Use swap theorems in pigeonhole principle proof.

Establish precedent for embedding LaTeX proof into code.

Bookshelf/Enderton/Set/Chapter_1.lean

@@ -126,4 +126,4 @@ theorem exercise_1_4 (x y : α) (hx : x ∈ B) (hy : y ∈ B)
       (Set.singleton_subset_iff.mpr hx)
       (Set.singleton_subset_iff.mpr hy)
 
-end Enderton.Set.Chapter_1
+end Enderton.Set.Chapter_1

Bookshelf/Enderton/Set/Chapter_2.lean

@@ -20,7 +20,8 @@ A ∩ B = B ∩ A
 -/
 
 theorem commutative_law_i (A B : Set α)
-  : A ∪ B = B ∪ A := calc A ∪ B
+  : A ∪ B = B ∪ A :=
+  calc A ∪ B
   _ = { x | x ∈ A ∨ x ∈ B } := rfl
   _ = { x | x ∈ B ∨ x ∈ A } := by
     ext

Bookshelf/Enderton/Set/Chapter_3.lean

@@ -23,8 +23,16 @@ If `x ∈ C` and `y ∈ C`, then `⟨x, y⟩ ∈ 𝒫 𝒫 C`.
 -/
 lemma lemma_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
   : OrderedPair x y ∈ 𝒫 𝒫 C := by
+/-
+> Let `C` be an arbitrary set and `x, y ∈ C`. Then by definition of the power
+> set, `{x}` and `{x, y}` are members of `𝒫 C`.
+-/
   have hxs : {x} ⊆ C := Set.singleton_subset_iff.mpr hx
   have hxys : {x, y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
+/-
+> Likewise `{{x}, {x, y}}` is a member of `𝒫 𝒫 C`. By definition of an ordered
+> pair, `⟨x, y⟩ = {{x}, {x, y}}`. This concludes our proof.
+-/
   exact Set.mem_mem_imp_pair_subset hxs hxys
 
 /-- #### Theorem 3D
@@ -33,10 +41,23 @@ If `⟨x, y⟩ ∈ A`, then `x` and `y` belong to `⋃ ⋃ A`.
 -/
 theorem theorem_3d {A : Set (Set (Set α))} (h : OrderedPair x y ∈ A)
   : x ∈ ⋃₀ (⋃₀ A) ∧ y ∈ ⋃₀ (⋃₀ A) := by
+/-
+> Let `A` be a set and `⟨x, y⟩ ∈ A`. By definition of an ordered pair,
+>
+> `⟨x, y⟩ = {{x}, {x, y}}`.
+>
+> By Exercise 2.3, `{{x}, {x, y}} ⊆ ∪ A`. Then `{x, y} ∈ ∪ A`.
+-/
   have hp := Chapter_2.exercise_2_3 (OrderedPair x y) h
-  unfold OrderedPair at hp  
+  unfold OrderedPair at hp
   have hq : {x, y} ∈ ⋃₀ A := hp (by simp)
+/-
+> Another application of Exercise 2.3 implies `{x, y} ∈ ∪ ∪ A`.
+-/
   have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq
+/-
+> Therefore `x, y ∈ ∪ ∪ A`.
+-/
   exact ⟨this (by simp), this (by simp)⟩
 
