Enderton (set). Finish equinumerosity theorems.
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@ -150,6 +150,9 @@
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\lean*{Mathlib/Init/Function}
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{Function.Bijective}
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\lean{Mathlib/Data/Set/Function}
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{Set.BijOn}
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\lean{Mathlib/Logic/Equiv/Defs}
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{Equiv}
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@ -8973,7 +8976,7 @@
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Hence no finite set is equinumerous to a proper subset of itself.
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\end{proof}
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\subsection{\pending{Corollary 6D}}%
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\subsection{\verified{Corollary 6D}}%
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\hyperlabel{sub:corollary-6d}
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\begin{corollary}[6D]
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@ -8984,6 +8987,12 @@
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\end{enumerate}
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\end{corollary}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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{Enderton.Set.Chapter\_6.corollary\_6d\_a}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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{Enderton.Set.Chapter\_6.corollary\_6d\_b}
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\begin{proof}
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\paragraph{(a)}%
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@ -9034,13 +9043,16 @@
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\end{proof}
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\subsection{\pending{Corollary 6E}}%
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\subsection{\verified{Corollary 6E}}%
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\hyperlabel{sub:corollary-6e}
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\begin{corollary}[6E]
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Any finite set is equinumerous to a unique natural number.
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\end{corollary}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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{Enderton.Set.Chapter\_6.corollary\_6e}
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\begin{proof}
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Let $S$ be a \nameref{ref:finite-set}.
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By definition $S$ is equinumerous to a natural number $n$.
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@ -9057,7 +9069,7 @@
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number.
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\end{proof}
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\subsection{\pending{Lemma 6F}}%
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\subsection{\verified{Lemma 6F}}%
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\hyperlabel{sub:lemma-6f}
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\begin{lemma}[6F]
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@ -9065,6 +9077,9 @@
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some $m$ less than $n$.
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\end{lemma}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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{Enderton.Set.Chapter\_6.lemma\_6f}
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\begin{proof}
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Let
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@ -9132,13 +9147,16 @@
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\end{proof}
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\subsection{\pending{Corollary 6G}}%
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\subsection{\verified{Corollary 6G}}%
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\hyperlabel{sub:corollary-6g}
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\begin{corollary}[6G]
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Any subset of a finite set is finite.
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\end{corollary}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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{Enderton.Set.Chapter\_6.corollary\_6g}
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\begin{proof}
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Let $S$ be a \nameref{ref:finite-set} and $S' \subseteq S$.
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Clearly, if $S' = S$, then $S'$ is finite.
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@ -1,7 +1,9 @@
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import Bookshelf.Enderton.Set.Chapter_4
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import Common.Logic.Basic
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import Common.Nat.Basic
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import Common.Set.Basic
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import Common.Set.Finite
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import Common.Set.Equinumerous
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import Common.Set.Intervals
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import Mathlib.Data.Finset.Card
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import Mathlib.Data.Set.Finite
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import Mathlib.Tactic.LibrarySearch
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@ -23,7 +25,8 @@ namespace Enderton.Set.Chapter_6
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No set is equinumerous to its powerset.
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-/
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theorem theorem_6b (A : Set α)
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: ∀ f, ¬ Set.BijOn f A (𝒫 A) := by
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: A ≉ 𝒫 A := by
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rw [Set.not_equinumerous_def]
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intro f hf
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unfold Set.BijOn at hf
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let φ := { a ∈ A | a ∉ f a }
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@ -287,82 +290,71 @@ lemma pigeonhole_principle_aux (n : ℕ)
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No natural number is equinumerous to a proper subset of itself.
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-/
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theorem pigeonhole_principle {n : ℕ}
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: ∀ M, M ⊂ Set.Iio n → ∀ f, ¬ Set.BijOn f M (Set.Iio n) := by
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intro M hM f nf
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have := pigeonhole_principle_aux n M hM f ⟨nf.left, nf.right.left⟩
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exact absurd nf.right.right this
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: ∀ {M}, M ⊂ Set.Iio n → M ≉ Set.Iio n := by
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intro M hM nM
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have ⟨f, hf⟩ := nM
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have := pigeonhole_principle_aux n M hM f ⟨hf.left, hf.right.left⟩
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exact absurd hf.right.right this
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/-- #### Corollary 6C
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No finite set is equinumerous to a proper subset of itself.
