bookshelf/Bookshelf/Enderton/Set/Chapter_6.lean

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import Bookshelf.Enderton.Set.Chapter_4
import Common.Logic.Basic
import Common.Nat.Basic
import Common.Set.Basic
import Common.Set.Equinumerous
import Common.Set.Intervals
import Mathlib.Data.Finset.Card
import Mathlib.Data.Set.Finite
import Mathlib.Tactic.LibrarySearch
/-! # Enderton.Set.Chapter_6
Cardinal Numbers and the Axiom of Choice
NOTE: We choose to use injectivity/surjectivity concepts found in
`Mathlib.Data.Set.Function` over those in `Mathlib.Init.Function` since the
former provides noncomputable utilities around obtaining inverse functions
(namely `Function.invFunOn`).
-/
namespace Enderton.Set.Chapter_6
/-- #### Theorem 6B
No set is equinumerous to its powerset.
-/
theorem theorem_6b (A : Set α)
: A ≉ 𝒫 A := by
rw [Set.not_equinumerous_def]
intro f hf
unfold Set.BijOn at hf
let φ := { a ∈ A | a ∉ f a }
suffices ∀ a ∈ A, f a ≠ φ by
have hφ := hf.right.right (show φ ∈ 𝒫 A by simp)
have ⟨a, ha⟩ := hφ
exact absurd ha.right (this a ha.left)
intro a ha hfa
by_cases h : a ∈ f a
· have h' := h
rw [hfa] at h
simp only [Set.mem_setOf_eq] at h
exact absurd h' h.right
· rw [Set.Subset.antisymm_iff] at hfa
have := hfa.right ⟨ha, h⟩
exact absurd this h
/-! ### Pigeonhole Principle -/
/--
A subset of a finite set of natural numbers has a max member.
-/
lemma subset_finite_max_nat {S' S : Set }
(hS : Set.Finite S) (hS' : Set.Nonempty S') (h : S' ⊆ S)
: ∃ m, m ∈ S' ∧ ∀ n, n ∈ S' → n ≤ m := by
have ⟨m, hm₁, hm₂⟩ :=
Set.Finite.exists_maximal_wrt id S' (Set.Finite.subset hS h) hS'
simp only [id_eq] at hm₂
refine ⟨m, hm₁, ?_⟩
intro n hn
match @trichotomous LT.lt _ m n with
| Or.inr (Or.inl r) => exact Nat.le_of_eq r.symm
| Or.inl r =>
have := hm₂ n hn (Nat.le_of_lt r)
exact Nat.le_of_eq this.symm
| Or.inr (Or.inr r) => exact Nat.le_of_lt r
/--
Auxiliary function to be proven by induction.
-/
lemma pigeonhole_principle_aux (n : )
: ∀ M, M ⊂ Set.Iio n →
∀ f : ,
Set.MapsTo f M (Set.Iio n) ∧ Set.InjOn f M →
¬ Set.SurjOn f M (Set.Iio n) := by
induction n with
| zero =>
intro _ hM
unfold Set.Iio at hM
simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hM
rw [Set.ssubset_empty_iff_false] at hM
exact False.elim hM
| succ n ih =>
intro M hM f ⟨hf_maps, hf_inj⟩ hf_surj
by_cases hM' : M = ∅
· unfold Set.SurjOn at hf_surj
rw [hM'] at hf_surj
simp only [Set.image_empty] at hf_surj
rw [Set.subset_def] at hf_surj
exact hf_surj n (show n < n + 1 by simp)
by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n
-- Trivial case. `f` must not be onto if this is the case.
