Drop Theorem 6A references.
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@ -9304,8 +9304,8 @@
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\begin{lemma}
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Let $A \equin m$ for some natural number $m$ and $B \subseteq A$.
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Then there exists some $n \in \omega$ such that $n \leq m$, $B \equin n$ and
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$A - B \equin m - n$.
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Then there exists some $n \in \omega$ such that $n \leq m$, $B \equin n$,
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and $A - B \equin m - n$.
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\end{lemma}
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\code{Bookshelf/Enderton/Set/Chapter\_6}
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@ -9384,7 +9384,7 @@
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& = (A - B) - \{a\} & \eqref{par:set-difference-size-ii-eq2} \\
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& = A - B \\
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& \equin m - n & \eqref{par:set-difference-size-ii-eq1} \\
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& \equin m^+ - n^+ & \textref{sub:theorem-6a}.
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& \equin m^+ - n^+.
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\end{align*}
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\subparagraph{Case 2}%
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@ -9400,8 +9400,7 @@
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between $(A - \{a\}) - B$ and $m - n$.
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Therefore $g \cup \{\tuple{a, m}\}$ is a one-to-one correspondence
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between $A - B$ and $(m - n) \cup \{m\}$.
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By \nameref{sub:theorem-6a},
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$$A - B \equin (m - n) \cup \{m\} \equin m^+ - n.$$
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Hence $$A - B \equin (m - n) \cup \{m\} \equin m^+ - n.$$
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\subparagraph{Subconclusion}%
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@ -1119,41 +1119,11 @@ lemma sdiff_size_aux [DecidableEq α] [Nonempty α]
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> `(A - {a} - (B - {a})) = (A - B) - {a}`
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> ` = A - B`
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> ` ≈ m - n` *(6.4)*
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> ` ≈ m⁺ - n⁺` *(Theorem 6A)*
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> ` ≈ m⁺ - n⁺`
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-/
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rw [hA₁, hA₂] at hn₃
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suffices (Set.Iio m) \ (Set.Iio n) ≈ (Set.Iio m.succ) \ (Set.Iio n.succ)
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from Set.equinumerous_trans hn₃ this
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-- `m - n ≈ m⁺ - n⁺`
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refine ⟨fun x => x + 1, ?_, ?_, ?_⟩
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· intro x ⟨hx₁, hx₂⟩
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simp at hx₁ hx₂ ⊢
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exact ⟨Nat.le_add_of_sub_le hx₂, Nat.add_lt_of_lt_sub hx₁⟩
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· intro _ _ _ _ h
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simp only [add_left_inj] at h
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exact h
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· unfold Set.SurjOn Set.image
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rw [Set.subset_def]
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intro x ⟨hx₁, hx₂⟩
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simp only [
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Set.Iio_diff_Iio,
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gt_iff_lt,
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not_lt,
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ge_iff_le,
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Set.mem_setOf_eq,
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Set.mem_Iio
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] at hx₁ hx₂ ⊢
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have ⟨p, hp⟩ : ∃ p : ℕ, x = p.succ := by
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refine Nat.exists_eq_succ_of_ne_zero ?_
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have := calc 0
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_ < n.succ := by simp
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_ ≤ x := hx₂
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exact Nat.pos_iff_ne_zero.mp this
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refine ⟨p, ⟨?_, ?_⟩, hp.symm⟩
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· rw [hp] at hx₂
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exact Nat.lt_succ.mp hx₂
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· rw [hp] at hx₁
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exact Nat.succ_lt_succ_iff.mp hx₁
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exact Set.equinumerous_trans hn₃
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(Set.equinumerous_symm Set.succ_diff_succ_equinumerous_diff)
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/-
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> ##### Case 2
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> Assume `a ∉ B`. Then `B - {a} = B` (i.e. `B ≈ n`) and
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@ -1223,49 +1193,11 @@ lemma sdiff_size_aux [DecidableEq α] [Nonempty α]
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· exact ⟨hy₁.left.left, hy₁.right⟩
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· rwa [if_neg hy₁.left.right]
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/-
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> By *Theorem 6A*
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> Hence
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>
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> `A - B ≈ (m - n) ∪ {m} ≈ m⁺ - n`.
