#check insert x A -- {x} ∪ A
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-- #check disjoint A B -- needs Mathlib.Order.Disjoint
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#check A ∆ B
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#check Nontrivial A
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#check ({2} : Set ℕ)
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example : ({2} : Set ℕ) ⊆ {4, 2, 3} := by
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simp only [mem_singleton_iff, mem_insert_iff, or_self, singleton_subset_iff, or_false, or_true]
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example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
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intro n hn
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simp only [mem_insert_iff, mem_singleton_iff] at *
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tauto
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example : {2, 3, 5} ⊆ (univ : Set ℕ) := by
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tauto
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example : 3 ∈ ({4, 2, 5, 7, 3} : Set ℕ) := by
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tauto
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example : ({4, 9} : Set ℕ) = Set.insert 4 {9} := by
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rfl
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#check Finset.card
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variable (A : Finset ℕ)
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#check A.card
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-- card_union_eq
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example (A B : Finset ℕ) (h : Disjoint A B) : (A ∪ B).card = A.card + B.card := by
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-- Not a suitable proof for the course.
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rw [← Finset.disjUnion_eq_union A B h, Finset.card_disjUnion _ _ _]
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example : ¬Disjoint {n : ℤ | ∃ k, n = 2 * k} {3, 5, 6, 9, 11} := by
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rw [not_disjoint_iff]
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use 6
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constructor
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rw [mem_setOf_eq]
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use 3
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tauto
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example : {n : ℕ | Even n ∨ Odd n} = univ := by
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rw [setOf_or]
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#check Set.eq_of_mem_singleton
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#check {n : ℤ | ∃ k, n = 2 * k}
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#check {n : ℤ // ∃ k, n = 2 * k}
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#check ℤ
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variables (C : Set ℤ)
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#check {n ∈ C | ∃ (k : ℤ), n = 2 * k}
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#check {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k}
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#check {x ∈ A | x = x}
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#check {y | y ∈ A}
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#check setOf_and
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#check setOf_or
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#check Disjoint C univ
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example : {n ∈ C | ∃ (k : ℤ), n = 2 * k} = {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k} := by
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rfl
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