2341 lines
71 KiB
Plaintext
2341 lines
71 KiB
Plaintext
import Bookshelf.Enderton.Set.Chapter_2
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import Bookshelf.Enderton.Set.OrderedPair
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import Bookshelf.Enderton.Set.Relation
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import Common.Logic.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Rel
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import Mathlib.Init.Algebra.Classes
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import Mathlib.Order.RelClasses
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import Mathlib.Tactic.CasesM
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/-! # Enderton.Set.Chapter_3
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Relations and Functions
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-/
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namespace Enderton.Set.Chapter_3
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/-- #### Theorem 3B
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If `x ∈ C` and `y ∈ C`, then `⟨x, y⟩ ∈ 𝒫 𝒫 C`.
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-/
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theorem theorem_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
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: OrderedPair x y ∈ 𝒫 𝒫 C := by
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have hxs : {x} ⊆ C := Set.singleton_subset_iff.mpr hx
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have hxys : {x, y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
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exact Set.mem_mem_imp_pair_subset hxs hxys
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/-- #### Exercise 3.1
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Suppose that we attempted to generalize the Kuratowski definitions of ordered
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pairs to ordered triples by defining
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```
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⟨x, y, z⟩* = {{x}, {x, y}, {x, y, z}}.open Set
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```
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Show that this definition is unsuccessful by giving examples of objects `u`,
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`v`, `w`, `x`, `y`, `z` with `⟨x, y, z⟩* = ⟨u, v, w⟩*` but with either `y ≠ v`
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or `z ≠ w` (or both).
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-/
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theorem exercise_3_1 {x y z u v w : ℕ}
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(hx : x = 1) (hy : y = 1) (hz : z = 2)
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(hu : u = 1) (hv : v = 2) (hw : w = 2)
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: ({{x}, {x, y}, {x, y, z}} : Set (Set ℕ)) = {{u}, {u, v}, {u, v, w}}
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∧ y ≠ v := by
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apply And.intro
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· rw [hx, hy, hz, hu, hv, hw]
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simp
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· rw [hy, hv]
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simp only
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/-- #### Exercise 3.2a
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Show that `A × (B ∪ C) = (A × B) ∪ (A × C)`.
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-/
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theorem exercise_3_2a {A : Set α} {B C : Set β}
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: Set.prod A (B ∪ C) = (Set.prod A B) ∪ (Set.prod A C) := by
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calc Set.prod A (B ∪ C)
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_ = { p | p.1 ∈ A ∧ p.2 ∈ B ∪ C } := rfl
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_ = { p | p.1 ∈ A ∧ (p.2 ∈ B ∨ p.2 ∈ C) } := rfl
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_ = { p | (p.1 ∈ A ∧ p.2 ∈ B) ∨ (p.1 ∈ A ∧ p.2 ∈ C) } := by
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ext x
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rw [Set.mem_setOf_eq]
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conv => lhs; rw [and_or_left]
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_ = { p | p ∈ Set.prod A B ∨ (p ∈ Set.prod A C) } := rfl
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_ = (Set.prod A B) ∪ (Set.prod A C) := rfl
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/-- #### Exercise 3.2b
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Show that if `A × B = A × C` and `A ≠ ∅`, then `B = C`.
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-/
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theorem exercise_3_2b {A : Set α} {B C : Set β}
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(h : Set.prod A B = Set.prod A C) (hA : Set.Nonempty A)
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: B = C := by
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by_cases hB : Set.Nonempty B
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· rw [Set.Subset.antisymm_iff]
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have ⟨a, ha⟩ := hA
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apply And.intro
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· show ∀ t, t ∈ B → t ∈ C
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intro t ht
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have : (a, t) ∈ Set.prod A B := ⟨ha, ht⟩
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rw [h] at this
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exact this.right
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· show ∀ t, t ∈ C → t ∈ B
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intro t ht
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have : (a, t) ∈ Set.prod A C := ⟨ha, ht⟩
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rw [← h] at this
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exact this.right
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· have nB : B = ∅ := Set.not_nonempty_iff_eq_empty.mp hB
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rw [nB, Set.prod_right_emptyset_eq_emptyset, Set.ext_iff] at h
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rw [nB]
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by_contra nC
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have ⟨a, ha⟩ := hA
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have ⟨c, hc⟩ := Set.nonempty_iff_ne_empty.mpr (Ne.symm nC)
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exact (h (a, c)).mpr ⟨ha, hc⟩
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/-- #### Exercise 3.3
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Show that `A × ⋃ 𝓑 = ⋃ {A × X | X ∈ 𝓑}`.
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-/
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theorem exercise_3_3 {A : Set (Set α)} {𝓑 : Set (Set β)}
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: Set.prod A (⋃₀ 𝓑) = ⋃₀ {Set.prod A X | X ∈ 𝓑} := by
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calc Set.prod A (⋃₀ 𝓑)
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_ = { p | p.1 ∈ A ∧ p.2 ∈ ⋃₀ 𝓑} := rfl
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_ = { p | p.1 ∈ A ∧ ∃ b ∈ 𝓑, p.2 ∈ b } := rfl
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_ = { p | ∃ b ∈ 𝓑, p.1 ∈ A ∧ p.2 ∈ b } := by
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ext x
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rw [Set.mem_setOf_eq]
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apply Iff.intro
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· intro ⟨h₁, b, h₂⟩
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exact ⟨b, h₂.left, h₁, h₂.right⟩
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· intro ⟨b, h₁, h₂, h₃⟩
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exact ⟨h₂, b, h₁, h₃⟩
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_ = ⋃₀ { Set.prod A p | p ∈ 𝓑 } := by
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ext x
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rw [Set.mem_setOf_eq]
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unfold Set.sUnion sSup Set.instSupSetSet
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simp only [Set.mem_setOf_eq, exists_exists_and_eq_and]
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apply Iff.intro
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· intro ⟨b, h₁, h₂, h₃⟩
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exact ⟨b, h₁, h₂, h₃⟩
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· intro ⟨b, h₁, h₂, h₃⟩
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exact ⟨b, h₁, h₂, h₃⟩
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/-- #### Exercise 3.5a
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Assume that `A` and `B` are given sets, and show that there exists a set `C`
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such that for any `y`,
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```
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y ∈ C ↔ y = {x} × B for some x in A.
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```
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In other words, show that `{{x} × B | x ∈ A}` is a set.
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-/
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theorem exercise_3_5a {A : Set α} {B : Set β}
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: ∃ C : Set (Set (α × β)),
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y ∈ C ↔ ∃ x ∈ A, y = Set.prod {x} B := by
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let C := {y ∈ 𝒫 (Set.prod A B) | ∃ a ∈ A, ∀ x, (x ∈ y ↔ ∃ b ∈ B, x = (a, b))}
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refine ⟨C, ?_⟩
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apply Iff.intro
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· intro hC
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simp only [Set.mem_setOf_eq] at hC
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have ⟨_, ⟨a, ⟨ha, h⟩⟩⟩ := hC
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refine ⟨a, ⟨ha, ?_⟩⟩
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ext x
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apply Iff.intro
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· intro hxy
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unfold Set.prod
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq]
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have ⟨b, ⟨hb, hx⟩⟩ := (h x).mp hxy
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rw [Prod.ext_iff] at hx
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simp only at hx
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rw [← hx.right] at hb
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exact ⟨hx.left, hb⟩
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· intro hx
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at hx
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have := (h (a, x.snd)).mpr ⟨x.snd, ⟨hx.right, rfl⟩⟩
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have hxab : x = (a, x.snd) := by
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ext <;> simp
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exact hx.left
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rwa [← hxab] at this
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· intro ⟨x, ⟨hx, hy⟩⟩
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show y ∈ 𝒫 Set.prod A B ∧ ∃ a, a ∈ A ∧
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∀ (x : α × β), x ∈ y ↔ ∃ b, b ∈ B ∧ x = (a, b)
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apply And.intro
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· simp only [Set.mem_powerset_iff]
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rw [hy]
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unfold Set.prod
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simp only [
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Set.mem_singleton_iff,
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Set.setOf_subset_setOf,
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and_imp,
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Prod.forall
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]
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intro a b ha hb
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exact ⟨by rw [ha]; exact hx, hb⟩
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· refine ⟨x, ⟨hx, ?_⟩⟩
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intro p
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apply Iff.intro
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· intro hab
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rw [hy] at hab
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unfold Set.prod at hab
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at hab
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exact ⟨p.2, ⟨hab.right, by ext; exact hab.left; simp⟩⟩
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· intro ⟨b, ⟨hb, hab⟩⟩
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rw [hy]
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unfold Set.prod
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq]
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rw [Prod.ext_iff] at hab
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simp only at hab
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rw [hab.right]
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exact ⟨hab.left, hb⟩
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/-- #### Exercise 3.5b
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With `A`, `B`, and `C` as above, show that `A × B = ∪ C`.
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-/
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theorem exercise_3_5b {A : Set α} (B : Set β)
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: Set.prod A B = ⋃₀ {Set.prod ({x} : Set α) B | x ∈ A} := by
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rw [Set.Subset.antisymm_iff]
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apply And.intro
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· show ∀ t, t ∈ Set.prod A B → t ∈ ⋃₀ {Set.prod {x} B | x ∈ A}
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intro t h
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simp only [Set.mem_setOf_eq] at h
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unfold Set.sUnion sSup Set.instSupSetSet
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simp only [Set.mem_setOf_eq, exists_exists_and_eq_and]
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unfold Set.prod at h
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simp only [Set.mem_setOf_eq] at h
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refine ⟨t.fst, ⟨h.left, ?_⟩⟩
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unfold Set.prod
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq, true_and]
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exact h.right
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· show ∀ t, t ∈ ⋃₀ {Set.prod {x} B | x ∈ A} → t ∈ Set.prod A B
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unfold Set.prod
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intro t ht
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simp only [
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Set.mem_singleton_iff,
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Set.mem_sUnion,
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Set.mem_setOf_eq,
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exists_exists_and_eq_and
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] at ht
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have ⟨a, ⟨h, ⟨ha, hb⟩⟩⟩ := ht
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simp only [Set.mem_setOf_eq]
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rw [← ha] at h
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exact ⟨h, hb⟩
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/-- #### Theorem 3D
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If `⟨x, y⟩ ∈ A`, then `x` and `y` belong to `⋃ ⋃ A`.
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-/
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theorem theorem_3d {A : Set (Set (Set α))} (h : OrderedPair x y ∈ A)
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: x ∈ ⋃₀ (⋃₀ A) ∧ y ∈ ⋃₀ (⋃₀ A) := by
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have hp := Chapter_2.exercise_2_3 (OrderedPair x y) h
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unfold OrderedPair at hp
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have hq : {x, y} ∈ ⋃₀ A := hp (by simp)
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have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq
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exact ⟨this (by simp), this (by simp)⟩
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section Relation
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open Set.Relation
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/-- #### Exercise 3.6
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Show that a set `A` is a relation **iff** `A ⊆ dom A × ran A`.
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-/
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theorem exercise_3_6 {A : Set.HRelation α β}
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: A ⊆ Set.prod (dom A) (ran A) := by
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show ∀ t, t ∈ A → t ∈ Set.prod (Prod.fst '' A) (Prod.snd '' A)
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intro (a, b) ht
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unfold Set.prod
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simp only [
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Set.mem_image,
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Prod.exists,
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exists_and_right,
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exists_eq_right,
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Set.mem_setOf_eq
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]
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exact ⟨⟨b, ht⟩, ⟨a, ht⟩⟩
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/-- #### Exercise 3.7
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Show that if `R` is a relation, then `fld R = ⋃ ⋃ R`.
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-/
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theorem exercise_3_7 {R : Set.Relation α}
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: R.fld = ⋃₀ ⋃₀ R.toOrderedPairs := by
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let img := R.toOrderedPairs
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rw [Set.Subset.antisymm_iff]
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apply And.intro
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· show ∀ x, x ∈ R.fld → x ∈ ⋃₀ ⋃₀ img
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intro x hx
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apply Or.elim hx
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· intro hd
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unfold Set.Relation.dom Prod.fst at hd
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simp only [
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Set.mem_image, Prod.exists, exists_and_right, exists_eq_right
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] at hd
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have ⟨y, hp⟩ := hd
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have hm : OrderedPair x y ∈ Set.image (fun p => OrderedPair p.1 p.2) R := by
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unfold Set.image
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simp only [Prod.exists, Set.mem_setOf_eq]
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exact ⟨x, ⟨y, ⟨hp, rfl⟩⟩⟩
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unfold OrderedPair at hm
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have : {x} ∈ ⋃₀ img := Chapter_2.exercise_2_3 {{x}, {x, y}} hm (by simp)
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exact (Chapter_2.exercise_2_3 {x} this) (show x ∈ {x} by rfl)
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· intro hr
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unfold Set.Relation.ran Prod.snd at hr
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simp only [Set.mem_image, Prod.exists, exists_eq_right] at hr
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have ⟨t, ht⟩ := hr
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have hm : OrderedPair t x ∈ Set.image (fun p => OrderedPair p.1 p.2) R := by
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simp only [Set.mem_image, Prod.exists]
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exact ⟨t, ⟨x, ⟨ht, rfl⟩⟩⟩
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unfold OrderedPair at hm
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have : {t, x} ∈ ⋃₀ img := Chapter_2.exercise_2_3 {{t}, {t, x}} hm
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(show {t, x} ∈ {{t}, {t, x}} by simp)
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exact Chapter_2.exercise_2_3 {t, x} this (show x ∈ {t, x} by simp)
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· show ∀ t, t ∈ ⋃₀ ⋃₀ img → t ∈ Set.Relation.fld R
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intro t ht
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have ⟨T, hT⟩ : ∃ T ∈ ⋃₀ img, t ∈ T := ht
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have ⟨T', hT'⟩ : ∃ T' ∈ img, T ∈ T' := hT.left
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dsimp only at hT'
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unfold Set.Relation.toOrderedPairs at hT'
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simp only [Set.mem_image, Prod.exists] at hT'
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have ⟨x, ⟨y, ⟨p, hp⟩⟩⟩ := hT'.left
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have hr := hT'.right
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rw [← hp] at hr
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unfold OrderedPair at hr
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simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at hr
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-- Use `exercise_6_6` to prove that if `t = x` then `t ∈ dom R` and if
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-- `t = y` then `t ∈ ran R`.
