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Enderton (set). Fixup the pigeonhole principle.

finite-set-exercises
Joshua Potter 2023-09-14 13:50:22 -06:00
parent f1a10c6877
commit 7959c474a0
6 changed files with 453 additions and 312 deletions
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80
Bookshelf/Enderton/Set.tex
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@ -217,7 +217,10 @@
A set is \textbf{finite} if and only if it is \nameref{ref:equinumerous} to a
\nameref{ref:natural-number}.
\lean*{Mathlib/Data/Set/Finite}
\code*{Common/Set/Finite}
{Set.finite\_iff\_equinumerous\_nat}
\lean{Mathlib/Data/Set/Finite}
{Set.Finite}
\section{\defined{Function}}%
@ -8770,14 +8773,14 @@
\end{enumerate}
\end{theorem}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.theorem\_6a\_a}
\code{Common/Set/Finite}
{Set.equinumerous\_refl}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.theorem\_6a\_b}
\code{Common/Set/Finite}
{Set.equinumerous\_symm}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.theorem\_6a\_c}
\code{Common/Set/Finite}
{Set.equinumerous\_trans}
\begin{proof}
@ -8857,14 +8860,17 @@
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.pigeonhole\_principle}
\lean{Mathlib/Data/Finset/Card}
{Finset.exists\_ne\_map\_eq\_of\_card\_lt\_of\_maps\_to}
\begin{proof}
Let
\begin{equation}
\hyperlabel{sub:pigeonhole-principle-eq1}
S = \{n \in \omega \mid
\forall m \in n, \text{every one-to-one function }
f \colon m \rightarrow n \text{ is not onto}\}.
\forall M \subset n, \text{every one-to-one function }
f \colon M \rightarrow n \text{ is not onto}\}.
\end{equation}
We show that (i) $0 \in S$ and (ii) if $n \in S$, then so is $n^+$.
Afterward we prove (iii) the theorem statement.
@ -8872,20 +8878,20 @@
\paragraph{(i)}%
\hyperlabel{par:pigeonhole-principle-i}
By \nameref{sub:zero-least-natural-number}, $0$ is the least natural
number.
Therefore, $0 \in S$ vacuously.
By definition, $0 = \emptyset$.
Then $0$ has no proper subsets.
Hence $0 \in S$ vacuously.
\paragraph{(ii)}%
\hyperlabel{par:pigeonhole-principle-ii}
Suppose $n \in S$ and let $m \in n^+$.
Furthermore, let $f \colon m \rightarrow n^+$ be a one-to-one
\nameref{ref:function} (as proof of a one-to-one function's existence,
just consider the identity function).
If $m = 0$, it vacuously holds that $f$ is not onto $n^+$.
If $m \neq 0$, \nameref{sub:theorem-4c} shows there exists some $p$ such
that $p^+ = m$.
Suppose $n \in S$ and $M \subset n^+$.
Furthermore, let $f \colon M \rightarrow n^+$ be a one-to-one
\nameref{ref:function}.
If $M = \emptyset$, it vacuously holds that $f$ is not onto $n^+$.
Otherwise $M \neq \emptyset$.
Because $M$ is finite, the \nameref{sub:trichotomy-law-natural-numbers}
implies the existence of a largest member $p \in M$.
There are two cases to consider:
\subparagraph{Case 1}%
@ -8896,8 +8902,8 @@
\subparagraph{Case 2}%
Suppose $n \in \ran{f}$.
Then there exists some $t \in m$ such that $\tuple{t, n} \in f$.
Define $f' \colon m \rightarrow n^+$ given by
Then there exists some $t \in M$ such that $\tuple{t, n} \in f$.
Define $f' \colon M \rightarrow n^+$ given by
\begin{align*}
f'(p) & = f(t) = n \\
f'(t) & = f(p) \\
@ -8906,18 +8912,17 @@
That is, $f'$ is a variant of $f$ in which the largest element of its
domain (i.e. $p$) corresponds to value $n$.
Next define $g = f' - \{\tuple{p, n}\}$.
Then $g$ is a function from $p$ to $n$.
Then $g$ is a function mapping $M - \{p\}$ to $n$.
Since $f$ is one-to-one, $f'$ and $g$ are also one-to-one.
Since $p \in m \in n$, \nameref{sub:theorem-4f} implies $p \in n$.
Thus \eqref{sub:pigeonhole-principle-eq1} indicates $g$ must not be onto
Then \eqref{sub:pigeonhole-principle-eq1} indicates $g$ must not be onto
$n$.
In other words, there exists some $a \in n$ such that
$a \not\in \ran{g}$.
Hence $a \not\in \ran{f'}$ and, consequently, $a \not\in \ran{f}$.