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-/
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theorem corollary_6c [DecidableEq α] [Nonempty α] {S S' : Finset α} (h : S' ⊂ S)
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: ∀ f, ¬ Set.BijOn f S.toSet S'.toSet := by
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have ⟨T, hT₁, hT₂⟩ : ∃ T, Disjoint S' T ∧ S = S' ∪ T := by
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refine ⟨S \ S', ?_, ?_⟩
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· intro X hX₁ hX₂
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show ∀ t, t ∈ X → t ∈ ⊥
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intro t ht
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have ht₂ := hX₂ ht
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simp only [Finset.mem_sdiff] at ht₂
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exact absurd (hX₁ ht) ht₂.right
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· simp only [
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Finset.union_sdiff_self_eq_union,
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Finset.right_eq_union_iff_subset
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]
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exact subset_of_ssubset h
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theorem corollary_6c [DecidableEq α] [Nonempty α]
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{S S' : Set α} (hS : Set.Finite S) (h : S' ⊂ S)
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: S ≉ S' := by
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let T := S \ S'
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have hT : S = S' ∪ (S \ S') := by
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simp only [Set.union_diff_self]
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exact (Set.left_subset_union_eq_self (subset_of_ssubset h)).symm
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-- `hF : S' ∪ T ≈ S`.
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-- `hG : S ≈ n`.
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-- `hH : S' ∪ T ≈ n`.
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have ⟨F, hF⟩ := Set.equinumerous_refl S.toSet
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conv at hF => arg 2; rw [hT₂]
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have ⟨n, G, hG⟩ := Set.finite_iff_equinumerous_nat.mp (Finset.finite_toSet S)
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have hF := Set.equinumerous_refl S
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conv at hF => arg 1; rw [hT]
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have ⟨n, hG⟩ := Set.finite_iff_equinumerous_nat.mp hS
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have ⟨H, hH⟩ := Set.equinumerous_trans hF hG
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-- Restrict `H` to `S'` to yield a bijection between `S'` and `m < n`.
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-- Restrict `H` to `S'` to yield a bijection between `S'` and a proper subset
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-- of `n`.
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let R := (Set.Iio n) \ (H '' T)
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have hR : Set.BijOn H S' R := by
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refine ⟨?_, ?_, ?_⟩
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· -- `Set.MapsTo H S' R`
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intro x hx
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refine ⟨hH.left $ Finset.mem_union_left T hx, ?_⟩
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refine ⟨hH.left $ Set.mem_union_left T hx, ?_⟩
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unfold Set.image
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by_contra nx
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simp only [Finset.mem_coe, Set.mem_setOf_eq] at nx
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have ⟨a, ha₁, ha₂⟩ := nx
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have hc₁ : a ∈ S' ∪ T := Finset.mem_union_right S' ha₁
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have hc₂ : x ∈ S' ∪ T := Finset.mem_union_left T hx
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have hc₁ : a ∈ S' ∪ T := Set.mem_union_right S' ha₁
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have hc₂ : x ∈ S' ∪ T := Set.mem_union_left T hx
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rw [hH.right.left hc₁ hc₂ ha₂] at ha₁
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have hx₁ : {x} ⊆ S' := Finset.singleton_subset_iff.mpr hx
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have hx₂ : {x} ⊆ T := Finset.singleton_subset_iff.mpr ha₁
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have hx₃ := hT₁ hx₁ hx₂
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have hx₁ : {x} ⊆ S' := Set.singleton_subset_iff.mpr hx
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have hx₂ : {x} ⊆ T := Set.singleton_subset_iff.mpr ha₁
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have hx₃ := Set.disjoint_sdiff_right hx₁ hx₂
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simp only [
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Finset.bot_eq_empty,
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Finset.le_eq_subset,
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Finset.singleton_subset_iff,
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Finset.not_mem_empty
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] at hx₃
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Set.bot_eq_empty,
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Set.le_eq_subset,
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Set.singleton_subset_iff,
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Set.mem_empty_iff_false
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] at hx₃
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· -- `Set.InjOn H S'`
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intro x₁ hx₁ x₂ hx₂ h
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have hc₁ : x₁ ∈ S' ∪ T := Finset.mem_union_left T hx₁
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have hc₂ : x₂ ∈ S' ∪ T := Finset.mem_union_left T hx₂
