Enderton. Verify theorem 3I.

Bookshelf/Enderton/Set.tex

@@ -3293,7 +3293,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
 \end{proof}
 
-\subsection{\partial{Theorem 3I}}%
+\subsection{\verified{Theorem 3I}}%
 \label{sub:theorem-3i}
 
 \begin{theorem}[3I]
@@ -3304,6 +3304,9 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
 \begin{proof}
 
+  \lean{Common/Set/Relation}
+    {Set.Relation.comp\_inv\_eq\_inv\_comp\_inv}
+
   By definition of the \nameref{ref:composition} of $F$ and $G$,
     $$F \circ G = \{\left< u, v \right> \mid \exists t(uGt \land tFv)\}.$$
   By definition of the \nameref{ref:inverse} of a function,

Bookshelf/Enderton/Set/Chapter_3.lean

@@ -66,7 +66,7 @@ theorem exercise_5_2b {A : Set α} {B C : Set β}
   (h : Set.prod A B = Set.prod A C) (hA : Set.Nonempty A)
   : B = C := by
   by_cases hB : Set.Nonempty B
-  · suffices B ⊆ C ∧ C ⊆ B from Set.Subset.antisymm_iff.mpr this
+  · rw [Set.Subset.antisymm_iff]
     have ⟨a, ha⟩ := hA
     apply And.intro
     · show ∀ t, t ∈ B → t ∈ C
@@ -189,9 +189,7 @@ With `A`, `B`, and `C` as above, show that `A × B = ∪ C`.
 -/
 theorem exercise_5_5b {A : Set α} (B : Set β)
   : Set.prod A B = ⋃₀ {Set.prod ({x} : Set α) B | x ∈ A} := by
-  suffices Set.prod A B ⊆ ⋃₀ {Set.prod {x} B | x ∈ A} ∧
+  rw [Set.Subset.antisymm_iff]
-           ⋃₀ {Set.prod {x} B | x ∈ A} ⊆ Set.prod A B from
-    Set.Subset.antisymm_iff.mpr this
   apply And.intro
   · show ∀ t, t ∈ Set.prod A B → t ∈ ⋃₀ {Set.prod {x} B | x ∈ A}
     intro t h
@@ -255,9 +253,7 @@ Show that if `R` is a relation, then `fld R = ⋃ ⋃ R`.
 theorem exercise_6_7 {R : Set.Relation α}
   : R.fld = ⋃₀ ⋃₀ R.toOrderedPairs := by
   let img := R.toOrderedPairs
-  suffices R.fld ⊆ ⋃₀ ⋃₀ img ∧ ⋃₀ ⋃₀ img ⊆ R.fld from
+  rw [Set.Subset.antisymm_iff]
-    Set.Subset.antisymm_iff.mpr this
-
   apply And.intro
   · show ∀ x, x ∈ R.fld → x ∈ ⋃₀ ⋃₀ img
     intro x hx
@@ -440,8 +436,8 @@ theorem exercise_6_9_ii {A : Set (Set.Relation α)}
 
 Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
 -/
-theorem theorem_3g_i {α : Type _} {x y : α} {F : Set.Relation α}
+theorem theorem_3g_i {F : Set.Relation α}
-  (hF : isOneToOne F) (hx : x ∈ dom F)
+  (hF : F.isOneToOne) (hx : x ∈ dom F)
   : ∃! y, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
   simp only [mem_self_comm_mem_inv, and_self]
   have ⟨y, hy⟩ := dom_exists hx
@@ -455,7 +451,7 @@ theorem theorem_3g_i {α : Type _} {x y : α} {F : Set.Relation α}
 Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`.
 -/
 theorem theorem_3g_ii {F : Set.Relation α}
-  (hF : isOneToOne F) (hy : y ∈ F.ran)
+  (hF : F.isOneToOne) (hy : y ∈ F.ran)
   : ∃! x, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
   simp only [mem_self_comm_mem_inv, and_self]
   have ⟨x, hx⟩ := ran_exists hy
@@ -472,13 +468,12 @@ dom (F ∘ G) = {x ∈ dom G | G(x) ∈ dom F}.
 ```
 -/
 theorem theorem_3h_dom {F G : Set.Relation α}
-  (_ : isSingleValued F) (hG : isSingleValued G)
+  (_ : F.isSingleValued) (hG : G.isSingleValued)
-  : dom (comp F G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F } := by
+  : dom (F.comp G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F } := by
   let rhs := {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F }
-  suffices dom (comp F G) ⊆ rhs ∧ rhs ⊆ dom (comp F G) from
+  rw [Set.Subset.antisymm_iff]
-    Set.Subset.antisymm_iff.mpr this
   apply And.intro
-  · show ∀ t, t ∈ dom (comp F G) → t ∈ rhs
+  · show ∀ t, t ∈ dom (F.comp G) → t ∈ rhs
     intro t ht
     simp only [Set.mem_setOf_eq]
     have ⟨z, hz⟩ := dom_exists ht
@@ -492,7 +487,7 @@ theorem theorem_3h_dom {F G : Set.Relation α}
     refine ⟨a, ⟨ha.left, z, ha.right⟩, ?_⟩
     intro y₁ hy₁
     exact fun _ _ => single_valued_eq_unique hG hy₁ ha.left
-  · show ∀ t, t ∈ rhs → t ∈ dom (comp F G)
+  · show ∀ t, t ∈ rhs → t ∈ dom (F.comp G)
     intro t ht
     simp only [Set.mem_setOf_eq] at ht
     unfold dom
@@ -505,6 +500,30 @@ theorem theorem_3h_dom {F G : Set.Relation α}
     simp only [Set.mem_setOf_eq]
     exact ⟨a, ha.left.left, hb⟩
 
+/-- ### Theorem 3J (a)
+
+Assume that `F : A → B`, and that `A` is nonempty. There exists a function
+`G : B → A` (a "left inverse") such that `G ∘ F` is the identity function on `A`
+**iff** `F` is one-to-one.
+-/
+theorem theorem_3j_a {F : Set.Relation α} {A B : Set α}
+  (hF : F.mapsInto A B) (hA : Set.Nonempty A)
+  : (∃ G : Set.Relation α,
+      G.mapsInto B A ∧ (∀ p ∈ G.comp F, p.1 = p.2)) ↔ F.isOneToOne := by
+  sorry
+
+/-- ### Theorem 3J (b)
+
+Assume that `F : A → B`, and that `A` is nonempty. There exists a function
+`H : B → A` (a "right inverse") such that `F ∘ H` is the identity function on
+`B` **iff** `F` maps `A` onto `B`.
+-/
+theorem theorem_3j_b {F : Set.Relation α} {A B : Set α}
+  (hF : F.mapsInto A B) (hA : Set.Nonempty A)
+  : (∃ H : Set.Relation α,
+      H.mapsInto B A ∧ (∀ p ∈ F.comp H, p.1 = p.2)) ↔ F.mapsOnto A B := by
+  sorry
+
 end
 
 end Enderton.Set.Chapter_3

Common/Set/Relation.lean

@@ -99,7 +99,7 @@ The inverse of the inverse of a `Relation` is the `Relation`.
 @[simp]