But $a \in n \in n^+$ meaning, by another application of
\nameref{sub:theorem-4f}, $a \in n^+$.
Therefore $f$ is not onto $n^+$.
That is, there exists some $a \in n$ such that $a \not\in \ran{g}$.
By the \nameref{sub:trichotomy-law-natural-numbers}, $a \neq n$.
Therefore $a \not\in \ran{f'}$.
$\ran{f'} = \ran{f}$ meaning $a \not\in \ran{f}$.
Because $a \in n \in n^+$, \nameref{sub:theorem-4f} implies $a \in n^+$.
Hence $f$ is not onto $n^+$.
\subparagraph{Subconclusion}%
@ -8937,26 +8942,29 @@
\end{proof}
\subsection{\pending{Corollary 6C}}%
\subsection{\verified{Corollary 6C}}%
\hyperlabel{sub:corollary-6c}
\begin{corollary}[6C]
No finite set is equinumerous to a proper subset of itself.
\end{corollary}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.corollary\_6c}
\begin{proof}
Let $S$ be a \nameref{ref:finite-set} and $S'$ be a
\nameref{ref:proper-subset} $S'$ of $S$.
Then there exists some nonempty set $T$, disjoint from $S'$, such that
Then there exists some set $T$, disjoint from $S'$, such that
$S' \cup T = S$.
By definition of a finite set, $S$ is \nameref{ref:equinumerous} to a
natural number $n$.
By definition of a \nameref{ref:finite-set}, $S$ is
\nameref{ref:equinumerous} to a natural number $n$.
By \nameref{sub:theorem-6a}, $\equinumerous{S' \cup T}{S}$ which, by the
same theorem, implies $\equinumerous{S' \cup T}{n}$.
Let $f$ be a one-to-one correspondence between $S' \cup T$ and $n$.
Since $T$ is nonempty, $f \restriction S'$ is a one-to-one correspondence
between $S'$ and a proper subset of $n$.
Then $f \restriction S'$ is a one-to-one correspondence between $S'$ and a
proper subset of $n$.
By the \nameref{sub:pigeonhole-principle}, $n$ is not equinumerous to any
proper subset of itself.
Therefore \nameref{sub:theorem-6a} implies $S'$ cannot be equinumerous to

4
Bookshelf/Enderton/Set/Chapter_4.lean
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@ -389,7 +389,7 @@ theorem corollary_4p_i (m n p : ℕ) (h : m + p = n + p)
Let `A` be a nonempty subset of `ω`. Then there is some `m ∈ A` such that
`m ≤ n` for all `n ∈ A`.
-/
theorem well_ordering_nat (A : Set ℕ) (hA : Set.Nonempty A)
theorem well_ordering_nat {A : Set ℕ} (hA : Set.Nonempty A)
: ∃ m ∈ A, ∀ n, n ∈ A → m ≤ n := by
-- Assume `A` does not have a least element.
by_contra nh
@ -453,7 +453,7 @@ theorem strong_induction_principle_nat (A : Set ℕ)
exact h'.symm
by_contra nh
have ⟨m, hm⟩ := well_ordering_nat A.compl (Set.nmem_singleton_empty.mp nh)
have ⟨m, hm⟩ := well_ordering_nat (Set.nmem_singleton_empty.mp nh)
refine absurd (h m ?_) hm.left
-- Show that every number less than `m` is in `A`.

603
Bookshelf/Enderton/Set/Chapter_6.lean
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@ -1,47 +1,23 @@
import Common.Logic.Basic
import Common.Nat.Basic
import Mathlib.Data.Finset.Basic
import Common.Set.Basic
import Common.Set.Finite
import Mathlib.Data.Finset.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Function
import Mathlib.Data.Rel
import Mathlib.Tactic.Ring
import Std.Data.Fin.Lemmas
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 6A
For any sets `A`, `B`, and `C`,
(a) `A ≈ A`.
(b) If `A ≈ B`, then `B ≈ A`.
(c) If `A ≈ B` and `B ≈ C`, then `A ≈ C`.
-/
theorem theorem_6a_a (A : Set α)
: ∃ F, Set.BijOn F A A := by
refine ⟨fun x => x, ?_⟩
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
simp only [imp_self, implies_true, Set.image_id', true_and]
exact Eq.subset rfl
theorem theorem_6a_b [Nonempty α] (A : Set α) (B : Set β)
(F : α → β) (hF : Set.BijOn F A B)
: ∃ G, Set.BijOn G B A := by
refine ⟨Function.invFunOn F A, ?_⟩
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
theorem theorem_6a_c (A : Set α) (B : Set β) (C : Set γ)
(F : α → β) (hF : Set.BijOn F A B)
(G : β → γ) (hG : Set.BijOn G B C)
: ∃ H, Set.BijOn H A C := by
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
/-- #### Theorem 6B
No set is equinumerous to its powerset.