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have hc₁ : x₁ ∈ S' ∪ T := Set.mem_union_left T hx₁
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have hc₂ : x₂ ∈ S' ∪ T := Set.mem_union_left T hx₂
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exact hH.right.left hc₁ hc₂ h
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· -- `Set.SurjOn H S' R`
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show ∀ r, r ∈ R → r ∈ H '' S'
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intro r hr
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unfold Set.image
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simp only [Finset.mem_coe, Set.mem_setOf_eq]
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simp only [Set.mem_setOf_eq]
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dsimp only at hr
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have := hH.right.right hr.left
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simp only [
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Finset.coe_union,
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Set.mem_image,
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Set.mem_union,
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Finset.mem_coe
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] at this
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simp only [Set.mem_image, Set.mem_union] at this
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have ⟨x, hx⟩ := this
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apply Or.elim hx.left
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· intro hx'
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@ -372,15 +364,14 @@ theorem corollary_6c [DecidableEq α] [Nonempty α] {S S' : Finset α} (h : S'
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rw [← hx.right]
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simp only [Set.mem_image, Finset.mem_coe]
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exact ⟨x, hx', rfl⟩
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intro f nf
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have ⟨f₁, hf₁⟩ : ∃ f₁ : α → ℕ, Set.BijOn f₁ S R :=
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Set.equinumerous_trans nf hR
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have ⟨f₂, hf₂⟩ : ∃ f₃ : ℕ → ℕ, Set.BijOn f₃ R (Set.Iio n) := by
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have ⟨k, hk₁⟩ := Set.equinumerous_symm hf₁
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exact Set.equinumerous_trans hk₁ hG
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intro hf
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have hf₁ : S ≈ R := Set.equinumerous_trans hf ⟨H, hR⟩
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have hf₂ : R ≈ Set.Iio n := by
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have ⟨k, hk⟩ := Set.equinumerous_symm hf₁
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exact Set.equinumerous_trans ⟨k, hk⟩ hG
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refine absurd hf₂ (pigeonhole_principle R ?_ f₂)
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refine absurd hf₂ (pigeonhole_principle ?_)
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show R ⊂ Set.Iio n
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apply And.intro
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· show ∀ r, r ∈ R → r ∈ Set.Iio n
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@ -388,17 +379,8 @@ theorem corollary_6c [DecidableEq α] [Nonempty α] {S S' : Finset α} (h : S'
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exact hr.left
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· show ¬ ∀ r, r ∈ Set.Iio n → r ∈ R
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intro nr
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have ⟨t, ht₁⟩ : Finset.Nonempty T := by
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rw [hT₂, Finset.ssubset_def] at h
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have : ¬ ∀ x, x ∈ S' ∪ T → x ∈ S' := h.right
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simp only [Finset.mem_union, not_forall, exists_prop] at this
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have ⟨x, hx⟩ := this
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apply Or.elim hx.left
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· intro nx
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exact absurd nx hx.right
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· intro hx
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exact ⟨x, hx⟩
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have ht₂ : H t ∈ Set.Iio n := hH.left (Finset.mem_union_right S' ht₁)
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have ⟨t, ht₁⟩ : Set.Nonempty T := Set.diff_ssubset_nonempty h
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have ht₂ : H t ∈ Set.Iio n := hH.left (Set.mem_union_right S' ht₁)
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have ht₃ : H t ∈ R := nr (H t) ht₂
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exact absurd ⟨t, ht₁, rfl⟩ ht₃.right
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@ -406,9 +388,12 @@ theorem corollary_6c [DecidableEq α] [Nonempty α] {S S' : Finset α} (h : S'
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Any set equinumerous to a proper subset of itself is infinite.
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-/
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theorem corollary_6d_a (S S' : Set α) (hS : S' ⊂ S) (hf : S' ≃ S)
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theorem corollary_6d_a [DecidableEq α] [Nonempty α]
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{S S' : Set α} (hS : S' ⊂ S) (hf : S ≈ S')
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: Set.Infinite S := by
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sorry
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by_contra nS
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simp only [Set.not_infinite] at nS
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exact absurd hf (corollary_6c nS hS)
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/-- #### Corollary 6D (b)
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@ -416,28 +401,279 @@ The set `ω` is infinite.