 theorem inv_inv_eq_self (R : Relation α)
   : R.inv.inv = R := by
-  unfold Set.Relation.inv
+  unfold inv
   simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
   ext x
   apply Iff.intro
@@ -117,10 +117,10 @@ theorem inv_inv_eq_self (R : Relation α)
 For a set `F`, `dom F⁻¹ = ran F`.
 -/
 @[simp]
-theorem dom_inv_eq_ran_self {F : Set.Relation α}
+theorem dom_inv_eq_ran_self {F : Relation α}
-  : Set.Relation.dom (F.inv) = Set.Relation.ran F := by
+  : dom (F.inv) = ran F := by
   ext x
-  unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
+  unfold dom ran inv
   simp only [
     Prod.exists,
     Set.mem_image,
@@ -140,10 +140,10 @@ theorem dom_inv_eq_ran_self {F : Set.Relation α}
 For a set `F`, `ran F⁻¹ = dom F`.
 -/
 @[simp]
-theorem ran_inv_eq_dom_self {F : Set.Relation α}
+theorem ran_inv_eq_dom_self {F : Relation α}
-  : Set.Relation.ran (F.inv) = Set.Relation.dom F := by
+  : ran (F.inv) = dom F := by
   ext x
-  unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
+  unfold dom ran inv
   simp only [
     Prod.exists,
     Set.mem_image,
@@ -188,7 +188,7 @@ def isSingleRooted (R : Relation α) : Prop :=
 A single-rooted `Relation` should map the same output to the same input.
 -/
 theorem single_rooted_eq_unique {R : Relation α} {x₁ x₂ y : α}
-  (hR : isSingleRooted R)
+  (hR : R.isSingleRooted)
   : (x₁, y) ∈ R → (x₂, y) ∈ R → x₁ = x₂ := by
   intro hx₁ hx₂
   unfold isSingleRooted at hR
@@ -212,7 +212,7 @@ def isSingleValued (R : Relation α) : Prop :=
 A single-valued `Relation` should map the same input to the same output.
 -/
 theorem single_valued_eq_unique {R : Relation α} {x y₁ y₂ : α}
-  (hR : isSingleValued R)
+  (hR : R.isSingleValued)
   : (x, y₁) ∈ R → (x, y₂) ∈ R → y₁ = y₂ := by
   intro hy₁ hy₂
   unfold isSingleValued at hR
@@ -227,7 +227,7 @@ theorem single_valued_eq_unique {R : Relation α} {x y₁ y₂ : α}
 For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted.
 -/
 theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
-  : isSingleValued F.inv ↔ isSingleRooted F := by
+  : F.inv.isSingleValued ↔ F.isSingleRooted := by
   apply Iff.intro
   · intro hF
     unfold isSingleValued at hF
@@ -261,7 +261,7 @@ theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
 For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted.
 -/
 theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α}
-  : Set.Relation.isSingleValued F ↔ Set.Relation.isSingleRooted F.inv := by
+  : F.isSingleValued ↔ F.inv.isSingleRooted := by
   conv => lhs; rw [← inv_inv_eq_self F]
   rw [single_valued_inv_iff_single_rooted_self]
 
@@ -269,13 +269,13 @@ theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α}
 A `Relation` `R` is one-to-one if it is a single-rooted function.
 -/
 def isOneToOne (R : Relation α) : Prop :=
-  isSingleValued R ∧ isSingleRooted R
+  R.isSingleValued ∧ R.isSingleRooted
 
 /--
 A `Relation` is one-to-one **iff** it's inverse is.
 -/
 theorem one_to_one_self_iff_one_to_one_inv {R : Relation α}
-  : isOneToOne R ↔ isOneToOne R.inv := by
+  : R.isOneToOne ↔ R.inv.isOneToOne := by