· have ⟨t, ht⟩ := hf_surj (show n ∈ _ by simp)
exact absurd ⟨t, ht⟩ h
-- Continue under the assumption `n ∈ ran f`.
simp only [not_not] at h
have ⟨t, ht₁, ht₂⟩ := h
-- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`.
have ⟨p, hp₁, hp₂⟩ : ∃ p ∈ M, ∀ x, x ∈ M → p ≥ x := by
refine subset_finite_max_nat (show Set.Finite M from ?_) ?_ ?_
· have := Set.finite_lt_nat (n + 1)
exact Set.Finite.subset this (subset_of_ssubset hM)
· exact Set.nmem_singleton_empty.mp hM'
· show ∀ t, t ∈ M → t ∈ M
simp only [imp_self, forall_const]
-- `g` is a variant of `f` in which the largest element of its domain
-- (i.e. `p`) corresponds to value `n`.
let g x := if x = p then n else if x = t then f p else f x
have hg_maps : Set.MapsTo g M (Set.Iio (n + 1)) := by
intro x hx
dsimp only
by_cases hx₁ : x = p
· rw [hx₁]
simp
· rw [if_neg hx₁]
by_cases hx₂ : x = t
· rw [hx₂]
simp only [ite_true, Set.mem_Iio]
exact hf_maps hp₁
· rw [if_neg hx₂]
simp only [Set.mem_Iio]
exact hf_maps hx
have hg_inj : Set.InjOn g M := by
intro x₁ hx₁ x₂ hx₂ hf'
by_cases hc₁ : x₁ = p
· by_cases hc₂ : x₂ = p
· rw [hc₁, hc₂]
· dsimp at hf'
rw [hc₁] at hf'
simp only [ite_self, ite_true] at hf'
by_cases hc₃ : x₂ = t
· rw [if_neg hc₂, if_pos hc₃, ← ht₂] at hf'
rw [hc₁] at hx₁ ⊢
rw [hc₃] at hx₂ ⊢
exact hf_inj hx₁ hx₂ hf'.symm
· rw [if_neg hc₂, if_neg hc₃, ← ht₂] at hf'
have := hf_inj ht₁ hx₂ hf'
exact absurd this.symm hc₃
· by_cases hc₂ : x₂ = p
· rw [hc₂] at hf'
simp only [ite_self, ite_true] at hf'
by_cases hc₃ : x₁ = t
· rw [if_neg hc₁, if_pos hc₃, ← ht₂] at hf'
rw [hc₃] at hx₁ ⊢
rw [hc₂] at hx₂ ⊢
have := hf_inj hx₂ hx₁ hf'
exact this.symm
· rw [if_neg hc₁, if_neg hc₃, ← ht₂] at hf'
have := hf_inj hx₁ ht₁ hf'
exact absurd this hc₃
· dsimp only at hf'
rw [if_neg hc₁, if_neg hc₂] at hf'
by_cases hc₃ : x₁ = t
· by_cases hc₄ : x₂ = t
· rw [hc₃, hc₄]
· rw [if_pos hc₃, if_neg hc₄] at hf'
have := hf_inj hp₁ hx₂ hf'
exact absurd this.symm hc₂
· by_cases hc₄ : x₂ = t
· rw [if_neg hc₃, if_pos hc₄] at hf'
have := hf_inj hx₁ hp₁ hf'
exact absurd this hc₁
· rw [if_neg hc₃, if_neg hc₄] at hf'
exact hf_inj hx₁ hx₂ hf'
let M' := M \ {p}
have hM' : M' ⊂ Set.Iio n := by
by_cases hc : p = n
· suffices Set.Iio (n + 1) \ {n} = Set.Iio n by
have h₁ := Set.diff_ssubset_diff_left hM hp₁
conv at h₁ => right; rw [hc]
rwa [← this]
ext x
apply Iff.intro
· intro hx₁
refine Or.elim (Nat.lt_or_eq_of_lt hx₁.left) (by simp) ?_
intro hx₂
rw [hx₂] at hx₁
simp at hx₁
· intro hx₁
exact ⟨Nat.lt_trans hx₁ (by simp), Nat.ne_of_lt hx₁⟩
have hp_lt_n : p < n := by
have := subset_of_ssubset hM
have hp' : p < n + 1 := this hp₁
exact Or.elim (Nat.lt_or_eq_of_lt hp') id (absurd · hc)
rw [Set.ssubset_def]
apply And.intro
· show ∀ x, x ∈ M' → x < n
intro x hx
simp only [Set.mem_diff, Set.mem_singleton_iff] at hx
calc x
_ ≤ p := hp₂ x hx.left
_ < n := hp_lt_n
· show ¬ ∀ x, x < n → x ∈ M'
by_contra np
have := np p hp_lt_n
simp at this
-- Consider `g = f' - {⟨p, n⟩}`. This restriction will allow us to use
-- the induction hypothesis to prove `g` isn't surjective.