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-/
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theorem corollary_6d_b
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: Set.Infinite (@Set.univ ℕ) := by
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sorry
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let S : Set ℕ := { 2 * n | n ∈ @Set.univ ℕ }
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let f x := 2 * x
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suffices Set.BijOn f (@Set.univ ℕ) S by
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refine corollary_6d_a ?_ ⟨f, this⟩
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rw [Set.ssubset_def]
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apply And.intro
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· simp
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· show ¬ ∀ x, x ∈ Set.univ → x ∈ S
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simp only [
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Set.mem_univ,
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true_and,
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Set.mem_setOf_eq,
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forall_true_left,
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not_forall,
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not_exists
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]
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refine ⟨1, ?_⟩
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intro x nx
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simp only [mul_eq_one, false_and] at nx
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refine ⟨by simp, ?_, ?_⟩
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· -- `Set.InjOn f Set.univ`
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intro n₁ _ n₂ _ hf
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match @trichotomous ℕ LT.lt _ n₁ n₂ with
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| Or.inr (Or.inl r) => exact r
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| Or.inl r =>
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have := (Chapter_4.theorem_4n_ii n₁ n₂ 1).mp r
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conv at this => left; rw [mul_comm]
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conv at this => right; rw [mul_comm]
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exact absurd hf (Nat.ne_of_lt this)
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| Or.inr (Or.inr r) =>
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have := (Chapter_4.theorem_4n_ii n₂ n₁ 1).mp r
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conv at this => left; rw [mul_comm]
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conv at this => right; rw [mul_comm]
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exact absurd hf.symm (Nat.ne_of_lt this)
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· -- `Set.SurjOn f Set.univ S`
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show ∀ x, x ∈ S → x ∈ f '' Set.univ
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intro x hx
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unfold Set.image
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simp only [Set.mem_univ, true_and, Set.mem_setOf_eq] at hx ⊢
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exact hx
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/-- #### Corollary 6E
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Any finite set is equinumerous to a unique natural number.
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-/
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theorem corollary_6e (S : Set α) (hn : S ≃ Fin n) (hm : S ≃ Fin m)
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: m = n := by
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sorry
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theorem corollary_6e [Nonempty α] (S : Set α) (hS : Set.Finite S)
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: ∃! n : ℕ, S ≈ Set.Iio n := by
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have ⟨n, hf⟩ := Set.finite_iff_equinumerous_nat.mp hS
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refine ⟨n, hf, ?_⟩
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intro m hg
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match @trichotomous ℕ LT.lt _ m n with
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| Or.inr (Or.inl r) => exact r
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| Or.inl r =>
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have hh := Set.equinumerous_symm hg
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have hk := Set.equinumerous_trans hh hf
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have hmn : Set.Iio m ⊂ Set.Iio n := Set.Iio_nat_lt_ssubset r
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exact absurd hk (pigeonhole_principle hmn)
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| Or.inr (Or.inr r) =>
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have hh := Set.equinumerous_symm hf
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have hk := Set.equinumerous_trans hh hg
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have hnm : Set.Iio n ⊂ Set.Iio m := Set.Iio_nat_lt_ssubset r
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exact absurd hk (pigeonhole_principle hnm)
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/-- #### Lemma 6F
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If `C` is a proper subset of a natural number `n`, then `C ≈ m` for some `m`
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less than `n`.
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-/
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lemma lemma_6f {n : ℕ} (hC : C ⊂ Finset.range n)
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: ∃ m : ℕ, m < n ∧ ∃ f : C → Fin m, Function.Bijective f := by
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sorry
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lemma lemma_6f {n : ℕ}
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: ∀ {C}, C ⊂ Set.Iio n → ∃ m, m < n ∧ C ≈ Set.Iio m := by
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induction n with
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| zero =>
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intro C hC
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unfold Set.Iio at hC
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simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hC
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rw [Set.ssubset_empty_iff_false] at hC
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exact False.elim hC
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| succ n ih =>
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have h_subset_equinumerous
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: ∀ S, S ⊆ Set.Iio n →
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∃ m, m < n + 1 ∧ S ≈ Set.Iio m := by
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intro S hS
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rw [subset_iff_ssubset_or_eq] at hS
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apply Or.elim hS
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· -- `S ⊂ Set.Iio n`
|
||||
intro h
|
||||
have ⟨m, hm⟩ := ih h
|
||||
exact ⟨m, calc m
|
||||
_ < n := hm.left
|
||||
_ < n + 1 := by simp, hm.right⟩
|
||||
· -- `S = Set.Iio n`
|
||||
intro h
|
||||
exact ⟨n, by simp, Set.eq_imp_equinumerous h⟩
|
||||
|
||||
theorem corollary_6g (S S' : Set α) (hS : Finite S) (hS' : S' ⊆ S)
|
||||
: Finite S' := by
|
||||
sorry
|
||||
intro C hC
|
||||
by_cases hn : n ∈ C
|
||||
· -- Since `C` is a proper subset of `n⁺`, the set `n⁺ - C` is nonempty.