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have hxy_mem : t = x ∨ t = y → t ∈ Set.Relation.fld R := by
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intro ht
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have hz : R ⊆ Set.prod (dom R) (ran R) := exercise_3_6
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have : (x, y) ∈ Set.prod (dom R) (ran R) := hz p
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unfold Set.prod at this
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simp at this
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apply Or.elim ht
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· intro ht'
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rw [← ht'] at this
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exact Or.inl this.left
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· intro ht'
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rw [← ht'] at this
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exact Or.inr this.right
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-- Eliminate `T = {x} ∨ T = {x, y}`.
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apply Or.elim hr
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· intro hx
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have := hT.right
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rw [hx] at this
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simp only [Set.mem_singleton_iff] at this
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exact hxy_mem (Or.inl this)
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· intro hxy
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have := hT.right
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rw [hxy] at this
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simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
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exact hxy_mem this
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/-- #### Exercise 3.8 (i)
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Show that for any set `𝓐`:
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```
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dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_8_i {A : Set (Set.HRelation α β)}
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: dom (⋃₀ A) = ⋃₀ { dom R | R ∈ A } := by
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ext x
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unfold dom Prod.fst
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simp only [
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Set.mem_image,
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Set.mem_sUnion,
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Prod.exists,
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exists_and_right,
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exists_eq_right,
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Set.mem_setOf_eq,
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exists_exists_and_eq_and
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]
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apply Iff.intro
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· intro ⟨y, t, ht, hx⟩
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exact ⟨t, ht, y, hx⟩
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· intro ⟨t, ht, y, hx⟩
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exact ⟨y, t, ht, hx⟩
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/-- #### Exercise 3.8 (ii)
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Show that for any set `𝓐`:
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```
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ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_8_ii {A : Set (Set.HRelation α β)}
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: ran (⋃₀ A) = ⋃₀ { ran R | R ∈ A } := by
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ext x
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unfold ran Prod.snd
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simp only [
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Set.mem_image,
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Set.mem_sUnion,
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Prod.exists,
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exists_eq_right,
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Set.mem_setOf_eq,
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exists_exists_and_eq_and
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]
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apply Iff.intro
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· intro ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
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exact ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
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· intro ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
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exact ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
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/-- #### Exercise 3.9 (i)
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Discuss the result of replacing the union operation by the intersection
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operation in the preceding problem.
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```
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dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_9_i {A : Set (Set.HRelation α β)}
|
||
: dom (⋂₀ A) ⊆ ⋂₀ { dom R | R ∈ A } := by
|
||
show ∀ x, x ∈ dom (⋂₀ A) → x ∈ ⋂₀ { dom R | R ∈ A }
|
||
unfold dom Prod.fst
|
||
simp only [
|
||
Set.mem_image,
|
||
Set.mem_sInter,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right,
|
||
Set.mem_setOf_eq,
|
||
forall_exists_index,
|
||
and_imp,
|
||
forall_apply_eq_imp_iff₂
|
||
]
|
||
intro _ y hy R hR
|
||
exact ⟨y, hy R hR⟩
|
||
|
||
/-- #### Exercise 3.9 (ii)
|
||
|
||
Discuss the result of replacing the union operation by the intersection
|
||
operation in the preceding problem.
|
||
```
|
||
ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
|
||
```
|
||
-/
|
||
theorem exercise_3_9_ii {A : Set (Set.HRelation α β)}
|
||
: ran (⋂₀ A) ⊆ ⋂₀ { ran R | R ∈ A } := by
|
||
show ∀ x, x ∈ ran (⋂₀ A) → x ∈ ⋂₀ { ran R | R ∈ A }
|
||
unfold ran Prod.snd
|
||
simp only [
|
||
Set.mem_image,
|
||
Set.mem_sInter,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right,
|
||
Set.mem_setOf_eq,
|
||
forall_exists_index,
|
||
and_imp,
|
||
forall_apply_eq_imp_iff₂
|
||
]
|
||
intro _ y hy R hR
|
||
exact ⟨y, hy R hR⟩
|
||
|
||
/-- #### Theorem 3G (i)
|
||
|
||
Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
|
||
-/
|
||
theorem theorem_3g_i {F : Set.HRelation α β}
|
||
(hF : isOneToOne F) (hx : x ∈ dom F)
|
||
: ∃! y, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
|
||
simp only [mem_self_comm_mem_inv, and_self]
|
||
have ⟨y, hy⟩ := dom_exists hx
|
||
refine ⟨y, hy, ?_⟩
|
||
intro y₁ hy₁
|
||
unfold isOneToOne at hF
|
||
exact (single_valued_eq_unique hF.left hy hy₁).symm
|
||
|
||
/-- #### Theorem 3G (ii)
|
||
|
||
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.HRelation α β}
|
||
(hF : isOneToOne F) (hy : y ∈ ran F)
|
||
: ∃! x, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
|
||
simp only [mem_self_comm_mem_inv, and_self]
|
||
have ⟨x, hx⟩ := ran_exists hy
|
||
refine ⟨x, hx, ?_⟩
|
||
intro x₁ hx₁
|
||
unfold isOneToOne at hF
|
||
exact (single_rooted_eq_unique hF.right hx hx₁).symm
|
||
|
||
/-- #### Theorem 3H
|
||
|
||
Assume that `F` and `G` are functions. Then
|
||
```
|
||
dom (F ∘ G) = {x ∈ dom G | G(x) ∈ dom F}.
|
||
```
|
||
-/
|
||
theorem theorem_3h_dom {F : Set.HRelation β γ} {G : Set.HRelation α β}
|
||
(_ : isSingleValued F) (hG : isSingleValued G)
|
||
: dom (comp F G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F} := by
|
||
let rhs := {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F }
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· show ∀ t, t ∈ dom (comp F G) → t ∈ rhs
|
||
intro t ht
|
||
simp only [Set.mem_setOf_eq]
|
||
have ⟨z, hz⟩ := dom_exists ht
|
||
refine ⟨dom_comp_imp_dom_self ht, ?_⟩
|
||
simp only [Set.mem_setOf_eq] at hz
|
||
have ⟨a, ha⟩ := hz
|
||
unfold dom
|
||
simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right]
|
||
unfold ExistsUnique
|
||
simp only [and_imp, forall_exists_index]
|
||
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)
|
||
intro t ht
|
||
simp only [Set.mem_setOf_eq] at ht
|
||
unfold dom
|
||
simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right]
|
||
have ⟨a, ha⟩ := ht.right
|
||
simp at ha
|
||
have ⟨b, hb⟩ := dom_exists ha.left.right
|
||
refine ⟨b, ?_⟩
|
||
unfold comp
|
||
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.HRelation α β} {A : Set α} {B : Set β}
|
||
(hF : mapsInto F A B) (hA : Set.Nonempty A)
|
||
: (∃ G : Set.HRelation β α,
|
||
mapsInto G B A ∧
|
||
(comp G F = { p | p.1 ∈ A ∧ p.1 = p.2 })) ↔ isOneToOne F := by
|
||
apply Iff.intro
|
||
· intro ⟨G, hG⟩
|
||
refine ⟨hF.is_func, ?_⟩
|
||
intro y hy
|
||
have ⟨x₁, hx₁⟩ := ran_exists hy
|
||
refine ⟨x₁, ⟨mem_pair_imp_fst_mem_dom hx₁, hx₁⟩, ?_⟩
|
||
intro x₂ hx₂
|
||
|
||
have hG' : y ∈ dom G := by
|
||
rw [hG.left.dom_eq]
|
||
exact hF.ran_ss hy
|
||
have ⟨z, hz⟩ := dom_exists hG'
|
||
|
||
have := hG.right
|
||
unfold comp at this
|
||
rw [Set.ext_iff] at this
|
||
have h₁ := (this (x₁, z)).mp ⟨y, hx₁, hz⟩
|
||
have h₂ := (this (x₂, z)).mp ⟨y, hx₂.right, hz⟩
|
||
simp only [Set.mem_setOf_eq] at h₁ h₂
|
||
rw [h₁.right, h₂.right]
|
||
· 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.HRelation α β} {A : Set α} {B : Set β}
|
||
(hF : mapsInto F A B) (hA : Set.Nonempty A)
|
||
: (∃ H : Set.HRelation β α,
|
||
mapsInto H B A ∧
|
||
(comp F H = { p | p.1 ∈ B ∧ p.1 = p.2 })) ↔ mapsOnto F A B := by
|
||
sorry
|
||
|
||
/-- #### Theorem 3K (a)
|
||
|
||
The following hold for any sets. (`F` need not be a function.)
|
||
The image of a union is the union of the images:
|
||
```
|
||
F⟦⋃ 𝓐⟧ = ⋃ {F⟦A⟧ | A ∈ 𝓐}
|
||
```
|
||
-/
|
||
theorem theorem_3k_a {F : Set.HRelation α β} {𝓐 : Set (Set α)}
|
||
: image F (⋃₀ 𝓐) = ⋃₀ { image F A | A ∈ 𝓐 } := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· show ∀ v, v ∈ image F (⋃₀ 𝓐) → v ∈ ⋃₀ { image F A | A ∈ 𝓐 }
|
||
intro v hv
|
||
unfold image at hv
|
||
simp only [Set.mem_sUnion, Set.mem_setOf_eq] at hv
|
||
have ⟨u, hu⟩ := hv
|
||
have ⟨A, hA⟩ := hu.left
|
||
simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and]
|
||
refine ⟨A, hA.left, ?_⟩
|
||
show v ∈ image F A
|
||
unfold image
|
||
simp only [Set.mem_setOf_eq]
|
||
exact ⟨u, hA.right, hu.right⟩
|
||
· show ∀ v, v ∈ ⋃₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋃₀ 𝓐)
|
||
intro v hv
|
||
simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and] at hv
|
||
have ⟨A, hA⟩ := hv
|
||
unfold image at hA
|
||
simp only [Set.mem_setOf_eq] at hA
|
||
have ⟨u, hu⟩ := hA.right
|
||
unfold image
|
||
simp only [Set.mem_sUnion, Set.mem_setOf_eq]
|
||
exact ⟨u, ⟨A, hA.left, hu.left⟩, hu.right⟩
|
||
|
||
/-! #### Theorem 3K (b)
|
||
|
||
The following hold for any sets. (`F` need not be a function.)
|
||
The image of an intersection is included in the intersection of the images:
|
||
```
|
||
F⟦⋂ 𝓐⟧ ⊆ ⋂ {F⟦A⟧ | A ∈ 𝓐}
|
||
```
|
||
Equality holds if `F` is single-rooted.
|
||
-/
|
||
|
||
theorem theorem_3k_b_i {F : Set.HRelation α β} {𝓐 : Set (Set α)}
|
||
: image F (⋂₀ 𝓐) ⊆ ⋂₀ { image F A | A ∈ 𝓐} := by
|
||
show ∀ v, v ∈ image F (⋂₀ 𝓐) → v ∈ ⋂₀ { image F A | A ∈ 𝓐}
|
||
intro v hv
|
||
unfold image at hv
|
||
simp only [Set.mem_sInter, Set.mem_setOf_eq] at hv
|
||
have ⟨u, hu⟩ := hv
|
||
simp only [
|
||
Set.mem_sInter,
|
||
Set.mem_setOf_eq,
|
||
forall_exists_index,
|
||
and_imp,
|
||
forall_apply_eq_imp_iff₂
|
||
]
|
||
intro A hA
|
||
unfold image
|
||
simp only [Set.mem_setOf_eq]
|
||
exact ⟨u, hu.left A hA, hu.right⟩
|
||
|
||
theorem theorem_3k_b_ii {F : Set.HRelation α β} {𝓐 : Set (Set α)}
|
||
(hF : isSingleRooted F) (h𝓐 : Set.Nonempty 𝓐)
|
||
: image F (⋂₀ 𝓐) = ⋂₀ { image F A | A ∈ 𝓐} := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
refine ⟨theorem_3k_b_i, ?_⟩
|
||
show ∀ v, v ∈ ⋂₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋂₀ 𝓐)
|
||
intro v hv
|
||
simp only [
|
||
Set.mem_sInter,
|
||
Set.mem_setOf_eq,
|
||
forall_exists_index,
|
||
and_imp,
|
||
forall_apply_eq_imp_iff₂
|
||
] at hv
|
||
unfold image at hv
|
||
simp only [Set.mem_setOf_eq] at hv
|
||
have ⟨u, hu⟩ : ∃ u, (∀ (a : Set α), a ∈ 𝓐 → u ∈ a) ∧ (u, v) ∈ F := by
|
||
have ⟨A, hA⟩ := h𝓐
|
||
have ⟨_, ⟨_, hv'⟩⟩ := hv A hA
|
||
have ⟨u, hu⟩ := hF v (mem_pair_imp_snd_mem_ran hv')
|
||
simp only [and_imp] at hu
|
||
refine ⟨u, ?_, hu.left.right⟩
|
||
intro a ha
|
||
have ⟨u₁, hu₁⟩ := hv a ha
|
||
have := hu.right u₁ (mem_pair_imp_fst_mem_dom hu₁.right) hu₁.right
|
||
rw [← this]
|
||
exact hu₁.left
|
||
unfold image
|
||
simp only [Set.mem_sInter, Set.mem_setOf_eq]
|
||
exact ⟨u, hu⟩
|
||
|
||
/-! #### Theorem 3K (c)
|
||
|
||
The following hold for any sets. (`F` need not be a function.)