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-/
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suffices (Set.Iio m) \ (Set.Iio n) ∪ {m} ≈ (Set.Iio m.succ) \ (Set.Iio n)
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from Set.equinumerous_trans hAB this
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refine ⟨fun x => x, ?_, ?_, ?_⟩
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· intro x hx
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simp at hx ⊢
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apply Or.elim hx
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· intro hx₁
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rw [hx₁]
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exact ⟨hn₁, by simp⟩
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· intro ⟨hx₁, hx₂⟩
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exact ⟨hx₁, calc x
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_ < m := hx₂
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_ < m + 1 := by simp⟩
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· intro _ _ _ _ h
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exact h
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· unfold Set.SurjOn Set.image
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rw [Set.subset_def]
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simp only [
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Set.Iio_diff_Iio,
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gt_iff_lt,
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not_lt,
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ge_iff_le,
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Set.mem_Ico,
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Set.union_singleton,
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lt_self_iff_false,
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and_false,
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Set.mem_insert_iff,
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exists_eq_right,
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Set.mem_setOf_eq,
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and_imp
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]
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intro x hn hm
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apply Or.elim (Nat.lt_or_eq_of_lt hm)
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· intro hx
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right
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exact ⟨hn, hx⟩
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· intro hx
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left
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exact hx
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exact Set.equinumerous_trans hAB (Set.diff_union_equinumerous_succ_diff hn₁)
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/-
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> ##### Subconclusion
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> The above two cases are exhaustive and both conclude the existence of some
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@ -1,3 +1,5 @@
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import Common.Nat.Basic
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import Common.Set.Basic
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import Mathlib.Data.Finset.Card
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import Mathlib.Data.Set.Finite
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@ -128,4 +130,82 @@ theorem equinumerous_emptyset_emptyset [Bot β]
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· unfold SurjOn
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simp
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/--
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For all natural numbers `m, n`, `m⁺ - n⁺ ≈ m - n`.
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-/
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theorem succ_diff_succ_equinumerous_diff {m n : ℕ}
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: Set.Iio m.succ \ Set.Iio n.succ ≈ Set.Iio m \ Set.Iio n := by
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refine Set.equinumerous_symm ⟨fun x => x + 1, ?_, ?_, ?_⟩
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· intro x ⟨hx₁, hx₂⟩
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simp at hx₁ hx₂ ⊢
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exact ⟨Nat.le_add_of_sub_le hx₂, Nat.add_lt_of_lt_sub hx₁⟩
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· intro _ _ _ _ h
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simp only [add_left_inj] at h
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exact h
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· unfold Set.SurjOn Set.image
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rw [Set.subset_def]
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intro x ⟨hx₁, hx₂⟩
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simp only [
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Set.Iio_diff_Iio,
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gt_iff_lt,
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not_lt,
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ge_iff_le,
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Set.mem_setOf_eq,
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Set.mem_Iio
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] at hx₁ hx₂ ⊢
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have ⟨p, hp⟩ : ∃ p : ℕ, x = p.succ := by
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refine Nat.exists_eq_succ_of_ne_zero ?_
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have := calc 0
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_ < n.succ := by simp
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_ ≤ x := hx₂
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exact Nat.pos_iff_ne_zero.mp this
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refine ⟨p, ⟨?_, ?_⟩, hp.symm⟩
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· rw [hp] at hx₂
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exact Nat.lt_succ.mp hx₂
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· rw [hp] at hx₁
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exact Nat.succ_lt_succ_iff.mp hx₁
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/--
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For all natural numbers `m, n`, `m - n ∪ {m} ≈ m - n`.
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-/
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theorem diff_union_equinumerous_succ_diff {m n : ℕ} (hn: n ≤ m)
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: Set.Iio m \ Set.Iio n ∪ {m} ≈ Set.Iio (Nat.succ m) \ Set.Iio n := by
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refine ⟨fun x => x, ?_, ?_, ?_⟩
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· intro x hx
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simp at hx ⊢
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apply Or.elim hx
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· intro hx₁
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rw [hx₁]
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exact ⟨hn, by simp⟩
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· intro ⟨hx₁, hx₂⟩
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exact ⟨hx₁, calc x
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_ < m := hx₂
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_ < m + 1 := by simp⟩
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· intro _ _ _ _ h
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exact h
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· unfold Set.SurjOn Set.image
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rw [Set.subset_def]
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simp only [
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Set.Iio_diff_Iio,
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gt_iff_lt,
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not_lt,
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ge_iff_le,
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Set.mem_Ico,
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Set.union_singleton,
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lt_self_iff_false,
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and_false,
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Set.mem_insert_iff,
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exists_eq_right,
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Set.mem_setOf_eq,
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and_imp
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]
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intro x hn hm
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apply Or.elim (Nat.lt_or_eq_of_lt hm)
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· intro hx
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right
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exact ⟨hn, hx⟩
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· intro hx
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left
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exact hx
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end Set
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