have ng_surj : ¬ Set.SurjOn g M' (Set.Iio n) := by
refine ih _ hM' g ⟨?_, ?_⟩
· -- `Set.MapsTo g M' (Set.Iio n)`
intro x hx
have hx₁ : x ∈ M := Set.mem_of_mem_diff hx
apply Or.elim (Nat.lt_or_eq_of_lt $ hg_maps hx₁)
· exact id
· intro hx₂
rw [← show g p = n by simp] at hx₂
exact absurd (hg_inj hx₁ hp₁ hx₂) hx.right
· -- `Set.InjOn g M'`
intro x₁ hx₁ x₂ hx₂ hg
have hx₁' : x₁ ∈ M := (Set.diff_subset M {p}) hx₁
have hx₂' : x₂ ∈ M := (Set.diff_subset M {p}) hx₂
exact hg_inj hx₁' hx₂' hg
-- We have shown `g` isn't surjective. This is another way of saying that.
have ⟨a, ha₁, ha₂⟩ : ∃ a, a < n ∧ a ∉ g '' M' := by
unfold Set.SurjOn at ng_surj
rw [Set.subset_def] at ng_surj
simp only [
Set.mem_Iio,
Set.mem_image,
not_forall,
not_exists,
not_and,
exists_prop
] at ng_surj
unfold Set.image
simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and]
exact ng_surj
-- If `g` isn't surjective then neither is `f`.
refine absurd (hf_surj $ calc a
_ < n := ha₁
_ < n + 1 := by simp) (show ↑a ∉ f '' M from ?_)
suffices g '' M = f '' M by
rw [← this]
show a ∉ g '' M
unfold Set.image at ha₂ ⊢
simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and] at ha₂ ⊢
intro x hx
by_cases hxp : x = p
· rw [if_pos hxp]
exact (Nat.ne_of_lt ha₁).symm
· refine ha₂ x ?_
exact Set.mem_diff_of_mem hx hxp
ext x
simp only [Set.mem_image, Set.mem_Iio]
apply Iff.intro
· intro ⟨y, hy₁, hy₂⟩
by_cases hc₁ : y = p
· rw [if_pos hc₁] at hy₂
rw [hy₂] at ht₂
exact ⟨t, ht₁, ht₂⟩
· rw [if_neg hc₁] at hy₂
by_cases hc₂ : y = t
· rw [if_pos hc₂] at hy₂
exact ⟨p, hp₁, hy₂⟩
· rw [if_neg hc₂] at hy₂
exact ⟨y, hy₁, hy₂⟩
· intro ⟨y, hy₁, hy₂⟩
by_cases hc₁ : y = p
· refine ⟨t, ht₁, ?_⟩
by_cases hc₂ : y = t
· rw [hc₂, ht₂] at hy₂
rw [← hc₁, ← hc₂]
simp only [ite_self, ite_true]
exact hy₂
· rw [hc₁, ← Ne.def] at hc₂
rwa [if_neg hc₂.symm, if_pos rfl, ← hc₁]
· by_cases hc₂ : y = t
· refine ⟨p, hp₁, ?_⟩
simp only [ite_self, ite_true]
rwa [hc₂, ht₂] at hy₂
· refine ⟨y, hy₁, ?_⟩
rwa [if_neg hc₁, if_neg hc₂]
/--
No natural number is equinumerous to a proper subset of itself.
-/
theorem pigeonhole_principle {n : }
: ∀ {M}, M ⊂ Set.Iio n → M ≉ Set.Iio n := by
intro M hM nM
have ⟨f, hf⟩ := nM
have := pigeonhole_principle_aux n M hM f ⟨hf.left, hf.right.left⟩
exact absurd hf.right.right this
/-- #### Corollary 6C
No finite set is equinumerous to a proper subset of itself.
