Drop Theorem 6A references.

Bookshelf/Enderton/Set.tex

@@ -9304,8 +9304,8 @@
 
   \begin{lemma}
     Let $A \equin m$ for some natural number $m$ and $B \subseteq A$.
-    Then there exists some $n \in \omega$ such that $n \leq m$, $B \equin n$ and
+    Then there exists some $n \in \omega$ such that $n \leq m$, $B \equin n$,
-      $A - B \equin m - n$.
+      and $A - B \equin m - n$.
   \end{lemma}
 
   \code{Bookshelf/Enderton/Set/Chapter\_6}
@@ -9384,7 +9384,7 @@
               & = (A - B) - \{a\} & \eqref{par:set-difference-size-ii-eq2} \\
               & = A - B \\
               & \equin m - n & \eqref{par:set-difference-size-ii-eq1} \\
-              & \equin m^+ - n^+ & \textref{sub:theorem-6a}.
+              & \equin m^+ - n^+.
           \end{align*}
 
       \subparagraph{Case 2}%
@@ -9400,8 +9400,7 @@
           between $(A - \{a\}) - B$ and $m - n$.
         Therefore $g \cup \{\tuple{a, m}\}$ is a one-to-one correspondence
           between $A - B$ and $(m - n) \cup \{m\}$.
-        By \nameref{sub:theorem-6a},
+        Hence $$A - B \equin (m - n) \cup \{m\} \equin m^+ - n.$$
-          $$A - B \equin (m - n) \cup \{m\} \equin m^+ - n.$$
 
       \subparagraph{Subconclusion}%
 

Bookshelf/Enderton/Set/Chapter_6.lean

@@ -1119,41 +1119,11 @@ lemma sdiff_size_aux [DecidableEq α] [Nonempty α]
 > `(A - {a} - (B - {a})) = (A - B) - {a}`
 >                      ` = A - B`
 >                      ` ≈ m - n`          *(6.4)*
->                      ` ≈ m⁺ - n⁺`        *(Theorem 6A)*
+>                      ` ≈ m⁺ - n⁺`
 -/
         rw [hA₁, hA₂] at hn₃
-        suffices (Set.Iio m) \ (Set.Iio n) ≈ (Set.Iio m.succ) \ (Set.Iio n.succ)
+        exact Set.equinumerous_trans hn₃
-          from Set.equinumerous_trans hn₃ this
+          (Set.equinumerous_symm Set.succ_diff_succ_equinumerous_diff)
-        -- `m - n ≈ m⁺ - n⁺`
-        refine ⟨fun x => x + 1, ?_, ?_, ?_⟩
-        · intro x ⟨hx₁, hx₂⟩
-          simp at hx₁ hx₂ ⊢
-          exact ⟨Nat.le_add_of_sub_le hx₂, Nat.add_lt_of_lt_sub hx₁⟩
-        · intro _ _ _ _ h
-          simp only [add_left_inj] at h
-          exact h
-        · unfold Set.SurjOn Set.image
-          rw [Set.subset_def]
-          intro x ⟨hx₁, hx₂⟩
-          simp only [
-            Set.Iio_diff_Iio,
-            gt_iff_lt,
-            not_lt,
-            ge_iff_le,
-            Set.mem_setOf_eq,
-            Set.mem_Iio
-          ] at hx₁ hx₂ ⊢
-          have ⟨p, hp⟩ : ∃ p : ℕ, x = p.succ := by
-            refine Nat.exists_eq_succ_of_ne_zero ?_
-            have := calc 0
-              _ < n.succ := by simp
-              _ ≤ x := hx₂
-            exact Nat.pos_iff_ne_zero.mp this
-          refine ⟨p, ⟨?_, ?_⟩, hp.symm⟩
-          · rw [hp] at hx₂
-            exact Nat.lt_succ.mp hx₂
-          · rw [hp] at hx₁
-            exact Nat.succ_lt_succ_iff.mp hx₁
 /-
 > ##### Case 2
 > Assume `a ∉ B`. Then `B - {a} = B` (i.e. `B ≈ n`) and
@@ -1223,49 +1193,11 @@ lemma sdiff_size_aux [DecidableEq α] [Nonempty α]
               · exact ⟨hy₁.left.left, hy₁.right⟩
               · rwa [if_neg hy₁.left.right]
 /-
-> By *Theorem 6A*
+> Hence
 >
 > `A - B ≈ (m - n) ∪ {m} ≈ m⁺ - n`.
 -/
-        suffices (Set.Iio m) \ (Set.Iio n) ∪ {m} ≈ (Set.Iio m.succ) \ (Set.Iio n)
+        exact Set.equinumerous_trans hAB (Set.diff_union_equinumerous_succ_diff hn₁)
-          from Set.equinumerous_trans hAB this
-        refine ⟨fun x => x, ?_, ?_, ?_⟩
-        · intro x hx
-          simp at hx ⊢
-          apply Or.elim hx
-          · intro hx₁
-            rw [hx₁]
-            exact ⟨hn₁, by simp⟩
-          · intro ⟨hx₁, hx₂⟩
-            exact ⟨hx₁, calc x
-              _ < m := hx₂
-              _ < m + 1 := by simp⟩
-        · intro _ _ _ _ h
-          exact h
-        · unfold Set.SurjOn Set.image
-          rw [Set.subset_def]
-          simp only [
-            Set.Iio_diff_Iio,
-            gt_iff_lt,
-            not_lt,
-            ge_iff_le,
-            Set.mem_Ico,
-            Set.union_singleton,
-            lt_self_iff_false,
-            and_false,
-            Set.mem_insert_iff,
-            exists_eq_right,
-            Set.mem_setOf_eq,
-            and_imp
-          ]
-          intro x hn hm
-          apply Or.elim (Nat.lt_or_eq_of_lt hm)
-          · intro hx
-            right
-            exact ⟨hn, hx⟩
-          · intro hx
-            left
-            exact hx
 /-
 > ##### Subconclusion
 > The above two cases are exhaustive and both conclude the existence of some