|
||||
have hC₁ : Set.Nonempty (Set.Iio (n + 1) \ C) := by
|
||||
rw [Set.ssubset_def] at hC
|
||||
have : ¬ ∀ x, x ∈ Set.Iio (n + 1) → x ∈ C := hC.right
|
||||
simp only [Set.mem_Iio, not_forall, exists_prop] at this
|
||||
exact this
|
||||
-- `p` is the least element of `n⁺ - C`.
|
||||
have ⟨p, hp⟩ := Chapter_4.well_ordering_nat hC₁
|
||||
|
||||
let C' := (C \ {n}) ∪ {p}
|
||||
have hC'₁ : C' ⊆ Set.Iio n := by
|
||||
show ∀ x, x ∈ C' → x ∈ Set.Iio n
|
||||
intro x hx
|
||||
match @trichotomous ℕ LT.lt _ x n with
|
||||
| Or.inl r => exact r
|
||||
| Or.inr (Or.inl r) =>
|
||||
rw [r] at hx
|
||||
apply Or.elim hx
|
||||
· intro nx
|
||||
simp at nx
|
||||
· intro nx
|
||||
simp only [Set.mem_singleton_iff] at nx
|
||||
rw [nx] at hn
|
||||
exact absurd hn hp.left.right
|
||||
| Or.inr (Or.inr r) =>
|
||||
apply Or.elim hx
|
||||
· intro ⟨h₁, h₂⟩
|
||||
have h₃ := subset_of_ssubset hC h₁
|
||||
simp only [Set.mem_singleton_iff, Set.mem_Iio] at h₂ h₃
|
||||
exact Or.elim (Nat.lt_or_eq_of_lt h₃) id (absurd · h₂)
|
||||
· intro h
|
||||
simp only [Set.mem_singleton_iff] at h
|
||||
have := hp.left.left
|
||||
rw [← h] at this
|
||||
exact Or.elim (Nat.lt_or_eq_of_lt this)
|
||||
id (absurd · (Nat.ne_of_lt r).symm)
|
||||
have ⟨m, hm₁, hm₂⟩ := h_subset_equinumerous C' hC'₁
|
||||
|
||||
suffices C' ≈ C from
|
||||
⟨m, hm₁, Set.equinumerous_trans (Set.equinumerous_symm this) hm₂⟩
|
||||
|
||||
-- Proves `f` is a one-to-one correspondence between `C'` and `C`.
|
||||
let f x := if x = p then n else x
|
||||
refine ⟨f, ?_, ?_, ?_⟩
|
||||
· -- `Set.MapsTo f C' C`
|
||||
intro x hx
|
||||
dsimp only
|
||||
by_cases hxp : x = p
|
||||
· rw [if_pos hxp]
|
||||
exact hn
|
||||
· rw [if_neg hxp]
|
||||
apply Or.elim hx
|
||||
· exact fun x => x.left
|
||||
· intro hx₁
|
||||
simp only [Set.mem_singleton_iff] at hx₁
|
||||
exact absurd hx₁ hxp
|
||||
· -- `Set.InjOn f C'`
|
||||
intro x₁ hx₁ x₂ hx₂ hf
|
||||
dsimp only at hf
|
||||
by_cases hx₁p : x₁ = p
|
||||
· by_cases hx₂p : x₂ = p
|
||||
· rw [hx₁p, hx₂p]
|
||||
· rw [if_pos hx₁p, if_neg hx₂p] at hf
|
||||
apply Or.elim hx₂
|
||||
· intro nx
|
||||
exact absurd hf.symm nx.right
|
||||
· intro nx
|
||||
simp only [Set.mem_singleton_iff] at nx
|
||||
exact absurd nx hx₂p
|
||||
· by_cases hx₂p : x₂ = p
|
||||
· rw [if_neg hx₁p, if_pos hx₂p] at hf
|
||||
apply Or.elim hx₁
|
||||
· intro nx
|
||||
exact absurd hf nx.right
|
||||
· intro nx
|
||||
simp only [Set.mem_singleton_iff] at nx
|
||||
exact absurd nx hx₁p
|
||||
· rwa [if_neg hx₁p, if_neg hx₂p] at hf
|
||||
· -- `Set.SurjOn f C' C`
|
||||
show ∀ x, x ∈ C → x ∈ f '' C'
|
||||
intro x hx
|
||||
simp only [
|
||||
Set.union_singleton,
|
||||
Set.mem_diff,
|
||||
Set.mem_singleton_iff,
|
||||
Set.mem_image,
|
||||
Set.mem_insert_iff,
|
||||
exists_eq_or_imp,
|
||||
ite_true
|
||||
]
|
||||
by_cases nx₁ : x = n
|
||||
· left
|
||||
exact nx₁.symm
|
||||
· right
|
||||
by_cases nx₂ : x = p
|
||||
· have := hp.left.right
|
||||
rw [← nx₂] at this
|
||||
exact absurd hx this
|
||||
· exact ⟨x, ⟨hx, nx₁⟩, by rwa [if_neg]⟩
|
||||
|
||||
· refine h_subset_equinumerous C ?_
|
||||
show ∀ x, x ∈ C → x ∈ Set.Iio n
|
||||
intro x hx
|
||||
unfold Set.Iio
|
||||
apply Or.elim (Nat.lt_or_eq_of_lt (subset_of_ssubset hC hx))
|
||||
· exact id
|
||||
· intro hx₁
|
||||
rw [hx₁] at hx
|
||||
exact absurd hx hn
|
||||
|
||||
/-- #### Corollary 6G
|
||||
|
||||
Any subset of a finite set is finite.