-/
theorem corollary_6c [DecidableEq α] [Nonempty α]
{S S' : Set α} (hS : Set.Finite S) (h : S' ⊂ S)
: S ≉ S' := by
let T := S \ S'
have hT : S = S' (S \ S') := by
simp only [Set.union_diff_self]
exact (Set.left_subset_union_eq_self (subset_of_ssubset h)).symm
-- `hF : S' T ≈ S`.
-- `hG : S ≈ n`.
-- `hH : S' T ≈ n`.
have hF := Set.equinumerous_refl S
conv at hF => arg 1; rw [hT]
have ⟨n, hG⟩ := Set.finite_iff_equinumerous_nat.mp hS
have ⟨H, hH⟩ := Set.equinumerous_trans hF hG
-- Restrict `H` to `S'` to yield a bijection between `S'` and a proper subset
-- of `n`.
let R := (Set.Iio n) \ (H '' T)
have hR : Set.BijOn H S' R := by
refine ⟨?_, ?_, ?_⟩
· -- `Set.MapsTo H S' R`
intro x hx
refine ⟨hH.left $ Set.mem_union_left T hx, ?_⟩
unfold Set.image
by_contra nx
simp only [Finset.mem_coe, Set.mem_setOf_eq] at nx
have ⟨a, ha₁, ha₂⟩ := nx
have hc₁ : a ∈ S' T := Set.mem_union_right S' ha₁
have hc₂ : x ∈ S' T := Set.mem_union_left T hx
rw [hH.right.left hc₁ hc₂ ha₂] at ha₁
have hx₁ : {x} ⊆ S' := Set.singleton_subset_iff.mpr hx
have hx₂ : {x} ⊆ T := Set.singleton_subset_iff.mpr ha₁
have hx₃ := Set.disjoint_sdiff_right hx₁ hx₂
simp only [
Set.bot_eq_empty,
Set.le_eq_subset,
Set.singleton_subset_iff,
Set.mem_empty_iff_false
] at hx₃
· -- `Set.InjOn H S'`
intro x₁ hx₁ x₂ hx₂ h
have hc₁ : x₁ ∈ S' T := Set.mem_union_left T hx₁
have hc₂ : x₂ ∈ S' T := Set.mem_union_left T hx₂
exact hH.right.left hc₁ hc₂ h
· -- `Set.SurjOn H S' R`
show ∀ r, r ∈ R → r ∈ H '' S'
intro r hr
unfold Set.image
simp only [Set.mem_setOf_eq]
dsimp only at hr
have := hH.right.right hr.left
simp only [Set.mem_image, Set.mem_union] at this
have ⟨x, hx⟩ := this
apply Or.elim hx.left
· intro hx'
exact ⟨x, hx', hx.right⟩
· intro hx'
refine absurd ?_ hr.right
rw [← hx.right]
simp only [Set.mem_image, Finset.mem_coe]
exact ⟨x, hx', rfl⟩
intro hf
have hf₁ : S ≈ R := Set.equinumerous_trans hf ⟨H, hR⟩
have hf₂ : R ≈ Set.Iio n := by
have ⟨k, hk⟩ := Set.equinumerous_symm hf₁
exact Set.equinumerous_trans ⟨k, hk⟩ hG
refine absurd hf₂ (pigeonhole_principle ?_)
show R ⊂ Set.Iio n
apply And.intro
· show ∀ r, r ∈ R → r ∈ Set.Iio n
intro _ hr
exact hr.left
· show ¬ ∀ r, r ∈ Set.Iio n → r ∈ R
intro nr
have ⟨t, ht₁⟩ : Set.Nonempty T := Set.diff_ssubset_nonempty h
have ht₂ : H t ∈ Set.Iio n := hH.left (Set.mem_union_right S' ht₁)
have ht₃ : H t ∈ R := nr (H t) ht₂
exact absurd ⟨t, ht₁, rfl⟩ ht₃.right
/-- #### Corollary 6D (a)
Any set equinumerous to a proper subset of itself is infinite.
-/
theorem corollary_6d_a [DecidableEq α] [Nonempty α]
{S S' : Set α} (hS : S' ⊂ S) (hf : S ≈ S')
: Set.Infinite S := by
by_contra nS
simp only [Set.not_infinite] at nS
exact absurd hf (corollary_6c nS hS)
/-- #### Corollary 6D (b)
The set `ω` is infinite.