|
||
The image of a difference includes the difference of the images:
|
||
```
|
||
F⟦A⟧ - F⟦B⟧ ⊆ F⟦A - B⟧.
|
||
```
|
||
Equality holds if `F` is single-rooted.
|
||
-/
|
||
|
||
theorem theorem_3k_c_i {F : Set.HRelation α β} {A B : Set α}
|
||
: image F A \ image F B ⊆ image F (A \ B) := by
|
||
show ∀ v, v ∈ image F A \ image F B → v ∈ image F (A \ B)
|
||
intro v hv
|
||
have hv' : v ∈ image F A ∧ v ∉ image F B := hv
|
||
conv at hv' => arg 1; unfold image; simp only [Set.mem_setOf_eq, eq_iff_iff]
|
||
have ⟨u, hu⟩ := hv'.left
|
||
have hw : ∀ w ∈ B, (w, v) ∉ F := by
|
||
intro w hw nw
|
||
have nv := hv'.right
|
||
unfold image at nv
|
||
simp only [Set.mem_setOf_eq, not_exists, not_and] at nv
|
||
exact absurd nw (nv w hw)
|
||
have hu' : u ∉ B := by
|
||
by_contra nu
|
||
exact absurd hu.right (hw u nu)
|
||
unfold image
|
||
simp only [Set.mem_diff, Set.mem_setOf_eq]
|
||
exact ⟨u, ⟨hu.left, hu'⟩, hu.right⟩
|
||
|
||
theorem theorem_3k_c_ii {F : Set.HRelation α β} {A B : Set α}
|
||
(hF : isSingleRooted F)
|
||
: image F A \ image F B = image F (A \ B) := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
refine ⟨theorem_3k_c_i, ?_⟩
|
||
show ∀ v, v ∈ image F (A \ B) → v ∈ image F A \ image F B
|
||
intro v hv
|
||
unfold image at hv
|
||
simp only [Set.mem_diff, Set.mem_setOf_eq] at hv
|
||
have ⟨u, hu⟩ := hv
|
||
have hv₁ : v ∈ image F A := by
|
||
unfold image
|
||
simp only [Set.mem_setOf_eq]
|
||
exact ⟨u, hu.left.left, hu.right⟩
|
||
have hv₂ : v ∉ image F B := by
|
||
intro nv
|
||
unfold image at nv
|
||
simp only [Set.mem_setOf_eq] at nv
|
||
have ⟨u₁, hu₁⟩ := nv
|
||
have := single_rooted_eq_unique hF hu.right hu₁.right
|
||
rw [← this] at hu₁
|
||
exact absurd hu₁.left hu.left.right
|
||
exact ⟨hv₁, hv₂⟩
|
||
|
||
/-! #### Corollary 3L
|
||
|
||
For any function `G` and sets `A`, `B`, and `𝓐`:
|
||
|
||
```
|
||
G⁻¹⟦⋃ 𝓐⟧ = ⋃ {G⁻¹⟦A⟧ | A ∈ 𝓐},
|
||
G⁻¹⟦𝓐⟧ = ⋂ {G⁻¹⟦A⟧ | A ∈ 𝓐} for 𝓐 ≠ ∅,
|
||
G⁻¹⟦A - B⟧ = G⁻¹⟦A⟧ - G⁻¹⟦B⟧.
|
||
```
|
||
-/
|
||
|
||
theorem corollary_3l_i {G : Set.HRelation β α} {𝓐 : Set (Set α)}
|
||
: image (inv G) (⋃₀ 𝓐) = ⋃₀ {image (inv G) A | A ∈ 𝓐} := theorem_3k_a
|
||
|
||
theorem corollary_3l_ii {G : Set.HRelation β α} {𝓐 : Set (Set α)}
|
||
(hG : isSingleValued G) (h𝓐 : Set.Nonempty 𝓐)
|
||
: image (inv G) (⋂₀ 𝓐) = ⋂₀ {image (inv G) A | A ∈ 𝓐} := by
|
||
have hG' : isSingleRooted (inv G) :=
|
||
single_valued_self_iff_single_rooted_inv.mp hG
|
||
exact theorem_3k_b_ii hG' h𝓐
|
||
|
||
theorem corollary_3l_iii {G : Set.HRelation β α} {A B : Set α}
|
||
(hG : isSingleValued G)
|
||
: image (inv G) (A \ B) = image (inv G) A \ image (inv G) B := by
|
||
have hG' : isSingleRooted (inv G) :=
|
||
single_valued_self_iff_single_rooted_inv.mp hG
|
||
exact (theorem_3k_c_ii hG').symm
|
||
|
||
/-- #### Exercise 3.12
|
||
|
||
Assume that `f` and `g` are functions and show that
|
||
```
|
||
f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f) f(x) = g(x).
|
||
```
|
||
-/
|
||
theorem exercise_3_12 {f g : Set.HRelation α β}
|
||
(hf : isSingleValued f) (_ : isSingleValued g)
|
||
: f ⊆ g ↔ dom f ⊆ dom g ∧
|
||
(∀ x ∈ dom f, ∃! y : β, (x, y) ∈ f ∧ (x, y) ∈ g) := by
|
||
apply Iff.intro
|
||
· intro h
|
||
apply And.intro
|
||
· show ∀ x, x ∈ dom f → x ∈ dom g
|
||
intro x hx
|
||
have ⟨y, hy⟩ := dom_exists hx
|
||
exact mem_pair_imp_fst_mem_dom (h hy)
|
||
· intro x hx
|
||
have ⟨y, hy⟩ := dom_exists hx
|
||
refine ⟨y, ⟨hy, h hy⟩, ?_⟩
|
||
intro y₁ hy₁
|
||
exact single_valued_eq_unique hf hy₁.left hy
|
||
· intro ⟨_, hx⟩
|
||
show ∀ p, p ∈ f → p ∈ g
|
||
intro (x, y) hp
|
||
have ⟨y₁, hy₁⟩ := hx x (mem_pair_imp_fst_mem_dom hp)
|
||
rw [single_valued_eq_unique hf hp hy₁.left.left]
|
||
exact hy₁.left.right
|
||
|
||
/-- #### Exercise 3.13
|
||
|
||
Assume that `f` and `g` are functions with `f ⊆ g` and `dom g ⊆ dom f`. Show
|
||
that `f = g`.
|
||
-/
|
||
theorem exercise_3_13 {f g : Set.HRelation α β}
|
||
(hf : isSingleValued f) (hg : isSingleValued g)
|
||
(h : f ⊆ g) (h₁ : dom g ⊆ dom f)
|
||
: f = g := by
|
||
have h₂ := (exercise_3_12 hf hg).mp h
|
||
have hfg := Set.Subset.antisymm_iff.mpr ⟨h₁, h₂.left⟩
|
||
ext p
|
||
have (a, b) := p
|
||
apply Iff.intro
|
||
· intro hp
|
||
have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp)
|
||
rw [single_valued_eq_unique hf hp hx.left.left]
|
||
exact hx.left.right
|
||
· intro hp
|
||
rw [← hfg] at h₂
|
||
have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp)
|
||
rw [single_valued_eq_unique hg hp hx.left.right]
|
||
exact hx.left.left
|
||
|
||
/-- #### Exercise 3.14 (a)
|
||
|
||
Assume that `f` and `g` are functions. Show that `f ∩ g` is a function.
|
||
-/
|
||
theorem exercise_3_14_a {f g : Set.HRelation α β}
|
||
(hf : isSingleValued f) (_ : isSingleValued g)
|
||
: isSingleValued (f ∩ g) :=
|
||
single_valued_subset hf (Set.inter_subset_left f g)
|
||
|
||
/-- #### Exercise 3.14 (b)
|
||
|
||
Assume that `f` and `g` are functions. Show that `f ∪ g` is a function **iff**
|
||
`f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`.
|
||
-/
|
||
theorem exercise_3_14_b {f g : Set.HRelation α β}
|
||
(hf : isSingleValued f) (hg : isSingleValued g)
|
||
: isSingleValued (f ∪ g) ↔
|
||
(∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by
|
||
apply Iff.intro
|
||
· intro h x hx
|
||
have ⟨y₁, hy₁⟩ := hf x hx.left
|
||
have ⟨y₂, hy₂⟩ := hg x hx.right
|
||
have : y₁ = y₂ := single_valued_eq_unique h
|
||
(Or.inl hy₁.left.right)
|
||
(Or.inr hy₂.left.right)
|
||
rw [← this] at hy₂
|
||
refine ⟨y₁, ⟨hy₁.left.right, hy₂.left.right⟩, ?_⟩
|
||
intro y₃ hfy₃
|
||
exact single_valued_eq_unique hf hfy₃.left hy₁.left.right
|
||
· intro h x hx
|
||
by_cases hfx : x ∈ dom f <;>
|
||
by_cases hgx : x ∈ dom g
|
||
· -- `x ∈ dom f ∧ x ∈ dom g`
|
||
have ⟨y₁, hy₁⟩ := hf x hfx
|
||
have ⟨y₂, hy₂⟩ := hg x hgx
|
||
refine ⟨y₁, ⟨?_, Or.inl hy₁.left.right⟩, ?_⟩
|
||
· unfold ran
|
||
simp only [Set.mem_image, Set.mem_union, Prod.exists, exists_eq_right]
|
||
exact ⟨x, Or.inl hy₁.left.right⟩
|
||
· intro y₃ hy₃
|
||
apply Or.elim hy₃.right
|
||
· intro hxy
|
||
exact single_valued_eq_unique hf hxy hy₁.left.right
|
||
· refine fun hxy => single_valued_eq_unique hg hxy ?_
|
||
have : y₁ = y₂ := by
|
||
have ⟨z, ⟨hz, _⟩⟩ := h x ⟨hfx, hgx⟩
|
||
rw [
|
||
single_valued_eq_unique hf hy₁.left.right hz.left,
|
||
single_valued_eq_unique hg hy₂.left.right hz.right
|
||
]
|
||
rw [this]
|
||
exact hy₂.left.right
|
||
· -- `x ∈ dom f ∧ x ∉ dom g`
|
||
have ⟨y, hy⟩ := dom_exists hfx
|
||
have hxy : (x, y) ∈ f ∪ g := (Set.subset_union_left f g) hy
|
||
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩
|
||
intro y₁ hy₁
|
||
apply Or.elim hy₁.right
|
||
· intro hx'
|
||
exact single_valued_eq_unique hf hx' hy
|
||
· intro hx'
|
||
exact absurd (mem_pair_imp_fst_mem_dom hx') hgx
|
||
· -- `x ∉ dom f ∧ x ∈ dom g`
|
||
have ⟨y, hy⟩ := dom_exists hgx
|
||
have hxy : (x, y) ∈ f ∪ g := (Set.subset_union_right f g) hy
|
||
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩
|
||
intro y₁ hy₁
|
||
apply Or.elim hy₁.right
|
||
· intro hx'
|
||
exact absurd (mem_pair_imp_fst_mem_dom hx') hfx
|
||
· intro hx'
|
||
exact single_valued_eq_unique hg hx' hy
|
||
· -- `x ∉ dom f ∧ x ∉ dom g`
|
||
exfalso
|
||
unfold dom at hx
|
||
simp only [
|
||
Set.mem_image,
|
||
Set.mem_union,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right
|
||
] at hx
|
||
have ⟨_, hy⟩ := hx
|
||
apply Or.elim hy
|
||
· intro hz
|
||
exact absurd (mem_pair_imp_fst_mem_dom hz) hfx
|
||
· intro hz
|
||
exact absurd (mem_pair_imp_fst_mem_dom hz) hgx
|
||
|
||
/-- #### Exercise 3.15
|
||
|
||
Let `𝓐` be a set of functions such that for any `f` and `g` in `𝓐`, either
|
||
`f ⊆ g` or `g ⊆ f`. Show that `⋃ 𝓐` is a function.