 
@@ -199,7 +220,7 @@ theorem theorem_3j_a {F : Set.HRelation α β}
         Set.mem_union,
         Prod.mk.injEq
       ] at ht
-      have ⟨a₁, ha₁⟩ := ht 
+      have ⟨a₁, ha₁⟩ := ht
       apply Or.elim ha₁
       · intro ⟨⟨a, b, hab⟩, _⟩
         have := mem_pair_imp_fst_mem_dom hab.left
@@ -1073,7 +1094,7 @@ theorem exercise_3_17_i {F : Set.HRelation β γ} {G : Set.HRelation α β}
   (hF : isSingleRooted F) (hG : isSingleRooted G)
   : isSingleRooted (comp F G):= by
   intro v hv
-  
+
   have ⟨u₁, hu₁⟩ := ran_exists hv
   have hu₁' := hu₁
   unfold comp at hu₁'
@@ -1081,13 +1102,13 @@ theorem exercise_3_17_i {F : Set.HRelation β γ} {G : Set.HRelation α β}
   have ⟨t₁, ht₁⟩ := hu₁'
   unfold ExistsUnique
   refine ⟨u₁, ⟨mem_pair_imp_fst_mem_dom hu₁, hu₁⟩, ?_⟩
-  
+
   intro u₂ hu₂
   have hu₂' := hu₂
   unfold comp at hu₂'
   simp only [Set.mem_setOf_eq] at hu₂'
   have ⟨_, ⟨t₂, ht₂⟩⟩ := hu₂'
-  
+
   have ht : t₁ = t₂ := single_rooted_eq_unique hF ht₁.right ht₂.right
   rw [ht] at ht₁
   exact single_rooted_eq_unique hG ht₂.left ht₁.left
@@ -1435,7 +1456,7 @@ theorem exercise_3_19_x
     apply Or.elim ht₁
     · intro ht
       rw [← ht] at ht₂
-      simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at ht₂ 
+      simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at ht₂
       apply Or.elim ht₂
       · intro ha
         rw [ha] at hx
@@ -1780,7 +1801,7 @@ theorem exercise_3_28 {A : Set α} {B : Set β}
       have hX₁sub := mem_pair_imp_fst_mem_dom hX₁
       rw [dG] at hX₁sub
       simp only [Set.mem_powerset_iff] at hX₁sub
-      
+
       have ht' := hX₁sub ht
       rw [← hf.right.dom_eq] at ht'
       have ⟨ft, hft⟩ := dom_exists ht'
@@ -1885,12 +1906,12 @@ theorem exercise_3_30_a : F B = B ∧ F C = C := by
         show ∀ t, t ∈ B → t ∈ X
         intro t ht
         rw [hB] at ht
-        simp only [Set.mem_sInter] at ht 
+        simp only [Set.mem_sInter] at ht
         exact ht X ⟨hX₁, hX₂⟩
       exact hX₂ (hMono B X ⟨hB₁, hX₁⟩ hx)
     rw [hB]
     exact this
-  
+
   have hC_supset : C ⊆ F C := by
     intro x hx
     rw [hC] at hx
@@ -1910,7 +1931,7 @@ theorem exercise_3_30_a : F B = B ∧ F C = C := by
       have ⟨T, hT⟩ := ht
       exact hT.left.left hT.right
     exact hMono X C ⟨hC₁, hC₂⟩ (hX.left.right hX.right)
-  
+
   have hC_sub_A : C ⊆ A := by
     show ∀ t, t ∈ C → t ∈ A
     intro t ht
@@ -1936,7 +1957,7 @@ theorem exercise_3_30_a : F B = B ∧ F C = C := by
     intro t ht
     simp only [Set.mem_sUnion, Set.mem_setOf_eq]
     exact ⟨X, hX, ht⟩
-  
+
   have hB_sub_A : B ⊆ A := by
     show ∀ t, t ∈ B → t ∈ A
     intro t ht
@@ -1951,11 +1972,11 @@ theorem exercise_3_30_a : F B = B ∧ F C = C := by
     · exact hB_subset
     · have hInter : ∀ X, X ∈ {X | X ⊆ A ∧ F X ⊆ X} → B ⊆ X := by
         intro X hX
-        simp only [Set.mem_setOf_eq] at hX 
+        simp only [Set.mem_setOf_eq] at hX
         rw [hB]
         show ∀ t, t ∈ ⋂₀ {X | X ⊆ A ∧ F X ⊆ X} → t ∈ X
         intro t ht
-        simp only [Set.mem_sInter, Set.mem_setOf_eq] at ht 
+        simp only [Set.mem_sInter, Set.mem_setOf_eq] at ht
         exact ht X hX
       refine hInter (F B) ⟨?_, ?_⟩
       · show ∀ t, t ∈ F B → t ∈ A
@@ -1978,7 +1999,7 @@ theorem exercise_3_30_b : ∀ X, X ⊆ A ∧ F X = X → B ⊆ X ∧ X ⊆ C :=
     rw [hB]
     show ∀ t, t ∈ ⋂₀ {X | X ⊆ A ∧ F X ⊆ X} → t ∈ X
     intro t ht
-    simp only [Set.mem_sInter, Set.mem_setOf_eq] at ht 
+    simp only [Set.mem_sInter, Set.mem_setOf_eq] at ht
     exact ht X ⟨hX₁, this⟩
   · have : X ⊆ F X := Eq.subset (id (Eq.symm hX₂))
     rw [hC]
@@ -2173,7 +2194,7 @@ theorem exercise_3_36 {f : Set.HRelation α β}
       Prod.exists,
       exists_and_right,
       exists_eq_right
-    ] at hx 
+    ] at hx
     apply Or.elim hx
     · intro ⟨_, _, hx₁⟩
       rw [← hf.dom_eq]
@@ -2595,4 +2616,4 @@ theorem exercise_3_45 {A : Rel α α} {B : Rel β β} {R : Rel (α × β) (α ×
         have := hB.trans b₁ b₂ b₃ ha₁.right hb₂
         exact (hR a₁ b₁ a₃ b₃).mpr (Or.inr ⟨ha₂, this⟩)
 