-/
theorem corollary_6d_b
: Set.Infinite (@Set.univ ) := by
let S : Set := { 2 * n | n ∈ @Set.univ }
let f x := 2 * x
suffices Set.BijOn f (@Set.univ ) S by
refine corollary_6d_a ?_ ⟨f, this⟩
rw [Set.ssubset_def]
apply And.intro
· simp
· show ¬ ∀ x, x ∈ Set.univ → x ∈ S
simp only [
Set.mem_univ,
true_and,
Set.mem_setOf_eq,
forall_true_left,
not_forall,
not_exists
]
refine ⟨1, ?_⟩
intro x nx
simp only [mul_eq_one, false_and] at nx
refine ⟨by simp, ?_, ?_⟩
· -- `Set.InjOn f Set.univ`
intro n₁ _ n₂ _ hf
match @trichotomous LT.lt _ n₁ n₂ with
| Or.inr (Or.inl r) => exact r
| Or.inl r =>
have := (Chapter_4.theorem_4n_ii n₁ n₂ 1).mp r
conv at this => left; rw [mul_comm]
conv at this => right; rw [mul_comm]
exact absurd hf (Nat.ne_of_lt this)
| Or.inr (Or.inr r) =>
have := (Chapter_4.theorem_4n_ii n₂ n₁ 1).mp r
conv at this => left; rw [mul_comm]
conv at this => right; rw [mul_comm]
exact absurd hf.symm (Nat.ne_of_lt this)
· -- `Set.SurjOn f Set.univ S`
show ∀ x, x ∈ S → x ∈ f '' Set.univ
intro x hx
unfold Set.image
simp only [Set.mem_univ, true_and, Set.mem_setOf_eq] at hx ⊢
exact hx
/-- #### Corollary 6E
Any finite set is equinumerous to a unique natural number.
-/
theorem corollary_6e [Nonempty α] (S : Set α) (hS : Set.Finite S)
: ∃! n : , S ≈ Set.Iio n := by
have ⟨n, hf⟩ := Set.finite_iff_equinumerous_nat.mp hS
refine ⟨n, hf, ?_⟩
intro m hg
match @trichotomous LT.lt _ m n with
| Or.inr (Or.inl r) => exact r
| Or.inl r =>
have hh := Set.equinumerous_symm hg
have hk := Set.equinumerous_trans hh hf
have hmn : Set.Iio m ⊂ Set.Iio n := Set.Iio_nat_lt_ssubset r
exact absurd hk (pigeonhole_principle hmn)
| Or.inr (Or.inr r) =>
have hh := Set.equinumerous_symm hf
have hk := Set.equinumerous_trans hh hg
have hnm : Set.Iio n ⊂ Set.Iio m := Set.Iio_nat_lt_ssubset r
exact absurd hk (pigeonhole_principle hnm)
/-- #### Lemma 6F
If `C` is a proper subset of a natural number `n`, then `C ≈ m` for some `m`
less than `n`.
-/
lemma lemma_6f {n : }
: ∀ {C}, C ⊂ Set.Iio n → ∃ m, m < n ∧ C ≈ Set.Iio m := by
induction n with
| zero =>
intro C hC
unfold Set.Iio at hC
simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hC
rw [Set.ssubset_empty_iff_false] at hC
exact False.elim hC
| succ n ih =>
have h_subset_equinumerous
: ∀ S, S ⊆ Set.Iio n →
∃ m, m < n + 1 ∧ S ≈ Set.Iio m := by
intro S hS
rw [subset_iff_ssubset_or_eq] at hS
apply Or.elim hS
· -- `S ⊂ Set.Iio n`
intro h
have ⟨m, hm⟩ := ih h
exact ⟨m, calc m
_ < n := hm.left
_ < n + 1 := by simp, hm.right⟩
· -- `S = Set.Iio n`
intro h
exact ⟨n, by simp, Set.eq_imp_equinumerous h⟩
intro C hC
by_cases hn : n ∈ C
· -- Since `C` is a proper subset of `n⁺`, the set `n⁺ - C` is nonempty.
