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import Mathlib
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lemma not_odd {n : ℕ} : ¬ Odd n ↔ Even n := by
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sorry
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lemma not_even {n : ℕ} : ¬ Even n ↔ Odd n := by
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sorry
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lemma even_square (n : ℕ) : Even n → Even (n ^ 2) := by
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intro ⟨x, hx⟩
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unfold Even at *
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use 2 * x ^ 2
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rw [hx]
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ring
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section powerset
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open Set
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namespace Finset
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theorem powerset_singleton {U : Type _} [DecidableEq U] (x : U) :
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Finset.powerset {x} = {∅, {x}} := by
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ext y
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rw [mem_powerset, subset_singleton_iff, mem_insert, mem_singleton]
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end Finset
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/- The powerset of a singleton contains only `∅` and the singleton. -/
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theorem powerset_singleton {U : Type _} (x : U) :
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𝒫 ({x} : Set U) = {∅, {x}} := by
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ext y
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rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff]
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theorem subset_insert_iff_of_not_mem {U : Type _ } {s t : Set U} {a : U} (h : a ∉ s)
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: s ⊆ insert a t ↔ s ⊆ t := by
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constructor
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· intro g y hy
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specialize g hy
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rw [mem_insert_iff] at g
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rcases g with g | g
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· rw [g] at hy
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contradiction
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· assumption
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· intro g y hy
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specialize g hy
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exact mem_insert_of_mem _ g
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theorem subset_insert_iff_of_not_mem' {U : Type _ } {s t : Set U} {a : U} (h : a ∉ s)
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(g : s ⊆ t) : s ⊆ insert a t := by
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intro y hy
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specialize g hy
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exact mem_insert_of_mem _ g
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lemma mem_powerset_insert_iff {U : Type _} (A S : Set U) (x : U) :
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S ∈ 𝒫 (insert x A) ↔ S ∈ 𝒫 A ∨ ∃ B ∈ 𝒫 A , insert x B = S := by
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simp_rw [mem_powerset_iff]
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constructor
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· intro h
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by_cases hs : x ∈ S
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· right
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use S \ {x}
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rw [insert_diff_singleton, insert_eq_of_mem hs, diff_singleton_subset_iff]
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exact ⟨h, rfl⟩
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· left
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exact (subset_insert_iff_of_not_mem hs).mp h
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· intro h
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rcases h with h | ⟨B, h₁, h₂⟩
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· exact le_trans h (subset_insert x A)
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· rw [←h₂]
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exact insert_subset_insert h₁
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lemma mem_powerset_insert_iff' {U : Type _} (A S : Set U) (x : U) :
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S ∈ 𝒫 (insert x A) ↔ S \ {x} ∈ 𝒫 A := by
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rw [mem_powerset_iff, mem_powerset_iff, diff_singleton_subset_iff]
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lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
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𝒫 (insert x A) = A.powerset ∪ A.powerset.image (insert x) := by
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ext y
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rw [mem_powerset_insert_iff, mem_union, mem_image]
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end powerset
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