-end Enderton.Set.Chapter_3
+end Enderton.Set.Chapter_3

Bookshelf/Enderton/Set/Chapter_6.lean

@@ -72,28 +72,72 @@ lemma pigeonhole_principle_aux (n : ℕ)
       ∀ f : ℕ → ℕ,
         Set.MapsTo f M (Set.Iio n) ∧ Set.InjOn f M →
         ¬ Set.SurjOn f M (Set.Iio n) := by
+/-
+> Let
+>
+> `S = {n ∈ ω | ∀ M ⊂ n, every one-to-one function f: M → n is not onto}`. (1)
+>
+> We show that (i) `0 ∈ S` and (ii) if `n ∈ S`, then so is `n⁺`. Afterward we
+> prove (iii) the theorem statement.
+-/
   induction n with
+/-
+## (i)
+> By definition, `0 = ∅`.
+-/
   | zero =>
     intro _ hM
     unfold Set.Iio at hM
     simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hM
+/-
+> Then `0` has no proper subsets.
+-/
     rw [Set.ssubset_empty_iff_false] at hM
+/-
+> Hence `0 ∈ S` vacuously.
+-/
     exact False.elim hM
+/-
+## (ii)
+> Suppose `n ∈ S` and `M ⊂ n⁺`. Furthermore, let `f: M → n⁺` be a one-to-one
+> function.
+-/
   | succ n ih =>
     intro M hM f ⟨hf_maps, hf_inj⟩ hf_surj
-
+/-
+> If `M = ∅`, it vacuously holds that `f` is not onto `n⁺`.
+-/
     by_cases hM' : M = ∅
     · rw [hM', Set.SurjOn_emptyset_Iio_iff_eq_zero] at hf_surj
       simp at hf_surj
-
+/-
+> Otherwise `M ≠ 0`. Because `M` is finite, the trichotomy law for `ω` implies
+> the existence of a largest member `p ∈ M`. There are two cases to consider:
+-/
     by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n
-    -- Trivial case. `f` must not be onto if this is the case.
+/-
+### Case 1
+> `n ∉ ran f`.
+> Then `f` is not onto `n⁺`.
+-/
     · have ⟨t, ht⟩ := hf_surj (show n ∈ _ by simp)
       exact absurd ⟨t, ht⟩ h
-
-    -- Continue under the assumption `n ∈ ran f`.
+/-
+### Case 2
+> `n ∈ ran f`.
+> Then there exists some `t ∈ M` such that `⟨t, n⟩ ∈ f`.
+-/
     have ⟨t, ht₁, ht₂⟩ := not_not.mp h
-
+/-
+> Define `f': M → n⁺` given by
+>
+> `f'(p) = f(t) = n`
+> `f'(t) = f(p)`
+> `f'(x) = f(x)` for all other `x`.
+>
+> That is, `f'` is a variant of `f` in which the largest element of its domain
+> (i.e. `p`) corresponds to value `n`.
+-/
     -- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`, i.e. a maximum member.
     have ⟨p, hp₁, hp₂⟩ : ∃ p ∈ M, ∀ x, x ∈ M → p ≥ x := by
       refine subset_finite_max_nat (show Set.Finite M from ?_) ?_ ?_
@@ -104,69 +148,19 @@ lemma pigeonhole_principle_aux (n : ℕ)
         exact Set.nmem_singleton_empty.mp hM'
       · show M ⊆ M
         exact Eq.subset rfl
-
-    -- `g` is a variant of `f` in which the largest element of its domain
-    -- (i.e. `p`) corresponds to value `n`.
-    let g x := if x = p then n else if x = t then f p else f x
-
-    have hg_maps : Set.MapsTo g M (Set.Iio (n + 1)) := by
-      intro x hx
-      dsimp only
-      by_cases hx₁ : x = p
-      · rw [hx₁]
-        simp
-      · rw [if_neg hx₁]
-        by_cases hx₂ : x = t
-        · rw [hx₂]
-          simp only [ite_true, Set.mem_Iio]
-          exact hf_maps hp₁
-        · rw [if_neg hx₂]
-          simp only [Set.mem_Iio]
-          exact hf_maps hx
-
-    have hg_inj : Set.InjOn g M := by
-      intro x₁ hx₁ x₂ hx₂ hf'
-      by_cases hc₁ : x₁ = p
-      · by_cases hc₂ : x₂ = p
-        · rw [hc₁, hc₂]
-        · dsimp at hf'
-          rw [hc₁] at hf'
-          simp only [ite_self, ite_true] at hf'
-          by_cases hc₃ : x₂ = t
-          · rw [if_neg hc₂, if_pos hc₃, ← ht₂] at hf'
-            rw [hc₁] at hx₁ ⊢
-            rw [hc₃] at hx₂ ⊢
-            exact hf_inj hx₁ hx₂ hf'.symm
-          · rw [if_neg hc₂, if_neg hc₃, ← ht₂] at hf'
-            have := hf_inj ht₁ hx₂ hf'
-            exact absurd this.symm hc₃
-      · by_cases hc₂ : x₂ = p
-        · rw [hc₂] at hf'
-          simp only [ite_self, ite_true] at hf'
-          by_cases hc₃ : x₁ = t