have hC₁ : Set.Nonempty (Set.Iio (n + 1) \ C) := by
rw [Set.ssubset_def] at hC
have : ¬ ∀ x, x ∈ Set.Iio (n + 1) → x ∈ C := hC.right
simp only [Set.mem_Iio, not_forall, exists_prop] at this
exact this
-- `p` is the least element of `n⁺ - C`.
have ⟨p, hp⟩ := Chapter_4.well_ordering_nat hC₁
let C' := (C \ {n}) {p}
have hC'₁ : C' ⊆ Set.Iio n := by
show ∀ x, x ∈ C' → x ∈ Set.Iio n
intro x hx
match @trichotomous LT.lt _ x n with
| Or.inl r => exact r
| Or.inr (Or.inl r) =>
rw [r] at hx
apply Or.elim hx
· intro nx
simp at nx
· intro nx
simp only [Set.mem_singleton_iff] at nx
rw [nx] at hn
exact absurd hn hp.left.right
| Or.inr (Or.inr r) =>
apply Or.elim hx
· intro ⟨h₁, h₂⟩
have h₃ := subset_of_ssubset hC h₁
simp only [Set.mem_singleton_iff, Set.mem_Iio] at h₂ h₃
exact Or.elim (Nat.lt_or_eq_of_lt h₃) id (absurd · h₂)
· intro h
simp only [Set.mem_singleton_iff] at h
have := hp.left.left
rw [← h] at this
exact Or.elim (Nat.lt_or_eq_of_lt this)
id (absurd · (Nat.ne_of_lt r).symm)
have ⟨m, hm₁, hm₂⟩ := h_subset_equinumerous C' hC'₁
suffices C' ≈ C from
⟨m, hm₁, Set.equinumerous_trans (Set.equinumerous_symm this) hm₂⟩
-- Proves `f` is a one-to-one correspondence between `C'` and `C`.
let f x := if x = p then n else x
refine ⟨f, ?_, ?_, ?_⟩
· -- `Set.MapsTo f C' C`
intro x hx
dsimp only
by_cases hxp : x = p
· rw [if_pos hxp]
exact hn
· rw [if_neg hxp]
apply Or.elim hx
· exact fun x => x.left
· intro hx₁
simp only [Set.mem_singleton_iff] at hx₁
exact absurd hx₁ hxp
· -- `Set.InjOn f C'`
intro x₁ hx₁ x₂ hx₂ hf
dsimp only at hf
by_cases hx₁p : x₁ = p
· by_cases hx₂p : x₂ = p
· rw [hx₁p, hx₂p]
· rw [if_pos hx₁p, if_neg hx₂p] at hf
apply Or.elim hx₂
· intro nx
exact absurd hf.symm nx.right
· intro nx
simp only [Set.mem_singleton_iff] at nx
exact absurd nx hx₂p
· by_cases hx₂p : x₂ = p
· rw [if_neg hx₁p, if_pos hx₂p] at hf
apply Or.elim hx₁
· intro nx
exact absurd hf nx.right
· intro nx
simp only [Set.mem_singleton_iff] at nx
exact absurd nx hx₁p
· rwa [if_neg hx₁p, if_neg hx₂p] at hf
· -- `Set.SurjOn f C' C`
show ∀ x, x ∈ C → x ∈ f '' C'
intro x hx
simp only [
Set.union_singleton,
Set.mem_diff,
Set.mem_singleton_iff,
Set.mem_image,
Set.mem_insert_iff,
exists_eq_or_imp,
ite_true
]
by_cases nx₁ : x = n
· left
exact nx₁.symm
· right
by_cases nx₂ : x = p
· have := hp.left.right
rw [← nx₂] at this
exact absurd hx this
· exact ⟨x, ⟨hx, nx₁⟩, by rwa [if_neg]⟩
· refine h_subset_equinumerous C ?_
show ∀ x, x ∈ C → x ∈ Set.Iio n
intro x hx
unfold Set.Iio
apply Or.elim (Nat.lt_or_eq_of_lt (subset_of_ssubset hC hx))
· exact id
· intro hx₁
rw [hx₁] at hx
exact absurd hx hn
/-- #### Corollary 6G
Any subset of a finite set is finite.