   unfold isOneToOne isSingleValued isSingleRooted
   conv => rhs; simp only [
     dom_inv_eq_ran_self,
@@ -300,7 +300,7 @@ def comp (F G : Relation α) : Relation α :=
 If `x ∈ dom (F ∘ G)`, then `x ∈ dom G`.
 -/
 theorem dom_comp_imp_dom_self {F G : Relation α}
-  : x ∈ dom (comp F G) → x ∈ dom G := by
+  : x ∈ dom (F.comp G) → x ∈ dom G := by
   unfold dom comp
   simp only [
     mem_image,
@@ -317,7 +317,7 @@ theorem dom_comp_imp_dom_self {F G : Relation α}
 If `y ∈ ran (F ∘ G)`, then `y ∈ ran F`.
 -/
 theorem ran_comp_imp_ran_self {F G : Relation α}
-  : y ∈ ran (comp F G) → y ∈ ran F := by
+  : y ∈ ran (F.comp G) → y ∈ ran F := by
   unfold ran comp
   simp only [
     mem_image,
@@ -333,8 +333,8 @@ theorem ran_comp_imp_ran_self {F G : Relation α}
 The composition of two functions is itself a function.
 -/
 theorem single_valued_comp_is_single_valued {F G : Relation α}
-  (hF : isSingleValued F) (hG : isSingleValued G)
+  (hF : F.isSingleValued) (hG : G.isSingleValued)
-  : isSingleValued (comp F G) := by
+  : (F.comp G).isSingleValued := by
   unfold isSingleValued
   intro x hx
   have ⟨y, hxy⟩ := dom_exists hx
@@ -365,6 +365,63 @@ theorem single_valued_comp_is_single_valued {F G : Relation α}
   have hk₂ := hy'.right y₁ (mem_pair_imp_snd_mem_ran ht₂.right) ht₂.right
   rw [hk₁, hk₂]
 
+/--
+For `Relation`s `F` and `G`, `(F ∘ G)⁻¹ = G⁻¹ ∘ F⁻¹`.
+-/
+theorem comp_inv_eq_inv_comp_inv {F G : Relation α}
+  : (F.comp G).inv = G.inv.comp F.inv := by
+  calc (F.comp G).inv
+  _ = {p | ∃ t, (p.2, t) ∈ G ∧ (t, p.1) ∈ F} := by
+    rw [Set.Subset.antisymm_iff]
+    apply And.intro
+    · unfold inv comp
+      intro t ht
+      simp only [mem_setOf_eq, Prod.exists] at ht
+      have ⟨a, b, ⟨⟨p, hp⟩, hab⟩⟩ := ht
+      rw [← hab]
+      exact ⟨p, hp⟩
+    · unfold inv comp
+      intro (a, b) ⟨p, hp⟩
+      simp only [mem_setOf_eq, Prod.exists, Prod.mk.injEq]
+      exact ⟨b, a, ⟨p, hp⟩, rfl, rfl⟩
+  _ = {p | ∃ t, (t, p.1) ∈ F ∧ (p.2, t) ∈ G} := by
+    rw [Set.Subset.antisymm_iff]
+    apply And.intro
+    · intro (a, b) ht
+      simp only [mem_setOf_eq] at *
+      have ⟨t, p, q⟩ := ht
+      exact ⟨t, q, p⟩
+    · intro (a, b) ht
+      simp only [mem_setOf_eq] at *
+      have ⟨t, p, q⟩ := ht
+      exact ⟨t, q, p⟩
+  _ = {p | ∃ t, (p.1, t) ∈ F.inv ∧ (t, p.2) ∈ G.inv } := by
+    rw [Set.Subset.antisymm_iff]
+    apply And.intro
+    · intro (a, b) ht
+      simp only [mem_setOf_eq] at *
+      have ⟨t, p, q⟩ := ht
+      refine ⟨t, ?_, ?_⟩ <;> rwa [mem_self_comm_mem_inv]
+    · intro (a, b) ht
+      simp only [mem_setOf_eq] at *
+      have ⟨t, p, q⟩ := ht
+      refine ⟨t, ?_, ?_⟩ <;> rwa [← mem_self_comm_mem_inv]
+  _ = comp (inv G) (inv F) := rfl
+
+/--
+Indicates `Relation` `F` is a function from `A` to `B`.
+
+This is usually denoted as `F : A → B`.
+-/
+def mapsInto (F : Relation α) (A B : Set α) :=
+  F.isSingleValued ∧ dom F = A ∧ ran F ⊆ B
+
+/--
+Indicates `Relation` `F` is a function from `A` to `ran F = B`.
+-/
+def mapsOnto (F : Relation α) (A B : Set α) :=
+  F.isSingleValued ∧ dom F = A ∧ ran F = B
+
 /-! ## Ordered Pairs -/
 
 /--