-          · rw [if_neg hc₁, if_pos hc₃, ← ht₂] at hf'
-            rw [hc₃] at hx₁ ⊢
-            rw [hc₂] at hx₂ ⊢
-            have := hf_inj hx₂ hx₁ hf'
-            exact this.symm
-          · rw [if_neg hc₁, if_neg hc₃, ← ht₂] at hf'
-            have := hf_inj hx₁ ht₁ hf'
-            exact absurd this hc₃
-        · dsimp only at hf'
-          rw [if_neg hc₁, if_neg hc₂] at hf'
-          by_cases hc₃ : x₁ = t
-          · by_cases hc₄ : x₂ = t
-            · rw [hc₃, hc₄]
-            · rw [if_pos hc₃, if_neg hc₄] at hf'
-              have := hf_inj hp₁ hx₂ hf'
-              exact absurd this.symm hc₂
-          · by_cases hc₄ : x₂ = t
-            · rw [if_neg hc₃, if_pos hc₄] at hf'
-              have := hf_inj hx₁ hp₁ hf'
-              exact absurd this hc₁
-            · rw [if_neg hc₃, if_neg hc₄] at hf'
-              exact hf_inj hx₁ hx₂ hf'
-
+/-
+> Next define `g = f' - {⟨p, n⟩}`. Then `g` is a function mapping `M - {p}` to
+> `n`.
+-/
+    let g := Set.Function.swap f p t
+/-
+> Since `f` is one-to-one, `f'` and `g` are also one-to-one.
+-/
+    have hg_maps := Set.Function.swap_MapsTo_self hp₁ ht₁ hf_maps
+    have hg_inj := Set.Function.swap_InjOn_self hp₁ ht₁ hf_inj
+/-
+> Then (1) indicates `g` must not be onto `n`.
+-/
     let M' := M \ {p}
     have hM' : M' ⊂ Set.Iio n := by
       by_cases hc : p = n
@@ -209,18 +203,28 @@ lemma pigeonhole_principle_aux (n : ℕ)
       · -- `Set.MapsTo g M' (Set.Iio n)`
         intro x hx
         have hx₁ : x ∈ M := Set.mem_of_mem_diff hx
-        apply Or.elim (Nat.lt_or_eq_of_lt $ hg_maps hx₁)
-        · exact id
-        · intro hx₂
-          rw [← show g p = n by simp] at hx₂
-          exact absurd (hg_inj hx₁ hp₁ hx₂) hx.right
+        apply Or.elim (Nat.lt_or_eq_of_lt $ hg_maps hx₁) id
+        intro hx₂
+        unfold Set.Function.swap at hx₂
+        by_cases hc₁ : x = p
+        · exact absurd hc₁ hx.right
+        · rw [if_neg hc₁] at hx₂
+          by_cases hc₂ : x = t
+          · rw [if_pos hc₂, ← ht₂] at hx₂
+            have := hf_inj hp₁ ht₁ hx₂
+            rw [← hc₂] at this
+            exact absurd this.symm hc₁
+          · rw [if_neg hc₂, ← ht₂] at hx₂
+            have := hf_inj hx₁ ht₁ hx₂
+            exact absurd this hc₂
       · -- `Set.InjOn g M'`
         intro x₁ hx₁ x₂ hx₂ hg
         have hx₁' : x₁ ∈ M := (Set.diff_subset M {p}) hx₁
         have hx₂' : x₂ ∈ M := (Set.diff_subset M {p}) hx₂
         exact hg_inj hx₁' hx₂' hg
-
-    -- We have shown `g` isn't surjective. This is another way of saying that.
+/-
+> That is, there exists some `a ∈ n` such that `a ∉ ran g`.
+-/
     have ⟨a, ha₁, ha₂⟩ : ∃ a, a < n ∧ a ∉ g '' M' := by
       unfold Set.SurjOn at ng_surj
       rw [Set.subset_def] at ng_surj
@@ -231,12 +235,15 @@ lemma pigeonhole_principle_aux (n : ℕ)
         not_exists,
         not_and,
         exists_prop
-      ] at ng_surj 
+      ] at ng_surj
       unfold Set.image
       simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and]
       exact ng_surj
-
-    -- If `g` isn't surjective then neither is `f`.
+/-
+> By the trichotomy law for `ω`, `a ≠ n`. Therefore `a ∉ ran f'`.
+> `ran f' = ran f` meaning `a ∉ ran f`. Because `a ∈ n ∈n⁺`, Theorem 4F implies
+> `a ∈ n⁺`. Hence `f` is not onto `n⁺`.
+-/
     refine absurd (hf_surj $ calc a
       _ < n := ha₁
       _ < n + 1 := by simp) (show ↑a ∉ f '' M from ?_)
@@ -248,17 +255,20 @@ lemma pigeonhole_principle_aux (n : ℕ)
       simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and] at ha₂ ⊢
       intro x hx
       by_cases hxp : x = p
-      · rw [if_pos hxp]
+      · unfold Set.Function.swap
+        rw [if_pos hxp, ht₂]
         exact (Nat.ne_of_lt ha₁).symm
       · refine ha₂ x ?_
         exact Set.mem_diff_of_mem hx hxp
 