|
||||
-/
|
||||
theorem corollary_6g {S S' : Set α} (hS : Set.Finite S) (hS' : S' ⊆ S)
|
||||
: Set.Finite S' := by
|
||||
rw [subset_iff_ssubset_or_eq] at hS'
|
||||
apply Or.elim hS'
|
||||
· intro h
|
||||
rw [Set.finite_iff_equinumerous_nat] at hS
|
||||
have ⟨n, F, hF⟩ := hS
|
||||
|
||||
-- Mirrors logic found in `corollary_6c`.
|
||||
let T := S \ S'
|
||||
let R := (Set.Iio n) \ (F '' T)
|
||||
have hR : R ⊂ Set.Iio n := by
|
||||
rw [Set.ssubset_def]
|
||||
apply And.intro
|
||||
· show ∀ x, x ∈ R → x ∈ Set.Iio n
|
||||
intro _ hx
|
||||
exact hx.left
|
||||
· show ¬ ∀ x, x ∈ Set.Iio n → x ∈ R
|
||||
intro nr
|
||||
have ⟨t, ht₁⟩ : Set.Nonempty T := Set.diff_ssubset_nonempty h
|
||||
have ht₂ : F t ∈ Set.Iio n := hF.left ht₁.left
|
||||
have ht₃ : F t ∈ R := nr (F t) ht₂
|
||||
exact absurd ⟨t, ht₁, rfl⟩ ht₃.right
|
||||
|
||||
suffices Set.BijOn F S' R by
|
||||
have ⟨m, hm⟩ := lemma_6f hR
|
||||
have := Set.equinumerous_trans ⟨F, this⟩ hm.right
|
||||
exact Set.finite_iff_equinumerous_nat.mpr ⟨m, this⟩
|
||||
refine ⟨?_, ?_, ?_⟩
|
||||
· -- `Set.MapsTo f S' R`
|
||||
intro x hx
|
||||
dsimp only
|
||||
simp only [Set.mem_diff, Set.mem_Iio, Set.mem_image, not_exists, not_and]
|
||||
apply And.intro
|
||||
· exact hF.left (subset_of_ssubset h hx)
|
||||
· intro y hy
|
||||
by_contra nf
|
||||
have := hF.right.left (subset_of_ssubset h hx) hy.left nf.symm
|
||||
rw [this] at hx
|
||||
exact absurd hx hy.right
|
||||
· -- `Set.InjOn f S'`
|
||||
intro x₁ hx₁ x₂ hx₂ hf
|
||||
have h₁ : x₁ ∈ S := subset_of_ssubset h hx₁
|
||||
have h₂ : x₂ ∈ S := subset_of_ssubset h hx₂
|
||||
exact hF.right.left h₁ h₂ hf
|
||||
· -- `Set.SurjOn f S' R`
|
||||
show ∀ x, x ∈ R → x ∈ F '' S'
|
||||
intro x hx
|
||||
|
||||
have h₁ := hF.right.right
|
||||
unfold Set.SurjOn at h₁
|
||||
rw [Set.subset_def] at h₁
|
||||
have ⟨y, hy⟩ := h₁ x hx.left
|
||||
|
||||
refine ⟨y, ?_, hy.right⟩
|
||||
rw [← hy.right] at hx
|
||||
simp only [Set.mem_image, Set.mem_diff, not_exists, not_and] at hx
|
||||
by_contra ny
|
||||
exact (hx.right y ⟨hy.left, ny⟩) rfl
|
||||
|
||||
· intro h
|
||||
rwa [h]
|
||||
|
||||
/-- #### Exercise 6.1
|
||||
|
||||
|
|
|
|||
|
|
@ -1,2 +1,4 @@
|
|||
import Common.Set.Basic
|
||||
import Common.Set.Equinumerous
|
||||
import Common.Set.Intervals
|
||||
import Common.Set.Peano
|
||||
|
|
@ -175,6 +175,24 @@ theorem diff_ssubset_diff_left {A B : Set α} (h : A ⊂ B)
|
|||
rw [diff_subset_iff, union_diff_cancel this] at nh
|
||||
exact LT.lt.false (Set.ssubset_of_ssubset_of_subset h nh)
|
||||
|
||||
/--
|
||||
For any sets `A ⊂ B`, `B \ A` is nonempty.