|
||
-/
|
||
theorem exercise_3_15 {𝓐 : Set (Set.HRelation α β)}
|
||
(h𝓐 : ∀ F ∈ 𝓐, isSingleValued F)
|
||
(h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G ∨ G ⊆ F)
|
||
: isSingleValued (⋃₀ 𝓐) := by
|
||
intro x hx
|
||
have ⟨y₁, hy₁⟩ := dom_exists hx
|
||
refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩
|
||
intro y₂ hy₂
|
||
have ⟨f, hf⟩ : ∃ f : Set.HRelation α β, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁
|
||
have ⟨g, hg⟩ : ∃ g : Set.HRelation α β, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right
|
||
apply Or.elim (h f g hf.left hg.left)
|
||
· intro hf'
|
||
have := hf' hf.right
|
||
exact single_valued_eq_unique (h𝓐 g hg.left) hg.right this
|
||
· intro hg'
|
||
have := hg' hg.right
|
||
exact single_valued_eq_unique (h𝓐 f hf.left) this hf.right
|
||
|
||
/-! #### Exercise 3.17
|
||
|
||
Show that the composition of two single-rooted sets is again single-rooted.
|
||
Conclude that the composition of two one-to-one functions is again one-to-one.
|
||
-/
|
||
|
||
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'
|
||
simp only [Set.mem_setOf_eq] at hu₁'
|
||
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
|
||
|
||
theorem exercise_3_17_ii {F : Set.HRelation β γ} {G : Set.HRelation α β}
|
||
(hF : isOneToOne F) (hG : isOneToOne G)
|
||
: isOneToOne (comp F G) := And.intro
|
||
(single_valued_comp_is_single_valued hF.left hG.left)
|
||
(exercise_3_17_i hF.right hG.right)
|
||
|
||
/-! #### Exercise 3.18
|
||
|
||
Let `R` be the set
|
||
```
|
||
{⟨0, 1⟩, ⟨0, 2⟩, ⟨0, 3⟩, ⟨1, 2⟩, ⟨1, 3⟩, ⟨2, 3⟩}
|
||
```
|
||
Evaluate the following: `R ∘ R`, `R ↾ {1}`, `R⁻¹ ↾ {1}`, `R⟦{1}⟧`, and
|
||
`R⁻¹⟦{1}⟧`.
|
||
-/
|
||
|
||
section Exercise_3_18
|
||
|
||
variable {R : Set.Relation ℕ}
|
||
variable (hR : R = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)})
|
||
|
||
theorem exercise_3_18_i
|
||
: comp R R = {(0, 2), (0, 3), (1, 3)} := by
|
||
rw [hR]
|
||
unfold comp
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, or_self, Prod.mk.injEq]
|
||
ext x
|
||
have (a, b) := x
|
||
apply Iff.intro
|
||
· simp only [Set.mem_setOf_eq, Set.mem_singleton_iff, Set.mem_insert_iff]
|
||
intro ⟨t, ht₁, ht₂⟩
|
||
casesm* _ ∨ _
|
||
all_goals case _ hl hr => first
|
||
| {rw [hl.right] at hr; simp at hr}
|
||
| {rw [hl.left] at hr; simp at hr}
|
||
| {rw [hl.left, hr.right]; simp}
|
||
· simp only [
|
||
Set.mem_singleton_iff,
|
||
Set.mem_insert_iff,
|
||
Prod.mk.injEq,
|
||
Set.mem_setOf_eq
|
||
]
|
||
intro h
|
||
casesm* _ ∨ _
|
||
· case _ h =>
|
||
refine ⟨1, Or.inl ⟨h.left, rfl⟩, ?_⟩
|
||
iterate 3 right
|
||
left
|
||
exact ⟨rfl, h.right⟩
|
||
· case _ h =>
|
||
refine ⟨1, Or.inl ⟨h.left, rfl⟩, ?_⟩
|
||
iterate 4 right
|
||
left
|
||
exact ⟨rfl, h.right⟩
|
||
· case _ h =>
|
||
refine ⟨2, ?_, ?_⟩
|
||
· iterate 3 right
|
||
left
|
||
exact ⟨h.left, rfl⟩
|
||
· iterate 5 right
|
||
exact ⟨rfl, h.right⟩
|
||
|
||
theorem exercise_3_18_ii
|
||
: restriction R {1} = {(1, 2), (1, 3)} := by
|
||
rw [hR]
|
||
unfold restriction
|
||
ext p
|
||
have (a, b) := p
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Set.mem_insert_iff,
|
||
Set.mem_setOf_eq,
|
||
or_self
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨hp, ha⟩
|
||
rw [ha]
|
||
simp only [Prod.mk.injEq, true_and]
|
||
casesm* _ ∨ _
|
||
all_goals case _ h => first
|
||
| {rw [ha] at h; simp at h}
|
||
| {simp only [Prod.mk.injEq] at h; left; exact h.right}
|
||
| {simp only [Prod.mk.injEq] at h; right; exact h.right}
|
||
· intro h
|
||
apply Or.elim h
|
||
· intro hab
|
||
simp only [Prod.mk.injEq] at hab
|
||
refine ⟨?_, hab.left⟩
|
||
iterate 3 right
|
||
left
|
||
rw [hab.left, hab.right]
|
||
· intro hab
|
||
simp only [Prod.mk.injEq] at hab
|
||
refine ⟨?_, hab.left⟩
|
||
iterate 4 right
|
||
left
|
||
rw [hab.left, hab.right]
|
||
|
||
theorem exercise_3_18_iii
|
||
: restriction (inv R) {1} = {(1, 0)} := by
|
||
rw [hR]
|
||
unfold inv restriction
|
||
ext p
|
||
have (a, b) := p
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Set.mem_insert_iff,
|
||
or_self,
|
||
exists_eq_or_imp,
|
||
exists_eq_left,
|
||
Set.mem_setOf_eq,
|
||
Prod.mk.injEq
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨hb, ha⟩
|
||
casesm* _ ∨ _
|
||
all_goals case _ hr => first
|
||
| exact ⟨ha, hr.right.symm⟩
|
||
| rw [ha] at hr; simp at hr
|
||
· intro ⟨ha, hb⟩
|
||
rw [ha, hb]
|
||
simp
|
||
|
||
theorem exercise_3_18_iv
|
||
: image R {1} = {2, 3} := by
|
||
rw [hR]
|
||
unfold image
|
||
ext y
|
||
simp
|
||
|
||
theorem exercise_3_18_v
|
||
: image (inv R) {1} = {0} := by
|
||
rw [hR]
|
||
unfold inv image
|
||
ext y
|
||
simp
|
||
|
||
end Exercise_3_18
|
||
|
||
/-! #### Exercise 3.19
|
||
|
||
Let
|
||
```
|
||
A = {⟨∅, {∅, {∅}}⟩, ⟨{∅}, ∅⟩}.
|
||
```
|
||
Evaluate each of the following: `A(∅)`, `A⟦∅⟧`, `A⟦{∅}⟧`, `A⟦{∅, {∅}}⟧`,
|
||
`A⁻¹`, `A ∘ A`, `A ↾ ∅`, `A ↾ {∅, {∅}}`, `⋃ ⋃ A`.
|
||
-/
|
||
|
||
section Exercise_3_19
|
||
|
||
variable {A : Set.Relation (Set (Set (Set α)))}
|
||
variable (hA : A = {(∅, {∅, {∅}}), ({∅}, ∅)})
|
||
|
||
theorem exercise_3_19_i
|
||
: (∅, {∅, {∅}}) ∈ A := by
|
||
rw [hA]
|
||
simp
|
||
|
||
theorem exercise_3_19_ii
|
||
: image A ∅ = ∅ := by
|
||
unfold image
|
||
simp
|
||
|
||
theorem exercise_3_19_iii
|
||
: image A {∅} = {{∅, {∅}}} := by
|
||
unfold image
|
||
rw [hA]
|
||
ext x
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Prod.mk.injEq,
|
||
Set.mem_insert_iff,
|
||
exists_eq_left,
|
||
true_and
|
||
]
|
||
apply Iff.intro
|
||
· intro hx
|
||
simp at hx
|
||
apply Or.elim hx
|
||
· simp
|
||
· intro ⟨h, _⟩
|
||
exfalso
|
||
rw [Set.ext_iff] at h
|
||
have := h ∅
|
||
simp at this
|
||
· intro hx
|
||
rw [hx]
|
||
simp
|
||
|
||
theorem exercise_3_19_iv
|
||
: image A {∅, {∅}} = {{∅, {∅}}, ∅} := by
|
||
unfold image
|
||
rw [hA]
|
||
ext x
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Set.mem_insert_iff,
|
||
Prod.mk.injEq,
|
||
exists_eq_or_imp,
|
||
true_and,
|
||
exists_eq_left,
|
||
Set.singleton_ne_empty,
|
||
false_and,
|
||
false_or,
|
||
Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro h
|
||
apply Or.elim h
|
||
· intro hx₁
|
||
apply Or.elim hx₁
|
||
· intro hx₂; left ; exact hx₂
|
||
· intro hx₂; right; exact hx₂.right
|
||
· intro hx₂
|
||
right
|
||
exact hx₂
|
||
· intro h
|
||
apply Or.elim h
|
||
· intro hx₁; iterate 2 left
|
||
exact hx₁
|
||
· intro hx₁; right; exact hx₁
|
||
|
||
theorem exercise_3_19_v
|
||
: inv A = {({∅, {∅}}, ∅), (∅, {∅})} := by
|
||
unfold inv
|
||
rw [hA]
|
||
ext x
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Prod.mk.injEq,
|
||
Set.mem_insert_iff,
|
||
exists_eq_or_imp,
|
||
exists_eq_left,
|
||
Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro hx
|
||
apply Or.elim hx
|
||
· intro hx₁; left ; rw [← hx₁]
|
||
· intro hx₁; right; rw [← hx₁]
|
||
· intro hx
|
||
apply Or.elim hx
|
||
· intro hx₁; left ; rw [← hx₁]
|
||
· intro hx₁; right; rw [← hx₁]
|
||
|
||
theorem exercise_3_19_vi
|
||
: comp A A = {({∅}, {∅, {∅}})} := by
|
||
unfold comp
|
||
rw [hA]
|
||
ext x
|
||
have (a, b) := x
|
||
simp only [
|
||
Set.mem_singleton_iff, Prod.mk.injEq, Set.mem_insert_iff, Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨t, ht₁, ht₂⟩
|
||
casesm* _ ∨ _
|
||
all_goals case _ hl hr => first
|
||
| {
|
||
rw [hl.right] at hr
|
||
have := hr.left
|
||
rw [Set.ext_iff] at this
|
||
simp at this
|
||
}
|
||
| exact ⟨hl.left, hr.right⟩
|
||
· intro ⟨ha, hb⟩
|
||
refine ⟨∅, ?_, ?_⟩
|
||
· right; rw [ha]; simp
|
||
· left ; rw [hb]; simp
|
||
|
||
theorem exercise_3_19_vii
|
||
: restriction A ∅ = ∅ := by
|
||
unfold restriction
|
||
rw [hA]
|
||
simp
|
||
|
||
theorem exercise_3_19_viii
|
||
: restriction A {∅} = {(∅, {∅, {∅}})} := by
|
||
unfold restriction
|
||
rw [hA]
|
||
ext x
|
||
have (a, b) := x
|
||
simp only [
|
||
Set.mem_singleton_iff, Prod.mk.injEq, Set.mem_insert_iff, Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨h, ha⟩
|
||
apply Or.elim h
|
||
· simp
|
||
· intro ⟨ha', _⟩
|
||
exfalso
|
||
rw [ha', Set.ext_iff] at ha
|
||
simp at ha
|
||
· intro ⟨ha, hb⟩
|
||
rw [ha, hb]
|
||
simp
|
||
|
||
theorem exercise_3_19_ix
|
||
: restriction A {∅, {∅}} = A := by
|
||
unfold restriction
|
||
rw [hA]
|
||
ext x
|
||
have (a, b) := x
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Prod.mk.injEq,
|
||
Set.mem_insert_iff,
|
||
Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨h₁, h₂⟩
|
||
casesm* _ ∨ _
|
||
· case _ hl _ => left ; exact hl
|
||
· case _ hl _ => left ; exact hl
|
||
· case _ hl _ => right; exact hl
|
||
· case _ hl _ => right; exact hl
|
||
· intro h₁
|
||
apply Or.elim h₁ <;>
|
||
· intro ⟨ha, hb⟩
|
||
rw [ha, hb]
|
||
simp
|
||
|
||
theorem exercise_3_19_x
|
||
: ⋃₀ ⋃₀ A.toOrderedPairs = {∅, {∅}, {∅, {∅}}} := by
|
||
unfold toOrderedPairs OrderedPair Set.sUnion sSup Set.instSupSetSet
|
||
rw [hA]
|
||
ext x
|
||
simp only [
|
||
Set.mem_singleton_iff,
|
||
Prod.mk.injEq,
|
||
Set.mem_image,
|
||
Set.mem_insert_iff,
|
||
exists_eq_or_imp,
|
||
exists_eq_left,
|
||
Set.singleton_ne_empty,
|
||
Set.mem_setOf_eq
|
||
]
|
||
apply Iff.intro
|
||
· intro ⟨a, ⟨t, ht₁, ht₂⟩, hx⟩
|
||
apply Or.elim ht₁
|
||
· intro ht
|
||
rw [← ht] at ht₂
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at ht₂
|
||
apply Or.elim ht₂
|
||
· intro ha
|
||
rw [ha] at hx
|
||
simp only [Set.mem_singleton_iff] at hx
|
||
left
|
||
exact hx
|
||
· intro ha
|
||
rw [ha] at hx
|
||
simp at hx
|
||
apply Or.elim hx <;>
|
||
· intro hx'; rw [hx']; simp
|
||
· intro ht
|
||
rw [← ht] at ht₂
|
||
simp only [
|
||
Set.mem_singleton_iff, Set.singleton_ne_empty, Set.mem_insert_iff
|
||
] at ht₂
|
||
apply Or.elim ht₂
|
||
· intro ha
|
||
rw [ha] at hx
|
||
simp only [Set.mem_singleton_iff] at hx
|
||
rw [hx]
|
||
simp
|
||
· intro ha
|
||
rw [ha] at hx
|
||
simp only [
|
||
Set.mem_singleton_iff, Set.singleton_ne_empty, Set.mem_insert_iff
|
||
] at hx
|
||
apply Or.elim hx <;>
|
||
· intro hx'; rw [hx']; simp
|
||
· intro hx
|
||
apply Or.elim hx
|
||
· intro hx₁
|
||
rw [hx₁]
|
||
refine ⟨{{∅}, ∅}, ⟨{{{∅}}, {{∅}, ∅}}, ?_⟩, ?_⟩ <;> simp
|
||
· intro hx₁
|
||
apply Or.elim hx₁
|
||
· intro hx₂
|
||
rw [hx₂]
|
||
refine ⟨{{∅}, ∅}, ⟨{{{∅}}, {{∅}, ∅}}, ?_⟩, ?_⟩ <;> simp
|
||
· intro hx₂
|
||
rw [hx₂]
|
||
refine ⟨{∅, {∅, {∅}}}, ⟨{{∅}, {∅, {∅, {∅}}}}, ?_⟩, ?_⟩ <;> simp
|
||
|
||
end Exercise_3_19
|
||
|
||
/-- #### Exercise 3.20
|
||
|
||
Show that `F ↾ A = F ∩ (A × ran F)`.