     ext x
+    dsimp only
+    unfold Set.Function.swap
     simp only [Set.mem_image, Set.mem_Iio]
     apply Iff.intro
     · intro ⟨y, hy₁, hy₂⟩
       by_cases hc₁ : y = p
-      · rw [if_pos hc₁] at hy₂
+      · rw [if_pos hc₁, ht₂] at hy₂
         rw [hy₂] at ht₂
         exact ⟨t, ht₁, ht₂⟩
       · rw [if_neg hc₁] at hy₂
@@ -274,15 +284,25 @@ lemma pigeonhole_principle_aux (n : ℕ)
         · rw [hc₂, ht₂] at hy₂
           rw [← hc₁, ← hc₂]
           simp only [ite_self, ite_true]
-          exact hy₂
+          rwa [hc₂, ht₂]
         · rw [hc₁, ← Ne.def] at hc₂
           rwa [if_neg hc₂.symm, if_pos rfl, ← hc₁]
       · by_cases hc₂ : y = t
         · refine ⟨p, hp₁, ?_⟩
           simp only [ite_self, ite_true]
-          rwa [hc₂, ht₂] at hy₂
+          rwa [hc₂] at hy₂
         · refine ⟨y, hy₁, ?_⟩
           rwa [if_neg hc₁, if_neg hc₂]
+/-
+### Subconclusion
+> The foregoing cases are exhaustive. Hence `n⁺ ∈ S`.
+
+## (iii)
+> By (i) and (ii), `S` is an inductive set. By Theorem 4B, `S = ω`. Thus for all
+> natural numbers `n`, there is no one-to-one correspondence between `n` and a
+> proper subset of `n`. In other words, no natural number is equinumerous to a
+> proper subset of itself.
+-/
 