|
||||
-/
|
||||
theorem diff_ssubset_nonempty {A B : Set α} (h : A ⊂ B)
|
||||
: Set.Nonempty (B \ A) := by
|
||||
have : B = A ∪ (B \ A) := by
|
||||
simp only [Set.union_diff_self]
|
||||
exact (Set.left_subset_union_eq_self (subset_of_ssubset h)).symm
|
||||
rw [this, Set.ssubset_def] at h
|
||||
have : ¬ ∀ x, x ∈ A ∪ (B \ A) → x ∈ A := h.right
|
||||
simp only [Set.mem_union, not_forall, exists_prop] at this
|
||||
have ⟨x, hx⟩ := this
|
||||
apply Or.elim hx.left
|
||||
· intro nx
|
||||
exact absurd nx hx.right
|
||||
· intro hx
|
||||
exact ⟨x, hx⟩
|
||||
|
||||
/--
|
||||
For any set `A`, the difference between the sample space and `A` is the
|
||||
complement of `A`.
|
||||
|
|
|
|||
|
|
@ -0,0 +1,86 @@
|
|||
import Mathlib.Data.Finset.Card
|
||||
import Mathlib.Data.Set.Finite
|
||||
|
||||
/-! # Common.Set.Finite
|
||||
|
||||
Additional theorems around finite sets.
|
||||
-/
|
||||
|
||||
namespace Set
|
||||
|
||||
/--
|
||||
A set `A` is equinumerous to a set `B` (written `A ≈ B`) if and only if there is
|
||||
a one-to-one function from `A` onto `B`.
|
||||
-/
|
||||
def Equinumerous (A : Set α) (B : Set β) : Prop := ∃ F, Set.BijOn F A B
|
||||
|
||||
infix:50 " ≈ " => Equinumerous
|
||||
|
||||
theorem equinumerous_def (A : Set α) (B : Set β)
|
||||
: A ≈ B ↔ ∃ F, Set.BijOn F A B := Iff.rfl
|
||||
|
||||
/--
|
||||
For any set `A`, `A ≈ A`.
|
||||
-/
|
||||
theorem equinumerous_refl (A : Set α)
|
||||
: A ≈ A := by
|
||||
refine ⟨fun x => x, ?_⟩
|
||||
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
|
||||
simp only [imp_self, implies_true, Set.image_id', true_and]
|
||||
exact Eq.subset rfl
|
||||
|
||||
/--
|
||||
For any sets `A` and `B`, if `A ≈ B`. then `B ≈ A`.
|
||||
-/
|
||||
theorem equinumerous_symm [Nonempty α] {A : Set α} {B : Set β}
|
||||
(h : A ≈ B) : B ≈ A := by
|
||||
have ⟨F, hF⟩ := h
|
||||
refine ⟨Function.invFunOn F A, ?_⟩
|
||||
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
|
||||
|
||||
/--
|
||||
For any sets `A`, `B`, and `C`, if `A ≈ B` and `B ≈ C`, then `A ≈ C`.
|
||||
-/
|
||||
theorem equinumerous_trans {A : Set α} {B : Set β} {C : Set γ}
|
||||
(h₁ : A ≈ B) (h₂ : B ≈ C)
|
||||
: ∃ H, Set.BijOn H A C := by
|
||||
have ⟨F, hF⟩ := h₁
|
||||
have ⟨G, hG⟩ := h₂
|
||||
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
|
||||
|
||||
/--
|
||||
If two sets are equal, they are trivially equinumerous.
|
||||
-/
|
||||
theorem eq_imp_equinumerous {A B : Set α} (h : A = B)
|
||||
: A ≈ B := by
|
||||
have := equinumerous_refl A
|
||||
conv at this => right; rw [h]
|
||||
exact this
|
||||
|
||||
/--
|
||||
A set is finite if and only if it is equinumerous to a natural number.