|
||
-/
|
||
theorem exercise_3_20 {F : Set.HRelation α β} {A : Set α}
|
||
: restriction F A = F ∩ (Set.prod A (ran F)) := by
|
||
calc restriction F A
|
||
_ = {p | p ∈ F ∧ p.fst ∈ A} := rfl
|
||
_ = {p | p ∈ F ∧ p.fst ∈ A ∧ p.snd ∈ ran F} := by
|
||
ext x
|
||
have (a, b) := x
|
||
simp only [
|
||
Set.mem_setOf_eq, Set.sep_and, Set.mem_inter_iff, iff_self_and, and_imp
|
||
]
|
||
intro hF _
|
||
exact ⟨hF, mem_pair_imp_snd_mem_ran hF⟩
|
||
_ = {p | p ∈ F} ∩ {p | p.fst ∈ A ∧ p.snd ∈ ran F} := rfl
|
||
_ = F ∩ {p | p.fst ∈ A ∧ p.snd ∈ ran F} := rfl
|
||
_ = F ∩ (Set.prod A (ran F)) := rfl
|
||
|
||
/-- #### Exercise 3.22 (a)
|
||
|
||
Show that the following is correct for any sets.
|
||
```
|
||
A ⊆ B → F⟦A⟧ ⊆ F⟦B⟧
|
||
```
|
||
-/
|
||
theorem exercise_3_22_a {A B : Set α} {F : Set.HRelation α β} (h : A ⊆ B)
|
||
: image F A ⊆ image F B := by
|
||
show ∀ x, x ∈ image F A → x ∈ image F B
|
||
intro x hx
|
||
have ⟨u, hu⟩ := hx
|
||
have := h hu.left
|
||
exact ⟨u, this, hu.right⟩
|
||
|
||
/-- #### Exercise 3.22 (b)
|
||
|
||
Show that the following is correct for any sets.
|
||
```
|
||
(F ∘ G)⟦A⟧ = F⟦G⟦A⟧⟧
|
||
```
|
||
-/
|
||
theorem exercise_3_22_b {A B : Set α} {F : Set.HRelation α β}
|
||
: image (comp F G) A = image F (image G A) := by
|
||
calc image (comp F G) A
|
||
_ = { v | ∃ u ∈ A, (u, v) ∈ comp F G } := rfl
|
||
_ = { v | ∃ u ∈ A, ∃ a, (u, a) ∈ G ∧ (a, v) ∈ F } := rfl
|
||
_ = { v | ∃ a, ∃ u ∈ A, (u, a) ∈ G ∧ (a, v) ∈ F } := by
|
||
ext p
|
||
apply Iff.intro
|
||
· intro ⟨u, hu, a, ha⟩
|
||
exact ⟨a, u, hu, ha⟩
|
||
· intro ⟨a, u, hu, ha⟩
|
||
exact ⟨u, hu, a, ha⟩
|
||
_ = { v | ∃ a, (∃ u ∈ A, (u, a) ∈ G) ∧ (a, v) ∈ F } := by
|
||
ext p
|
||
apply Iff.intro
|
||
· intro ⟨a, u, h⟩
|
||
exact ⟨a, ⟨u, h.left, h.right.left⟩, h.right.right⟩
|
||
· intro ⟨a, ⟨u, hu⟩, ha⟩
|
||
exact ⟨a, u, hu.left, hu.right, ha⟩
|
||
_ = { v | ∃ a ∈ { w | ∃ u ∈ A, (u, w) ∈ G }, (a, v) ∈ F } := rfl
|
||
_ = { v | ∃ a ∈ image G A, (a, v) ∈ F } := rfl
|
||
_ = image F (image G A) := rfl
|
||
|
||
/-- #### Exercise 3.22 (c)
|
||
|
||
Show that the following is correct for any sets.
|
||
```
|
||
Q ↾ (A ∪ B) = (Q ↾ A) ∪ (Q ↾ B)
|
||
```
|
||
-/
|
||
theorem exercise_3_22_c {A B : Set α} {Q : Set.Relation α}
|
||
: restriction Q (A ∪ B) = (restriction Q A) ∪ (restriction Q B) := by
|
||
calc restriction Q (A ∪ B)
|
||
_ = { p | p ∈ Q ∧ p.1 ∈ A ∪ B } := rfl
|
||
_ = { p | p ∈ Q ∧ (p.1 ∈ A ∨ p.1 ∈ B) } := rfl
|
||
_ = { p | (p ∈ Q ∧ p.1 ∈ A) ∨ (p ∈ Q ∧ p.1 ∈ B) } := by
|
||
ext p
|
||
simp only [Set.sep_or, Set.mem_union, Set.mem_setOf_eq]
|
||
_ = { p | p ∈ Q ∧ p.1 ∈ A} ∪ { p | p ∈ Q ∧ p.1 ∈ B } := rfl
|
||
_ = (restriction Q A) ∪ (restriction Q B) := rfl
|
||
|
||
/-- #### Exercise 3.23 (i)
|
||
|
||
Let `I` be the identity function on the set `A`. Show that for any sets `B` and
|
||
`C`, `B ∘ I = B ↾ A`.
|
||
-/
|
||
theorem exercise_3_23_i {A : Set α} {B : Set.HRelation α β} {I : Set.Relation α}
|
||
(hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
|
||
: comp B I = restriction B A := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· show ∀ p, p ∈ comp B I → p ∈ restriction B A
|
||
intro (x, y) hp
|
||
have ⟨t, ht⟩ := hp
|
||
rw [hI] at ht
|
||
simp only [Set.mem_setOf_eq] at ht
|
||
show (x, y) ∈ B ∧ x ∈ A
|
||
rw [← ht.left.right] at ht
|
||
exact ⟨ht.right, ht.left.left⟩
|
||
· show ∀ p, p ∈ restriction B A → p ∈ comp B I
|
||
unfold restriction comp
|
||
rw [hI]
|
||
intro (x, y) hp
|
||
refine ⟨x, ⟨hp.right, rfl⟩, hp.left⟩
|
||
|
||
/-- #### Exercise 3.23 (ii)
|
||
|
||
Let `I` be the identity function on the set `A`. Show that for any sets `B` and
|
||
`C`, `I⟦C⟧ = A ∩ C`.
|
||
-/
|
||
theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α}
|
||
(hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
|
||
: image I C = A ∩ C := by
|
||
calc image I C
|
||
_ = { v | ∃ u ∈ C, (u, v) ∈ I } := rfl
|
||
_ = { v | ∃ u ∈ C, u ∈ A ∧ u = v } := by
|
||
ext v
|
||
apply Iff.intro
|
||
· intro ⟨u, h₁, h₂⟩
|
||
rw [hI] at h₂
|
||
exact ⟨u, h₁, h₂⟩
|
||
· intro ⟨u, h₁, h₂⟩
|
||
refine ⟨u, h₁, ?_⟩
|
||
· rw [hI]
|
||
exact h₂
|
||
_ = { v | v ∈ C ∧ v ∈ A } := by
|
||
ext v
|
||
simp only [Set.mem_setOf_eq, Set.sep_mem_eq, Set.mem_inter_iff]
|
||
apply Iff.intro
|
||
· intro ⟨u, hC, hA, hv⟩
|
||
rw [← hv]
|
||
exact ⟨hC, hA⟩
|
||
· intro ⟨hC, hA⟩
|
||
exact ⟨v, hC, hA, rfl⟩
|
||
_ = C ∩ A := rfl
|
||
_ = A ∩ C := Set.inter_comm C A
|
||
|
||
/-- #### Exercise 3.24
|
||
|
||
Show that for a function `F`, `F⁻¹⟦A⟧ = { x ∈ dom F | F(x) ∈ A }`.
|
||
-/
|
||
theorem exercise_3_24 {F : Set.HRelation α β} {A : Set β}
|
||
(hF : isSingleValued F)
|
||
: image (inv F) A = { x ∈ dom F | ∃! y : β, (x, y) ∈ F ∧ y ∈ A } := by
|
||
calc image (inv F) A
|
||
_ = { x | ∃ y ∈ A, (y, x) ∈ inv F } := rfl
|
||
_ = { x | ∃ y ∈ A, (x, y) ∈ F } := by simp only [mem_self_comm_mem_inv]
|
||
_ = { x | x ∈ dom F ∧ (∃ y : β, (x, y) ∈ F ∧ y ∈ A) } := by
|
||
ext x
|
||
apply Iff.intro
|
||
· intro ⟨y, hy, hyx⟩
|
||
exact ⟨mem_pair_imp_fst_mem_dom hyx, y, hyx, hy⟩
|
||
· intro ⟨_, y, hxy, hy⟩
|
||
exact ⟨y, hy, hxy⟩
|
||
_ = { x ∈ dom F | ∃ y : β, (x, y) ∈ F ∧ y ∈ A } := rfl
|
||
_ = { x ∈ dom F | ∃! y : β, (x, y) ∈ F ∧ y ∈ A } := by
|
||
ext x
|
||
simp only [Set.mem_setOf_eq, and_congr_right_iff]
|
||
intro _
|
||
apply Iff.intro
|
||
· intro ⟨y, hy⟩
|
||
refine ⟨y, hy, ?_⟩
|
||
intro y₁ hy₁
|
||
exact single_valued_eq_unique hF hy₁.left hy.left
|
||
· intro ⟨y, hy⟩
|
||
exact ⟨y, hy.left⟩
|
||
|
||
/-- #### Exercise 3.25 (b)
|
||
|
||
Show that the result of part (a) holds for any function `G`, not necessarily
|
||
one-to-one.
|
||
-/
|
||
theorem exercise_3_25_b {G : Set.HRelation α β} (hG : isSingleValued G)
|
||
: comp G (inv G) = { p | p.1 ∈ ran G ∧ p.1 = p.2 } := by
|
||
ext p
|
||
have (x, y) := p
|
||
apply Iff.intro
|
||
· unfold comp inv
|
||
intro h
|
||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq] at h
|
||
have ⟨t, ⟨a, b, ⟨hab, hb, ha⟩⟩, ht⟩ := h
|
||
rw [hb, ha] at hab
|
||
exact ⟨mem_pair_imp_snd_mem_ran hab, single_valued_eq_unique hG hab ht⟩
|
||
· intro h
|
||
simp only [Set.mem_setOf_eq] at h
|
||
unfold comp inv
|
||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
|
||
have ⟨t, ht⟩ := ran_exists h.left
|
||
exact ⟨t, ⟨t, x, ht, rfl, rfl⟩, by rwa [← h.right]⟩
|
||
|
||
/-- #### Exercise 3.25 (a)
|
||
|
||
Assume that `G` is a one-to-one function. Show that `G ∘ G⁻¹` is the identity
|
||
function on `ran G`.
|
||
-/
|
||
theorem exercise_3_25_a {G : Set.HRelation α β} (hG : isOneToOne G)
|
||
: comp G (inv G) = { p | p.1 ∈ ran G ∧ p.1 = p.2 } :=
|
||
exercise_3_25_b hG.left
|
||
|
||
/-- #### Exercise 3.27
|
||
|
||
Show that `dom (F ∘ G) = G⁻¹⟦dom F⟧` for any sets `F` and `G`. (`F` and `G` need
|
||
not be functions.)