-/
theorem corollary_6g {S S' : Set α} (hS : Set.Finite S) (hS' : S' ⊆ S)
: Set.Finite S' := by
rw [subset_iff_ssubset_or_eq] at hS'
apply Or.elim hS'
· intro h
rw [Set.finite_iff_equinumerous_nat] at hS
have ⟨n, F, hF⟩ := hS
-- Mirrors logic found in `corollary_6c`.
let T := S \ S'
let R := (Set.Iio n) \ (F '' T)
have hR : R ⊂ Set.Iio n := by
rw [Set.ssubset_def]
apply And.intro
· show ∀ x, x ∈ R → x ∈ Set.Iio n
intro _ hx
exact hx.left
· show ¬ ∀ x, x ∈ Set.Iio n → x ∈ R
intro nr
have ⟨t, ht₁⟩ : Set.Nonempty T := Set.diff_ssubset_nonempty h
have ht₂ : F t ∈ Set.Iio n := hF.left ht₁.left
have ht₃ : F t ∈ R := nr (F t) ht₂
exact absurd ⟨t, ht₁, rfl⟩ ht₃.right
suffices Set.BijOn F S' R by
have ⟨m, hm⟩ := lemma_6f hR
have := Set.equinumerous_trans ⟨F, this⟩ hm.right
exact Set.finite_iff_equinumerous_nat.mpr ⟨m, this⟩
refine ⟨?_, ?_, ?_⟩
· -- `Set.MapsTo f S' R`
intro x hx
dsimp only
simp only [Set.mem_diff, Set.mem_Iio, Set.mem_image, not_exists, not_and]
apply And.intro
· exact hF.left (subset_of_ssubset h hx)
· intro y hy
by_contra nf
have := hF.right.left (subset_of_ssubset h hx) hy.left nf.symm
rw [this] at hx
exact absurd hx hy.right
· -- `Set.InjOn f S'`
intro x₁ hx₁ x₂ hx₂ hf
have h₁ : x₁ ∈ S := subset_of_ssubset h hx₁
have h₂ : x₂ ∈ S := subset_of_ssubset h hx₂
exact hF.right.left h₁ h₂ hf
· -- `Set.SurjOn f S' R`
show ∀ x, x ∈ R → x ∈ F '' S'
intro x hx
have h₁ := hF.right.right
unfold Set.SurjOn at h₁
rw [Set.subset_def] at h₁
have ⟨y, hy⟩ := h₁ x hx.left
refine ⟨y, ?_, hy.right⟩
rw [← hy.right] at hx
simp only [Set.mem_image, Set.mem_diff, not_exists, not_and] at hx
by_contra ny
exact (hx.right y ⟨hy.left, ny⟩) rfl
· intro h
rwa [h]
/-- #### Exercise 6.1
Show that the equation
```
f(m, n) = 2ᵐ(2n + 1) - 1
```
defines a one-to-one correspondence between `ω × ω` and `ω`.
-/
theorem exercise_6_1
: Function.Bijective (fun p : × => 2 ^ p.1 * (2 * p.2 + 1) - 1) := by
sorry
/-- #### Exercise 6.2
Show that in Fig. 32 we have:
```
J(m, n) = [1 + 2 + ⋯ + (m + n)] + m
= (1 / 2)[(m + n)² + 3m + n].
```
-/
theorem exercise_6_2
: Function.Bijective
(fun p : × => (1 / 2) * ((p.1 + p.2) ^ 2 + 3 * p.1 + p.2)) := by
sorry
/-- #### Exercise 6.3
Find a one-to-one correspondence between the open unit interval `(0, 1)` and ``
that takes rationals to rationals and irrationals to irrationals.
-/
theorem exercise_6_3
: True := by
sorry
/-- #### Exercise 6.4
Construct a one-to-one correspondence between the closed unit interval
```
[0, 1] = {x ∈ | 0 ≤ x ≤ 1}
```
and the open unit interval `(0, 1)`.
-/
theorem exercise_6_4
: ∃ F, Set.BijOn F (Set.Ioo 0 1) (@Set.univ ) := by
sorry
end Enderton.Set.Chapter_6