|
||||
-/
|
||||
axiom finite_iff_equinumerous_nat {α : Type _} {S : Set α}
|
||||
: Set.Finite S ↔ ∃ n : ℕ, S ≈ Set.Iio n
|
||||
|
||||
/--
|
||||
A set `A` is not equinumerous to a set `B` (written `A ≉ B`) if and only if
|
||||
there is no one-to-one function from `A` onto `B`.
|
||||
-/
|
||||
def NotEquinumerous (A : Set α) (B : Set β) : Prop := ¬ Equinumerous A B
|
||||
|
||||
infix:50 " ≉ " => NotEquinumerous
|
||||
|
||||
@[simp]
|
||||
theorem not_equinumerous_def : A ≉ B ↔ ∀ F, ¬ Set.BijOn F A B := by
|
||||
apply Iff.intro
|
||||
· intro h
|
||||
unfold NotEquinumerous Equinumerous at h
|
||||
simp only [not_exists] at h
|
||||
exact h
|
||||
· intro h
|
||||
unfold NotEquinumerous Equinumerous
|
||||
simp only [not_exists]
|
||||
exact h
|
||||
|
||||
end Set
|
||||
|
|
@ -1,45 +0,0 @@
|
|||
import Mathlib.Data.Finset.Card
|
||||
import Mathlib.Data.Set.Finite
|
||||
|
||||
/-! # Common.Set.Finite
|
||||
|
||||
Additional theorems around finite sets.
|
||||
-/
|
||||
|
||||
namespace Set
|
||||
|
||||
/--
|
||||
For any set `A`, `A ≈ A`.
|
||||
-/
|
||||
theorem equinumerous_refl (A : Set α)
|
||||
: ∃ F, Set.BijOn F A A := by
|
||||
refine ⟨fun x => x, ?_⟩
|
||||
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
|
||||
simp only [imp_self, implies_true, Set.image_id', true_and]
|
||||
exact Eq.subset rfl
|
||||
|
||||
/--
|
||||
For any sets `A` and `B`, if `A ≈ B`. then `B ≈ A`.
|
||||
-/
|
||||
theorem equinumerous_symm [Nonempty α] {A : Set α} {B : Set β}
|
||||
{F : α → β} (hF : Set.BijOn F A B)
|
||||
: ∃ G, Set.BijOn G B A := by
|
||||
refine ⟨Function.invFunOn F A, ?_⟩
|
||||
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
|
||||
|
||||
/--
|
||||
For any sets `A`, `B`, and `C`, if `A ≈ B` and `B ≈ C`, then `A ≈ C`.
|
||||
-/
|
||||
theorem equinumerous_trans {A : Set α} {B : Set β} {C : Set γ}
|
||||
{F : α → β} (hF : Set.BijOn F A B)
|
||||
{G : β → γ} (hG : Set.BijOn G B C)
|
||||
: ∃ H, Set.BijOn H A C := by
|
||||
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
|
||||
|
||||
/--
|
||||
A set is finite if and only if it is equinumerous to a natural number.
|
||||
-/
|
||||
axiom finite_iff_equinumerous_nat {α : Type _} {S : Set α}
|
||||
: Set.Finite S ↔ ∃ n : ℕ, ∃ f, Set.BijOn f S (Set.Iio n)
|
||||
|
||||
end Set
|
||||
|
|
@ -0,0 +1,25 @@
|
|||
import Common.Logic.Basic
|
||||
import Mathlib.Data.Set.Intervals.Basic
|
||||
|
||||
namespace Set
|
||||
|
||||
/-! # Common.Set.Intervals
|
||||
|
||||
Additional theorems around intervals.
|
||||
-/
|
||||
|
||||
theorem Iio_nat_lt_ssubset {m n : ℕ} (h : m < n)
|
||||
: Iio m ⊂ Iio n := by
|
||||
rw [ssubset_def]
|
||||
apply And.intro
|
||||
· unfold Iio
|
||||
simp only [setOf_subset_setOf]
|
||||
intro x hx
|
||||
calc x
|
||||
_ < m := hx
|
||||
_ < n := h
|
||||
· show ¬ ∀ x, x < n → x < m
|
||||
simp only [not_forall, not_lt, exists_prop]
|
||||
exact ⟨m, h, by simp⟩
|
||||
|
||||
end Set
|
||||
Loading…
Reference in New Issue