|
||
-/
|
||
theorem exercise_3_27 {F : Set.HRelation β γ} {G : Set.HRelation α β}
|
||
: dom (comp F G) = image (inv G) (dom F) := by
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· show ∀ x, x ∈ dom (comp F G) → x ∈ image (inv G) (dom F)
|
||
intro x hx
|
||
have ⟨y, t, ht⟩ := dom_exists hx
|
||
have htF : t ∈ dom F := mem_pair_imp_fst_mem_dom ht.right
|
||
unfold image inv
|
||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
|
||
exact ⟨t, htF, x, t, ht.left, rfl, rfl⟩
|
||
|
||
· show ∀ x, x ∈ image (inv G) (dom F) → x ∈ dom (comp F G)
|
||
intro x hx
|
||
unfold image at hx
|
||
simp only [mem_self_comm_mem_inv, Set.mem_setOf_eq] at hx
|
||
have ⟨u, hu⟩ := hx
|
||
have ⟨t, ht⟩ := dom_exists hu.left
|
||
|
||
unfold dom comp
|
||
simp only [
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right
|
||
]
|
||
exact ⟨t, u, hu.right, ht⟩
|
||
|
||
/-- #### Exercise 3.28
|
||
|
||
Assume that `f` is a one-to-one function from `A` into `B`, and that `G` is the
|
||
function with `dom G = 𝒫 A` defined by the equation `G(X) = f⟦X⟧`. Show that `G`
|
||
maps `𝒫 A` one-to-one into `𝒫 B`.
|
||
-/
|
||
theorem exercise_3_28 {A : Set α} {B : Set β}
|
||
{f : Set.HRelation α β} {G : Set.HRelation (Set α) (Set β)}
|
||
(hf : isOneToOne f ∧ mapsInto f A B)
|
||
(hG : G = { p | p.1 ∈ 𝒫 A ∧ p.2 = image f p.1 })
|
||
: isOneToOne G ∧ mapsInto G (𝒫 A) (𝒫 B) := by
|
||
have dG : dom G = 𝒫 A := by
|
||
rw [hG]
|
||
ext p
|
||
unfold dom Prod.fst
|
||
simp
|
||
|
||
have hG₁ : isSingleValued G := by
|
||
intro x hx
|
||
have ⟨y, hy⟩ := dom_exists hx
|
||
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hy, hy⟩, ?_⟩
|
||
intro y₁ hy₁
|
||
rw [hG, Set.mem_setOf_eq] at hy
|
||
conv at hy₁ => rhs; rw [hG, Set.mem_setOf_eq]
|
||
simp only at *
|
||
rw [hy.right, hy₁.right.right]
|
||
|
||
apply And.intro
|
||
· show isOneToOne G
|
||
refine ⟨hG₁, ?_⟩
|
||
intro y hy
|
||
have ⟨X₁, hX₁⟩ := ran_exists hy
|
||
refine ⟨X₁, ⟨mem_pair_imp_fst_mem_dom hX₁, hX₁⟩, ?_⟩
|
||
intro X₂ hX₂
|
||
have hX₁' : y = image f X₁ := by
|
||
rw [hG] at hX₁
|
||
simp only [Set.mem_powerset_iff, Set.mem_setOf_eq] at hX₁
|
||
exact hX₁.right
|
||
have hX₂' : y = image f X₂ := by
|
||
have := hX₂.right
|
||
rw [hG] at this
|
||
simp only [Set.mem_powerset_iff, Set.mem_setOf_eq] at this
|
||
exact this.right
|
||
|
||
ext t
|
||
apply Iff.intro
|
||
· intro ht
|
||
rw [dG] at hX₂
|
||
simp only [Set.mem_powerset_iff] at hX₂
|
||
|
||
have ht' := hX₂.left ht
|
||
rw [← hf.right.dom_eq] at ht'
|
||
have ⟨ft, hft⟩ := dom_exists ht'
|
||
have hft' : ft ∈ image f X₂ := ⟨t, ht, hft⟩
|
||
|
||
rw [← hX₂', hX₁'] at hft'
|
||
have ⟨t₁, ht₁⟩ := hft'
|
||
rw [single_rooted_eq_unique hf.left.right hft ht₁.right]
|
||
exact ht₁.left
|
||
|
||
· intro ht
|
||
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'
|
||
have hft' : ft ∈ image f X₁ := ⟨t, ht, hft⟩
|
||
|
||
rw [← hX₁', hX₂'] at hft'
|
||
have ⟨t₁, ht₁⟩ := hft'
|
||
rw [single_rooted_eq_unique hf.left.right hft ht₁.right]
|
||
exact ht₁.left
|
||
|
||
· show mapsInto G (𝒫 A) (𝒫 B)
|
||
refine ⟨hG₁, dG, ?_⟩
|
||
show ∀ x, x ∈ ran G → x ∈ 𝒫 B
|
||
intro x hx
|
||
rw [hG] at hx
|
||
unfold ran Prod.snd at hx
|
||
simp only [
|
||
Set.mem_powerset_iff,
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_eq_right
|
||
] at hx
|
||
have ⟨a, ha⟩ := hx
|
||
rw [ha.right]
|
||
show ∀ y, y ∈ image f a → y ∈ B
|
||
intro y ⟨b, hb⟩
|
||
have hz := mem_pair_imp_snd_mem_ran hb.right
|
||
exact hf.right.ran_ss hz
|
||
|
||
/-- #### Exercise 3.29
|
||
|
||
Assume that `f : A → B` and define a function `G : B → 𝒫 A` by
|
||
```
|
||
G(b) = {x ∈ A | f(x) = b}
|
||
```
|
||
Show that if `f` maps `A` *onto* `B`, then `G` is one-to-one. Does the converse
|
||
hold?
|
||
-/
|
||
theorem exercise_3_29 {f : Set.HRelation α β} {G : Set.HRelation β (Set α)}
|
||
{A : Set α} {B : Set β} (hf : mapsOnto f A B)
|
||
(hG : mapsInto G B (𝒫 A) ∧ G = { p | p.1 ∈ B ∧ p.2 = {x ∈ A | (x, p.1) ∈ f} })
|
||
: isOneToOne G := by
|
||
unfold isOneToOne
|
||
refine ⟨hG.left.is_func, ?_⟩
|
||
intro y hy
|
||
have ⟨x₁, hx₁⟩ := ran_exists hy
|
||
refine ⟨x₁, ⟨mem_pair_imp_fst_mem_dom hx₁, hx₁⟩, ?_⟩
|
||
intro x₂ hx₂
|
||
|
||
have hG₁ : (x₁, {x ∈ A | (x, x₁) ∈ f}) ∈ G := by
|
||
rw [hG.right, ← hG.left.dom_eq]
|
||
simp only [Set.mem_setOf_eq, and_true]
|
||
exact mem_pair_imp_fst_mem_dom hx₁
|
||
have hG₂ : (x₂, {x ∈ A | (x, x₂) ∈ f}) ∈ G := by
|
||
rw [hG.right, ← hG.left.dom_eq]
|
||
simp only [Set.mem_setOf_eq, and_true]
|
||
exact hx₂.left
|
||
have heq : {x ∈ A | (x, x₁) ∈ f} = {x ∈ A | (x, x₂) ∈ f} := by
|
||
have h₁ := single_valued_eq_unique hG.left.is_func hx₁ hG₁
|
||
have h₂ := single_valued_eq_unique hG.left.is_func hx₂.right hG₂
|
||
rw [← h₁, ← h₂]
|
||
|
||
rw [hG.right, ← hf.ran_eq] at hG₁
|
||
simp only [Set.mem_setOf_eq, and_true] at hG₁
|
||
have ⟨t, ht⟩ := ran_exists hG₁
|
||
have : t ∈ {x ∈ A | (x, x₁) ∈ f} := by
|
||
refine ⟨?_, ht⟩
|
||
rw [← hf.dom_eq]
|
||
exact mem_pair_imp_fst_mem_dom ht
|
||
rw [heq] at this
|
||
exact single_valued_eq_unique hf.is_func this.right ht
|
||
|
||
/-- #### Theorem 3M
|
||
|
||
If `R` is a symmetric and transitive relation, then `R` is an equivalence
|
||
relation on `fld R`.
|
||
-/
|
||
theorem theorem_3m {R : Set.Relation α}
|
||
(hS : R.isSymmetric) (hT : R.isTransitive)
|
||
: R.isEquivalence (fld R) := by
|
||
refine ⟨Eq.subset rfl, ?_, hS, hT⟩
|
||
intro x hx
|
||
apply Or.elim hx
|
||
· intro h
|
||
have ⟨y, hy⟩ := dom_exists h
|
||
have := hS hy
|
||
exact hT hy this
|
||
· intro h
|
||
have ⟨t, ht⟩ := ran_exists h
|
||
have := hS ht
|
||
exact hT this ht
|
||
|
||
/-- #### Exercise 3.32 (a)
|
||
|
||
Show that `R` is symmetric **iff** `R⁻¹ ⊆ R`.
|
||
-/
|
||
theorem exercise_3_32_a {R : Set.Relation α}
|
||
: isSymmetric R ↔ inv R ⊆ R := by
|
||
apply Iff.intro
|
||
· intro hR
|
||
show ∀ p, p ∈ inv R → p ∈ R
|
||
intro (x, y) hp
|
||
simp only [mem_self_comm_mem_inv] at hp
|
||
exact hR hp
|
||
· intro hR
|
||
unfold isSymmetric
|
||
intro x y hp
|
||
rw [← mem_self_comm_mem_inv] at hp
|
||
exact hR hp
|
||
|
||
/-- #### Exercise 3.32 (b)
|
||
|
||
Show that `R` is transitive **iff** `R ∘ R ⊆ R`.
|
||
-/
|
||
theorem exercise_3_32_b {R : Set.Relation α}
|
||
: isTransitive R ↔ comp R R ⊆ R := by
|
||
apply Iff.intro
|
||
· intro hR
|
||
show ∀ p, p ∈ comp R R → p ∈ R
|
||
intro (x, y) hp
|
||
have ⟨t, ht⟩ := hp
|
||
exact hR ht.left ht.right
|
||
· intro hR
|
||
intro x y z hx hz
|
||
have : (x, z) ∈ comp R R := ⟨y, hx, hz⟩
|
||
exact hR this
|
||
|
||
/-- #### Exercise 3.33
|
||
|
||
Show that `R` is a symmetric and transitive relation **iff** `R = R⁻¹ ∘ R`.
|
||
-/
|
||
theorem exercise_3_33 {R : Set.Relation α}
|
||
: isSymmetric R ∧ isTransitive R ↔ R = comp (inv R) R := by
|
||
have hR : comp (inv R) R = { p | ∃ t, (p.1, t) ∈ R ∧ (p.2, t) ∈ R } := by
|
||
ext p
|
||
unfold comp inv
|
||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
|
||
apply Iff.intro
|
||
· intro ⟨t, ht, a, b, h⟩
|
||
refine ⟨t, ht, ?_⟩
|
||
rw [← h.right.right, ← h.right.left]
|
||
exact h.left
|
||
· intro ⟨t, ht⟩
|
||
exact ⟨t, ht.left, p.snd, t, ht.right, rfl, rfl⟩
|
||
|
||
apply Iff.intro
|
||
· intro h
|
||
rw [Set.Subset.antisymm_iff]
|
||
apply And.intro
|
||
· show ∀ p, p ∈ R → p ∈ comp (inv R) R
|
||
intro (x, y) hp
|
||
have hy := h.left hp
|
||
have hx := h.right hp hy
|
||
rw [hR]
|
||
exact ⟨x, hx, hy⟩
|
||
· show ∀ p, p ∈ comp (inv R) R → p ∈ R
|
||
intro (x, y) hp
|
||
rw [hR] at hp
|
||
have ⟨_, ht⟩ := hp
|
||
have := h.left ht.right
|
||
exact h.right ht.left this
|
||
· intro h
|
||
have hS : isSymmetric R := by
|
||
intro x y hp
|
||
have : inv R = R := by
|
||
calc inv R
|
||
_ = inv (comp (inv R) R) := by conv => lhs; rw [h]
|
||
_ = comp (inv R) (inv (inv R)) := by rw [comp_inv_eq_inv_comp_inv]
|
||
_ = comp (inv R) R := by rw [inv_inv_eq_self]
|
||
_ = R := h.symm
|
||
rwa [← this, mem_self_comm_mem_inv]
|
||
refine ⟨hS, ?_⟩
|
||
intro x y z hx hy
|
||
have : (z, y) ∈ R := hS hy
|
||
rw [h, hR]
|
||
exact ⟨y, hx, this⟩
|
||
|
||
/-- #### Exercise 3.34 (a)
|
||
|
||
Assume that `𝓐` is a nonempty set, every member of which is a transitive
|
||
relation. Is the set `⋂ 𝓐` a transitive relation?
|
||
-/
|
||
theorem exercise_3_34_a {𝓐 : Set (Set.Relation α)}
|
||
(_ : Set.Nonempty 𝓐) (h𝓐 : ∀ A ∈ 𝓐, isTransitive A)
|
||
: isTransitive (⋂₀ 𝓐) := by
|
||
intro x y z hx hy
|
||
simp only [Set.mem_sInter] at *
|
||
intro A hA
|
||
have hx' := hx A hA
|
||
have hy' := hy A hA
|
||
exact h𝓐 A hA hx' hy'
|
||
|
||
/-- #### Exercise 3.34 (b)
|
||
|
||
Assume that `𝓐` is a nonempty set, every member of which is a transitive
|
||
relation. Is `⋃ 𝓐` a transitive relation?