Common/Set/Equinumerous.lean

@@ -1,3 +1,5 @@
+import Common.Nat.Basic
+import Common.Set.Basic
 import Mathlib.Data.Finset.Card
 import Mathlib.Data.Set.Finite
 
@@ -128,4 +130,82 @@ theorem equinumerous_emptyset_emptyset [Bot β]
   · unfold SurjOn
     simp
 
+/--
+For all natural numbers `m, n`, `m⁺ - n⁺ ≈ m - n`.
+-/
+theorem succ_diff_succ_equinumerous_diff {m n : ℕ}
+  : Set.Iio m.succ \ Set.Iio n.succ ≈ Set.Iio m \ Set.Iio n := by
+  refine Set.equinumerous_symm ⟨fun x => x + 1, ?_, ?_, ?_⟩
+  · intro x ⟨hx₁, hx₂⟩
+    simp at hx₁ hx₂ ⊢
+    exact ⟨Nat.le_add_of_sub_le hx₂, Nat.add_lt_of_lt_sub hx₁⟩
+  · intro _ _ _ _ h
+    simp only [add_left_inj] at h
+    exact h
+  · unfold Set.SurjOn Set.image
+    rw [Set.subset_def]
+    intro x ⟨hx₁, hx₂⟩
+    simp only [
+      Set.Iio_diff_Iio,
+      gt_iff_lt,
+      not_lt,
+      ge_iff_le,
+      Set.mem_setOf_eq,
+      Set.mem_Iio
+    ] at hx₁ hx₂ ⊢
+    have ⟨p, hp⟩ : ∃ p : ℕ, x = p.succ := by
+      refine Nat.exists_eq_succ_of_ne_zero ?_
+      have := calc 0
+        _ < n.succ := by simp
+        _ ≤ x := hx₂
+      exact Nat.pos_iff_ne_zero.mp this
+    refine ⟨p, ⟨?_, ?_⟩, hp.symm⟩
+    · rw [hp] at hx₂
+      exact Nat.lt_succ.mp hx₂
+    · rw [hp] at hx₁
+      exact Nat.succ_lt_succ_iff.mp hx₁
+
+/--
+For all natural numbers `m, n`, `m - n ∪ {m} ≈ m - n`.
+-/
+theorem diff_union_equinumerous_succ_diff {m n : ℕ} (hn: n ≤ m)
+  : Set.Iio m \ Set.Iio n ∪ {m} ≈ Set.Iio (Nat.succ m) \ Set.Iio n := by
+  refine ⟨fun x => x, ?_, ?_, ?_⟩
+  · intro x hx
+    simp at hx ⊢
+    apply Or.elim hx
+    · intro hx₁
+      rw [hx₁]
+      exact ⟨hn, by simp⟩
+    · intro ⟨hx₁, hx₂⟩
+      exact ⟨hx₁, calc x
+        _ < m := hx₂
+        _ < m + 1 := by simp⟩
+  · intro _ _ _ _ h
+    exact h
+  · unfold Set.SurjOn Set.image
+    rw [Set.subset_def]
+    simp only [
+      Set.Iio_diff_Iio,
+      gt_iff_lt,
+      not_lt,
+      ge_iff_le,
+      Set.mem_Ico,
+      Set.union_singleton,
+      lt_self_iff_false,
+      and_false,
+      Set.mem_insert_iff,
+      exists_eq_right,
+      Set.mem_setOf_eq,
+      and_imp
+    ]
+    intro x hn hm
+    apply Or.elim (Nat.lt_or_eq_of_lt hm)
+    · intro hx
+      right
+      exact ⟨hn, hx⟩
+    · intro hx
+      left
+      exact hx
+
 end Set