132 lines
3.2 KiB
Plaintext
132 lines
3.2 KiB
Plaintext
import Mathlib.Data.Finset.Card
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import Mathlib.Data.Set.Finite
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/-! # Common.Set.Finite
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Additional theorems around finite sets.
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-/
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namespace Set
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/-! ## Definitions -/
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/--
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A set `A` is equinumerous to a set `B` (written `A ≈ B`) if and only if there is
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a one-to-one function from `A` onto `B`.
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-/
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def Equinumerous (A : Set α) (B : Set β) : Prop := ∃ F, Set.BijOn F A B
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infix:50 " ≈ " => Equinumerous
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theorem equinumerous_def (A : Set α) (B : Set β)
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: A ≈ B ↔ ∃ F, Set.BijOn F A B := Iff.rfl
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/--
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A set `A` is not equinumerous to a set `B` (written `A ≉ B`) if and only if
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there is no one-to-one function from `A` onto `B`.
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-/
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def NotEquinumerous (A : Set α) (B : Set β) : Prop := ¬ Equinumerous A B
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infix:50 " ≉ " => NotEquinumerous
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@[simp]
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theorem not_equinumerous_def : A ≉ B ↔ ∀ F, ¬ Set.BijOn F A B := by
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apply Iff.intro
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· intro h
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unfold NotEquinumerous Equinumerous at h
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simp only [not_exists] at h
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exact h
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· intro h
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unfold NotEquinumerous Equinumerous
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simp only [not_exists]
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exact h
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/--
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For any set `A`, `A ≈ A`.
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-/
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theorem equinumerous_refl (A : Set α)
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: A ≈ A := by
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refine ⟨fun x => x, ?_⟩
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unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
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simp only [imp_self, implies_true, Set.image_id', true_and]
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exact Eq.subset rfl
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/--
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For any sets `A` and `B`, if `A ≈ B`. then `B ≈ A`.
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-/
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theorem equinumerous_symm [Nonempty α] {A : Set α} {B : Set β}
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(h : A ≈ B) : B ≈ A := by
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have ⟨F, hF⟩ := h
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refine ⟨Function.invFunOn F A, ?_⟩
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exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
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/--
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For any sets `A`, `B`, and `C`, if `A ≈ B` and `B ≈ C`, then `A ≈ C`.
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-/
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theorem equinumerous_trans {A : Set α} {B : Set β} {C : Set γ}
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(h₁ : A ≈ B) (h₂ : B ≈ C)
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: ∃ H, Set.BijOn H A C := by
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have ⟨F, hF⟩ := h₁
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have ⟨G, hG⟩ := h₂
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exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
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/--
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If two sets are equal, they are trivially equinumerous.
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-/
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theorem eq_imp_equinumerous {A B : Set α} (h : A = B)
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: A ≈ B := by
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have := equinumerous_refl A
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conv at this => right; rw [h]
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exact this
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/-! ## Finite Sets -/
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/--
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A set is finite if and only if it is equinumerous to a natural number.
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-/
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axiom finite_iff_equinumerous_nat {α : Type _} {S : Set α}
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: Set.Finite S ↔ ∃ n : ℕ, S ≈ Set.Iio n
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/-! ## Emptyset -/
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/--
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Any set equinumerous to the emptyset is the emptyset.
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-/
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@[simp]
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theorem equinumerous_zero_iff_emptyset {S : Set α}
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: S ≈ Set.Iio 0 ↔ S = ∅ := by
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apply Iff.intro
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· intro ⟨f, hf⟩
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by_contra nh
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rw [← Ne.def, ← Set.nonempty_iff_ne_empty] at nh
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have ⟨x, hx⟩ := nh
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have := hf.left hx
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simp at this
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· intro h
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rw [h]
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refine ⟨fun _ => ⊥, ?_, ?_, ?_⟩
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· intro _ hx
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simp at hx
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· intro _ hx
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simp at hx
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· unfold SurjOn
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simp only [bot_eq_zero', image_empty]
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show ∀ x, x ∈ Set.Iio 0 → x ∈ ∅
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intro _ hx
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simp at hx
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/--
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Empty sets are always equinumerous, regardless of their underlying type.
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-/
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theorem equinumerous_emptyset_emptyset [Bot β]
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: (∅ : Set α) ≈ (∅ : Set β) := by
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refine ⟨fun _ => ⊥, ?_, ?_, ?_⟩
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· intro _ hx
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simp at hx
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· intro _ hx
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simp at hx
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· unfold SurjOn
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simp
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end Set
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