|
||
-/
|
||
theorem exercise_3_34_b {𝓐 : Set (Set.Relation ℕ)}
|
||
(_ : Set.Nonempty 𝓐) (h𝓐 : 𝓐 = {{(1, 2), (2, 3), (1, 3)}, {(2, 1)}})
|
||
: (∀ A ∈ 𝓐, isTransitive A) ∧ ¬ isTransitive (⋃₀ 𝓐) := by
|
||
apply And.intro
|
||
· intro A hA
|
||
rw [h𝓐] at hA
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at hA
|
||
apply Or.elim hA
|
||
· intro hA₁
|
||
rw [hA₁]
|
||
intro x y z hx hy
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, Prod.mk.injEq] at *
|
||
casesm* _ ∨ _
|
||
all_goals case _ hl hr => first
|
||
| {rw [hl.right] at hr; simp at hr}
|
||
| {rw [hl.left] at hr; simp at hr}
|
||
| {right; right; exact ⟨hl.left, hr.right⟩}
|
||
· intro hA₁
|
||
rw [hA₁]
|
||
intro x y z hx hy
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, Prod.mk.injEq] at *
|
||
rw [hx.right] at hy
|
||
simp at hy
|
||
· intro h
|
||
have h₁ : (1, 2) ∈ ⋃₀ 𝓐 := by
|
||
simp only [Set.mem_sUnion]
|
||
exact ⟨{(1, 2), (2, 3), (1, 3)}, by rw [h𝓐]; simp, by simp⟩
|
||
have h₂ : (2, 1) ∈ ⋃₀ 𝓐 := by
|
||
simp only [Set.mem_sUnion]
|
||
exact ⟨{(2, 1)}, by rw [h𝓐]; simp, by simp⟩
|
||
have h₃ : (1, 1) ∉ ⋃₀ 𝓐 := by
|
||
simp only [Set.mem_sUnion]
|
||
rw [h𝓐]
|
||
intro ⟨t, ht⟩
|
||
simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at ht
|
||
have := ht.right
|
||
apply Or.elim ht.left <;>
|
||
· intro ht₁
|
||
rw [ht₁] at this
|
||
simp at this
|
||
exact absurd (h h₁ h₂) h₃
|
||
|
||
/-- #### Exercise 3.35
|
||
|
||
Show that for any `R` and `x`, we have `[x]_R = R⟦{x}⟧`.
|
||
-/
|
||
theorem exercise_3_35 {R : Set.Relation α} {x : α}
|
||
: neighborhood R x = image R {x} := by
|
||
calc neighborhood R x
|
||
_ = { t | (x, t) ∈ R } := rfl
|
||
_ = { t | ∃ u ∈ ({x} : Set α), (u, t) ∈ R } := by simp
|
||
_ = image R {x} := rfl
|
||
|
||
/-- #### Exercise 3.36
|
||
|
||
Assume that `f : A → B` and that `R` is an equivalence relation on `B`. Define
|
||
`Q` to be the set `{⟨x, y⟩ ∈ A × A | ⟨f(x), f(y)⟩ ∈ R}`. Show that `Q` is an
|
||
equivalence relation on `A`.
|
||
-/
|
||
theorem exercise_3_36 {f : Set.HRelation α β}
|
||
{Q : Set.Relation α} {R : Set.Relation β} {A : Set α} {B : Set β}
|
||
(hf : mapsInto f A B) (hR : isEquivalence R B)
|
||
(hQ : Q = { p | ∃ fx fy : β, (p.1, fx) ∈ f ∧ (p.2, fy) ∈ f ∧ (fx, fy) ∈ R })
|
||
: isEquivalence Q A := by
|
||
refine ⟨?_, ?_, ?_, ?_⟩
|
||
|
||
· -- `fld Q ⊆ A`
|
||
show ∀ x, x ∈ fld Q → x ∈ A
|
||
intro x hx
|
||
rw [hQ] at hx
|
||
unfold fld dom ran at hx
|
||
simp only [
|
||
exists_and_left,
|
||
Set.mem_union,
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right
|
||
] at hx
|
||
apply Or.elim hx
|
||
· intro ⟨_, _, hx₁⟩
|
||
rw [← hf.dom_eq]
|
||
exact mem_pair_imp_fst_mem_dom hx₁.left
|
||
· intro ⟨_, _, _, _, hx₂⟩
|
||
rw [← hf.dom_eq]
|
||
exact mem_pair_imp_fst_mem_dom hx₂.left
|
||
|
||
· -- `isReflexive Q A`
|
||
intro x hx
|
||
rw [← hf.dom_eq] at hx
|
||
have ⟨fx, hfx⟩ := dom_exists hx
|
||
have := hR.refl fx (hf.ran_ss $ mem_pair_imp_snd_mem_ran hfx)
|
||
rw [hQ]
|
||
simp only [exists_and_left, Set.mem_setOf_eq]
|
||
exact ⟨fx, hfx, fx, hfx, this⟩
|
||
|
||
· -- `isSymmetric Q`
|
||
intro x y h
|
||
rw [hQ] at h
|
||
simp only [exists_and_left, Set.mem_setOf_eq] at h
|
||
have ⟨fx, hfx, fy, hfy, h'⟩ := h
|
||
have := hR.symm h'
|
||
rw [hQ]
|
||
simp only [exists_and_left, Set.mem_setOf_eq]
|
||
exact ⟨fy, hfy, fx, hfx, this⟩
|
||
|
||
· -- `isTransitive Q`
|
||
intro x y z hx hy
|
||
rw [hQ] at hx hy
|
||
simp only [exists_and_left, Set.mem_setOf_eq] at hx hy
|
||
have ⟨fx, hfx, fy, hfy, h₁⟩ := hx
|
||
have ⟨fy₁, hfy₁, fz, hfz, h₂⟩ := hy
|
||
have hfy' : fy = fy₁ := single_valued_eq_unique hf.is_func hfy hfy₁
|
||
rw [hfy'] at h₁
|
||
rw [hQ]
|
||
simp only [exists_and_left, Set.mem_setOf_eq]
|
||
exact ⟨fx, hfx, fz, hfz, hR.trans h₁ h₂⟩
|
||
|
||
/-- #### Exercise 3.37
|
||
|
||
Assume that `P` is a partition of a set `A`. Define the relation `Rₚ` as
|
||
follows:
|
||
```
|
||
xRₚy ↔ (∃ B ∈ Π)(x ∈ B ∧ y ∈ B).
|
||
```
|
||
Show that `Rₚ` is an equivalence relation on `A`. (This is a formalized version
|
||
of the discussion at the beginning of this section.)
|
||
-/
|
||
theorem exercise_3_37 {P : Set (Set α)} {A : Set α}
|
||
(hP : isPartition P A) (Rₚ : Set.Relation α)
|
||
(hRₚ : ∀ x y, (x, y) ∈ Rₚ ↔ ∃ B ∈ P, x ∈ B ∧ y ∈ B)
|
||
: isEquivalence Rₚ A := by
|
||
have hRₚ_eq : Rₚ = { p | ∃ B ∈ P, p.1 ∈ B ∧ p.2 ∈ B } := by
|
||
ext p
|
||
have (x, y) := p
|
||
exact hRₚ x y
|
||
|
||
refine ⟨?_, ?_, ?_, ?_⟩
|
||
· -- `fld Rₚ ⊆ A`
|
||
show ∀ x, x ∈ fld Rₚ → x ∈ A
|
||
rw [hRₚ_eq]
|
||
intro x hx
|
||
unfold fld dom ran at hx
|
||
simp only [
|
||
Set.mem_union,
|
||
Set.mem_image,
|
||
Set.mem_setOf_eq,
|
||
Prod.exists,
|
||
exists_and_right,
|
||
exists_eq_right
|
||
] at hx
|
||
apply Or.elim hx
|
||
· intro ⟨t, B, hB⟩
|
||
have := hP.p_subset B hB.left
|
||
exact this hB.right.left
|
||
· intro ⟨a, B, hB⟩
|
||
have := hP.p_subset B hB.left
|
||
exact this hB.right.right
|
||
|
||
· -- `isReflexive Rₚ A`
|
||
intro x hx
|
||
rw [hRₚ_eq]
|
||
simp only [Set.mem_setOf_eq, and_self]
|
||
exact hP.exhaustive x hx
|
||
|
||
· -- `isSymmetric Rₚ`
|
||
intro x y h
|
||
rw [hRₚ_eq] at h
|
||
simp only [Set.mem_setOf_eq] at h
|
||
have ⟨B, hB⟩ := h
|
||
rw [hRₚ_eq]
|
||
simp only [Set.mem_setOf_eq]
|
||
conv at hB => right; rw [and_comm]
|
||
exact ⟨B, hB⟩
|
||
|
||
· -- `isTransitive Rₚ`
|
||
intro x y z hx hy
|
||
rw [hRₚ_eq] at hx hy
|
||
simp only [Set.mem_setOf_eq] at hx hy
|
||
have ⟨B₁, hB₁⟩ := hx
|
||
have ⟨B₂, hB₂⟩ := hy
|
||
have hB : B₁ = B₂ := by
|
||
have hy₁ : y ∈ B₁ := hB₁.right.right
|
||
have hy₂ : y ∈ B₂ := hB₂.right.left
|
||
have hy := hP.disjoint B₁ hB₁.left B₂ hB₂.left
|
||
rw [contraposition] at hy
|
||
simp at hy
|
||
suffices B₁ ∩ B₂ ≠ ∅ from hy this
|
||
intro h'
|
||
rw [Set.ext_iff] at h'
|
||
exact (h' y).mp ⟨hy₁, hy₂⟩
|
||
rw [hRₚ_eq]
|
||
simp only [Set.mem_setOf_eq]
|
||
exact ⟨B₁, hB₁.left, hB₁.right.left, by rw [hB]; exact hB₂.right.right⟩
|
||
|
||
/-- #### Exercise 3.38
|
||
|
||
Theorem 3P shows that `A / R` is a partition of `A` whenever `R` is an
|
||
equivalence relation on `A`. Show that if we start with the equivalence relation
|
||
`Rₚ` of the preceding exercise, then the partition `A / Rₚ` is just `P`.
|
||
-/
|
||
theorem exercise_3_38 {P : Set (Set α)} {A : Set α}
|
||
(hP : isPartition P A) (Rₚ : Set.Relation α)
|
||
(hRₚ : ∀ x y, (x, y) ∈ Rₚ ↔ ∃ B ∈ P, x ∈ B ∧ y ∈ B)
|
||
: modEquiv (exercise_3_37 hP Rₚ hRₚ) = P := by
|
||
have hRₚ_eq : Rₚ = { p | ∃ B ∈ P, p.1 ∈ B ∧ p.2 ∈ B } := by
|
||
ext p
|
||
have (x, y) := p
|
||
exact hRₚ x y
|
||
|
||
ext B
|
||
apply Iff.intro
|
||
· intro ⟨x, hx⟩
|
||
have ⟨B', hB'⟩ := partition_mem_mem_eq hP hx.left
|
||
simp only at hB'
|
||
suffices B = B' by
|
||
rw [this]
|
||
exact hB'.left.left
|
||
|
||
ext y
|
||
apply Iff.intro
|
||
· intro hy
|
||
rw [← hx.right, hRₚ_eq] at hy
|
||
have ⟨B₁, hB₁⟩ := hy
|
||
have := hB'.right B₁ ⟨hB₁.left, hB₁.right.left⟩
|
||
rw [← this]
|
||
exact hB₁.right.right
|
||
· intro hy
|
||
rw [← hx.right, hRₚ_eq]
|
||
exact ⟨B', hB'.left.left, hB'.left.right, hy⟩
|
||
|
||
· intro hB
|
||
have ⟨x, hx⟩ := hP.nonempty B hB
|
||
have hx' : x ∈ A := hP.p_subset B hB hx
|
||
refine ⟨x, hx', Eq.symm ?_⟩
|
||
calc B
|
||
_ = {t | x ∈ B ∧ t ∈ B} := by
|
||
ext y
|
||
apply Iff.intro
|
||
· intro hy
|
||
exact ⟨hx, hy⟩
|
||
· intro hy
|
||
exact hy.right
|
||
_ = {t | ∃ B₁ ∈ P, x ∈ B₁ ∧ t ∈ B₁} := by
|
||
ext y
|
||
apply Iff.intro
|
||
· intro hy
|
||
exact ⟨B, hB, hy⟩
|
||
· intro hy
|
||
have ⟨B₁, hB₁⟩ := hy
|
||
have ⟨B', hB'⟩ := partition_mem_mem_eq hP hx'
|
||
simp only [Set.mem_setOf_eq] at *
|
||
have : B = B₁ := by
|
||
have lhs := hB'.right B ⟨hB, hx⟩
|
||
have rhs := hB'.right B₁ ⟨hB₁.left, hB₁.right.left⟩
|
||
rw [lhs, rhs]
|
||
rw [this]
|
||
exact hB₁.right
|
||
_ = {t | (x, t) ∈ Rₚ } := by
|
||
rw [hRₚ_eq]
|
||
simp only [Set.mem_setOf_eq]
|
||
_ = neighborhood Rₚ x := rfl
|
||
|
||
/-- #### Exercise 3.39
|
||
|
||
Assume that we start with an equivalence relation `R` on `A` and define `P` to
|
||
be the partition `A / R`. Show that `Rₚ`, as defined in Exercise 37, is just
|
||
`R`.