 /--
 No natural number is equinumerous to a proper subset of itself.
@@ -339,7 +359,7 @@ theorem corollary_6c [DecidableEq α] [Nonempty α]
         Set.le_eq_subset,
         Set.singleton_subset_iff,
         Set.mem_empty_iff_false
-      ] at hx₃ 
+      ] at hx₃
     · -- `Set.InjOn H S'`
       intro x₁ hx₁ x₂ hx₂ h
       have hc₁ : x₁ ∈ S' ∪ T := Set.mem_union_left T hx₁
@@ -368,7 +388,7 @@ theorem corollary_6c [DecidableEq α] [Nonempty α]
   have hf₂ : R ≈ Set.Iio n := by
     have ⟨k, hk⟩ := Set.equinumerous_symm hf₁
     exact Set.equinumerous_trans ⟨k, hk⟩ hG
-    
+
   refine absurd hf₂ (pigeonhole_principle ?_)
   show R ⊂ Set.Iio n
   apply And.intro
@@ -536,7 +556,7 @@ lemma lemma_6f {n : ℕ}
 
       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, ?_, ?_, ?_⟩
@@ -1053,14 +1073,14 @@ theorem exercise_6_7 [DecidableEq α] [Nonempty α] {A : Set α} {f : α → α}
       have hB₂ : x₂ ∈ B := sorry
       have hB₃ : B ⊆ A := sorry
       have ⟨m₁, n₁, hm₁, hn₁, hmn₁⟩ := subset_size hB₃ hA₁
-      
+
       have hf'₁ : Set.MapsTo f (A \ B) (A \ {y}) := sorry
       have hf'₂ : f '' (A \ B) = A \ {y} := sorry
       have hf'₃ : Set.Finite (A \ B) := sorry
       have hf'₄ : Set.Finite (A \ {y}) := sorry
 
       have ⟨m₂, n₂, hm₂, hn₂, hmn₂⟩ := finite_dom_ran_size hf'₃ hf'₄ hf'₁
-      
+
       have h₁ : A \ B ≈ Set.Iio (n₁ - m₁) := sorry
       have h₂ : A \ {y} ≈ Set.Iio (n₁ - 1) := sorry
       sorry

Common/Set/Function.lean

@@ -5,7 +5,7 @@ import Mathlib.Data.Set.Function
 Additional theorems around functions defined on sets.
 -/
 
-namespace Set
+namespace Set.Function
 
 /--
 Produce a new function that swaps the outputs of the two specified inputs.
@@ -232,4 +232,4 @@ theorem self_iff_swap_BijOn [DecidableEq α]
   : BijOn (swap f a₁ a₂) A B ↔ BijOn f A B :=
   ⟨self_BijOn_swap ha₁ ha₂, swap_BijOn_self ha₁ ha₂⟩
 
-end Set
+end Set.Function