|
||
-/
|
||
theorem exercise_3_39 {P : Set (Set α)} {R Rₚ : Set.Relation α} {A : Set α}
|
||
(hR : isEquivalence R A) (hP : P = modEquiv hR)
|
||
(hRₚ : ∀ x y, (x, y) ∈ Rₚ ↔ ∃ B ∈ P, x ∈ B ∧ y ∈ B)
|
||
: Rₚ = R := by
|
||
have hRₚ_eq : Rₚ = { p | ∃ B ∈ P, p.1 ∈ B ∧ p.2 ∈ B } := by
|
||
ext p
|
||
have (x, y) := p
|
||
exact hRₚ x y
|
||
|
||
ext p
|
||
have (x, y) := p
|
||
apply Iff.intro
|
||
· intro hp
|
||
rw [hRₚ_eq] at hp
|
||
have ⟨B, hB, hx, hy⟩ := hp
|
||
rw [hP] at hB
|
||
have ⟨z, hz⟩ := hB
|
||
rw [← hz.right] at hx hy
|
||
exact neighborhood_mem_imp_relate hR hx hy
|
||
· intro hp
|
||
have hxA : x ∈ A := hR.b_on (Or.inl (mem_pair_imp_fst_mem_dom hp))
|
||
have hyA : y ∈ A := hR.b_on (Or.inr (mem_pair_imp_snd_mem_ran hp))
|
||
rw [hRₚ_eq]
|
||
have hx : x ∈ neighborhood R x := neighborhood_self_mem hR hxA
|
||
have hy : y ∈ neighborhood R x := by
|
||
rw [← neighborhood_eq_iff_mem_relate hR hxA hyA] at hp
|
||
rw [hp]
|
||
exact neighborhood_self_mem hR hyA
|
||
refine ⟨neighborhood R x, ?_, ⟨hx, hy⟩⟩
|
||
rw [hP]
|
||
exact ⟨x, hxA, rfl⟩
|
||
|
||
/-- #### Exercise 3.41 (a)
|
||
|
||
Let `ℝ` be the set of real numbers and define the realtion `Q` on `ℝ × ℝ` by
|
||
`⟨u, v⟩ Q ⟨x, y⟩` **iff** `u + y = x + v`. Show that `Q` is an equivalence
|
||
relation on `ℝ × ℝ`.
|
||
-/
|
||
theorem exercise_3_41_a {Q : Set.Relation (ℝ × ℝ)}
|
||
(hQ : ∀ u v x y, ((u, v), (x, y)) ∈ Q ↔ u + y = x + v)
|
||
: isEquivalence Q (Set.univ : Set (ℝ × ℝ)) := by
|
||
have hQ_eq : Q = {p | p.1.1 + p.2.2 = p.2.1 + p.1.2} := by
|
||
ext p
|
||
apply Iff.intro <;>
|
||
· intro hp
|
||
rwa [hQ] at *
|
||
|
||
refine ⟨?_, ?_, ?_, ?_⟩
|
||
|
||
· -- `fld Q ⊆ Set.univ`
|
||
show ∀ p, p ∈ fld Q → p ∈ Set.univ
|
||
intro _ _
|
||
simp only [Set.mem_univ]
|
||
|
||
· -- `isReflexive Q Set.univ`
|
||
intro (x, y) _
|
||
rw [hQ_eq]
|
||
simp
|
||
|
||
· -- `isSymmetric Q`
|
||
intro (u, v) (x, y) hp
|
||
rw [hQ_eq] at *
|
||
exact Eq.symm hp
|
||
|
||
· -- `isTransitive Q`
|
||
unfold isTransitive
|
||
intro (u, v) (x, y) (a, b) h₁ h₂
|
||
rw [hQ_eq] at *
|
||
simp at h₁ h₂
|
||
simp
|
||
have h₁' : u - v = x - y := by
|
||
have := sub_eq_of_eq_add h₁
|
||
rw [add_sub_right_comm] at this
|
||
exact eq_sub_of_add_eq this
|
||
have h₂' : x - y = a - b := by
|
||
have := sub_eq_of_eq_add h₂
|
||
rw [add_sub_right_comm] at this
|
||
exact eq_sub_of_add_eq this
|
||
rw [h₂'] at h₁'
|
||
have := eq_add_of_sub_eq' h₁'
|
||
rw [← add_sub_assoc] at this
|
||
have := add_eq_of_eq_sub this
|
||
conv => right; rw [add_comm]
|
||
exact this
|
||
|
||
end Relation
|
||
|
||
/-- #### Theorem 3R
|
||
|
||
Let `R` be a linear ordering on `A`.
|
||
|
||
(i) There is no `x` for which `xRx`.
|
||
(ii) For distinct `x` and `y` in `A`, either `xRy` or `yRx`.
|
||
-/
|
||
theorem theorem_3r {R : Rel α α} (hR : IsStrictTotalOrder α R)
|
||
: (∀ x : α, ¬ R x x) ∧ (∀ x y : α, x ≠ y → R x y ∨ R y x) := by
|
||
apply And.intro
|
||
· exact hR.irrefl
|
||
· intro x y h
|
||
apply Or.elim (hR.trichotomous x y)
|
||
· intro h₁
|
||
left
|
||
exact h₁
|
||
· intro h₁
|
||
apply Or.elim h₁
|
||
· intro h₂
|
||
exact absurd h₂ h
|
||
· intro h₂
|
||
right
|
||
exact h₂
|
||
|
||
/-- #### Exercise 3.43
|
||
|
||
Assume that `R` is a linear ordering on a set `A`. Show that `R⁻¹` is also a
|
||
linear ordering on `A`.
|
||
-/
|
||
theorem exercise_3_43 {R : Rel α α} (hR : IsStrictTotalOrder α R)
|
||
: IsStrictTotalOrder α R.inv := by
|
||
refine { trichotomous := ?_, irrefl := ?_, trans := ?_ }
|
||
· intro a b
|
||
unfold Rel.inv flip
|
||
apply Or.elim (hR.trichotomous a b)
|
||
· intro h; right; right; exact h
|
||
· intro h
|
||
apply Or.elim h
|
||
· intro h; right; left; exact h
|
||
· intro h; left; exact h
|
||
· intro x h
|
||
unfold Rel.inv flip at h
|
||
exact absurd h (hR.irrefl x)
|
||
· intro a b c hab hac
|
||
unfold Rel.inv flip at *
|
||
exact hR.trans c b a hac hab
|
||
|
||
/-! #### Exercise 3.44
|
||
|
||
Assume that `<` is a linear ordering on a set `A`. Assume that `f : A → A` and
|
||
that `f` has the property that whenever `x < y`, then `f(x) < f(y)`. Show that
|
||
`f` is one-to-one and that whenever `f(x) < f(y)`, then `x < y`.
|
||
-/
|
||
|
||
theorem exercise_3_44_i {R : Rel α α} (hR : IsStrictTotalOrder α R)
|
||
(f : α → α) (hf : ∀ x y, R x y → R (f x) (f y))
|
||
: Function.Injective f := by
|
||
unfold Function.Injective
|
||
intro x₁ x₂ hx
|
||
apply Or.elim (hR.trichotomous x₁ x₂)
|
||
· -- `x₁ < x₂`
|
||
intro hx₁
|
||
have nh := hf x₁ x₂ hx₁
|
||
rw [hx] at nh
|
||
exact absurd nh (hR.irrefl (f x₂))
|
||
· intro hx₁
|
||
apply Or.elim hx₁
|
||
· simp -- `x₁ = x₂`
|
||
· -- `x₁ > x₂`
|
||
intro hx₂
|
||
have nh := hf x₂ x₁ hx₂
|
||
rw [← hx] at nh
|
||
exact absurd nh (hR.irrefl (f x₁))
|
||
|
||
theorem exercise_3_44_ii {R : Rel α α} (hR : IsStrictTotalOrder α R)
|
||
(f : α → α) (hf : ∀ x y, R x y → R (f x) (f y))
|
||
: R (f x) (f y) → R x y := by
|
||
intro h
|
||
apply Or.elim (hR.trichotomous x y)
|
||
· simp -- `x < y`
|
||
· intro h₁
|
||
apply Or.elim h₁
|
||
· -- `x = y`
|
||
intro h₂
|
||
rw [h₂] at h
|
||
exact absurd h (hR.irrefl (f y))
|
||
· -- `x > y`
|
||
intro h₂
|
||
have := hR.trans (f x) (f y) (f x) h (hf y x h₂)
|
||
exact absurd this (hR.irrefl (f x))
|
||
|
||
/-- #### Exercise 3.45
|
||
|
||
Assume that `<_A` and `<_B` are linear orderings on `A` and `B`, respectively.
|
||
Define the binary relation `<_L` on the Cartesian product `A × B` by:
|
||
```
|
||
⟨a₁, b₁⟩ <_L ⟨a₂, b₂⟩ iff either a₁ <_A a₂ or (a₁ = a₂ ∧ b₁ <_B b₂).
|
||
```
|
||
Show that `<_L` is a linear ordering on `A × B`. (The relation `<_L` is called
|
||
*lexicographic* ordering, being the ordering used in making dictionaries.)
|
||
-/
|
||
theorem exercise_3_45 {A : Rel α α} {B : Rel β β} {R : Rel (α × β) (α × β)}
|
||
(hA : IsStrictTotalOrder α A) (hB : IsStrictTotalOrder β B)
|
||
(hR : ∀ a₁ b₁ a₂ b₂, R (a₁, b₁) (a₂, b₂) ↔ A a₁ a₂ ∨ (a₁ = a₂ ∧ B b₁ b₂))
|
||
: IsStrictTotalOrder (α × β) R := by
|
||
refine { trichotomous := ?_, irrefl := ?_, trans := ?_ }
|
||
· intro (a₁, b₁) (a₂, b₂)
|
||
apply Or.elim (hA.trichotomous a₁ a₂)
|
||
· -- `a₁ <_A a₂`
|
||
intro ha
|
||
left
|
||
exact (hR a₁ b₁ a₂ b₂).mpr (Or.inl ha)
|
||
· intro ha
|
||
apply Or.elim ha
|
||
· -- `a₁ = a₂`
|
||
intro ha₁
|
||
apply Or.elim (hB.trichotomous b₁ b₂)
|
||
· -- `b₁ <_B b₂`
|
||
intro hb
|
||
left
|
||
exact (hR a₁ b₁ a₂ b₂).mpr (Or.inr ⟨ha₁, hb⟩)
|
||
· intro hb
|
||
apply Or.elim hb
|
||
· -- `b₁ = b₂`
|
||
intro hb₁
|
||
right; left
|
||
rw [ha₁, hb₁]
|
||
· -- `b₂ <_B b₁`
|
||
intro hb₁
|
||
right; right
|
||
exact (hR a₂ b₂ a₁ b₁).mpr (Or.inr ⟨ha₁.symm, hb₁⟩)
|
||
· -- `a₂ <_A a₁`
|
||
intro ha₁
|
||
right; right
|
||
exact (hR a₂ b₂ a₁ b₁).mpr (Or.inl ha₁)
|
||
· intro (a, b) h
|
||
have := (hR a b a b).mp h
|
||
apply Or.elim this
|
||
· intro ha₁
|
||
exact absurd ha₁ (hA.irrefl a)
|
||
· intro ⟨_, hb₁⟩
|
||
exact absurd hb₁ (hB.irrefl b)
|
||
· intro (a₁, b₁) (a₂, b₂) (a₃, b₃) h₁ h₂
|
||
have h₁' := (hR a₁ b₁ a₂ b₂).mp h₁
|
||
have h₂' := (hR a₂ b₂ a₃ b₃).mp h₂
|
||
apply Or.elim h₁'
|
||
· -- `a₁ <_A a₂`
|
||
intro ha₁
|
||
apply Or.elim h₂'
|
||
· -- `a₂ <_A a₃`
|
||
intro ha₂
|
||
have := hA.trans a₁ a₂ a₃ ha₁ ha₂
|
||
exact (hR a₁ b₁ a₃ b₃).mpr (Or.inl this)
|
||
· -- `a₂ = a₃ ∧ b₂ <_B b₃`
|
||
intro ha₂
|
||
rw [ha₂.left] at ha₁
|
||
exact (hR a₁ b₁ a₃ b₃).mpr (Or.inl ha₁)
|
||
· -- `a₁ = a₂ ∧ b₁ <_B b₂`
|
||
intro ha₁
|
||
apply Or.elim h₂'
|
||
· -- `a₂ <_A a₃`
|
||
intro ha₂
|
||
rw [← ha₁.left] at ha₂
|
||
exact (hR a₁ b₁ a₃ b₃).mpr (Or.inl ha₂)
|
||
· -- `a₂ = a₃ ∧ b₂ <_B b₃`
|
||
intro ⟨ha₂, hb₂⟩
|
||
rw [← ha₁.left] at ha₂
|
||
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 |