@ -65,273 +41,366 @@ theorem theorem_6b (A : Set α)
have := hfa.right ⟨ha, h⟩
exact absurd this h
/-! #### Pigeonhole Principle
/-! ### Pigeonhole Principle -/
No natural number is equinumerous to a proper subset of itself.
/--
A subset of a finite set of natural numbers has a max member.
-/
lemma subset_finite_max_nat {S' S : Set ℕ}
(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 ℕ 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 : ℕ)
: ∀ m : ℕ, m < n →
∀ f : Fin m → Fin n, Function.Injective f →
¬ Function.Surjective f := by
: ∀ M, M ⊂ Set.Iio n →
∀ f : ℕ → ℕ,
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
simp at hm
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_inj hf_surj
intro M hM f ⟨hf_maps, hf_inj⟩ hf_surj
by_cases hm' : m = 0
· have ⟨a, ha⟩ := hf_surj 0
rw [hm'] at a
have := a.isLt
simp only [not_lt_zero'] at this
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)
-- `m ≠ 0` so `∃ p, p + 1 = m`. Represent as both a `ℕ` and `Fin` type.
have ⟨nat_p, hnat_p⟩ := Nat.exists_eq_succ_of_ne_zero hm'
have hnat_p_lt_m : nat_p < m := calc nat_p
_ < nat_p + 1 := by simp
_ = m := hnat_p.symm
let fin_p : Fin m := ⟨nat_p, hnat_p_lt_m⟩
by_cases hn : ¬ ∃ t, f t = n
by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n
-- Trivial case. `f` must not be onto if this is the case.
· exact absurd (hf_surj n) hn
· 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 hn
have ⟨fin_t, hfin_t⟩ := hn
simp only [not_not] at h
have ⟨t, ht₁, ht₂⟩ := h
-- `f'` is a variant of `f` in which the largest element of its domain
-- `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 f' : Fin m → Fin (n + 1) := fun x =>
if x = fin_p then n
else if x = fin_t then f fin_p
else f x
let g x := if x = p then n else if x = t then f p else f x
have hf'_inj : Function.Injective f' := by
intro x₁ x₂ hf'
by_cases hx₁ : x₁ = fin_p
· by_cases hx₂ : x₂ = fin_p
· rw [hx₁, hx₂]
· rw [hx₁] at hf'
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 ht : x₂ = fin_t
· rw [if_neg hx₂, if_pos ht, ← hfin_t] at hf'
have := (hf_inj hf').symm
rwa [hx₁, ht]
· rw [if_neg hx₂, if_neg ht, ← hfin_t] at hf'
have := (hf_inj hf').symm
exact absurd this ht
· by_cases hx₂ : x₂ = fin_p
· rw [hx₂] 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 ht : x₁ = fin_t
· rw [if_neg hx₁, if_pos ht, ← hfin_t] at hf'
have := (hf_inj hf').symm
rw [← ht] at this
exact absurd this hx₁
· rw [if_neg hx₁, if_neg ht, ← hfin_t] at hf'
have := hf_inj hf'
exact absurd this ht
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 hx₁, if_neg hx₂] at hf'
by_cases ht₁ : x₁ = fin_t
· by_cases ht₂ : x₂ = fin_t
· rw [ht₁, ht₂]
· rw [if_pos ht₁, if_neg ht₂] at hf'
have := (hf_inj hf').symm
exact absurd this hx₂
· by_cases ht₂ : x₂ = fin_t
· rw [if_neg ht₁, if_pos ht₂] at hf'
have := hf_inj hf'
exact absurd this hx₁
· rw [if_neg ht₁, if_neg ht₂] at hf'
exact hf_inj 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'
-- `g = f' - {⟨p, n⟩}`. This restriction allows us to use the induction
-- hypothesis to prove `g` isn't surjective.
let g : Fin nat_p → Fin n := fun x =>
let hxm := calc ↑x
_ < nat_p := x.isLt
_ < m := hnat_p_lt_m
let y := f' ⟨x, hxm⟩
⟨y, by
suffices y ≠ ↑n by
apply Or.elim (Nat.lt_or_eq_of_lt y.isLt)
· simp
· intro hy
rw [← Fin.val_ne_iff] at this
refine absurd ?_ this
rw [hy]
simp only [Fin.coe_ofNat_eq_mod]
exact Eq.symm (Nat.mod_succ_eq_iff_lt.mpr (by simp))
by_contra ny
have hp₁ : f' fin_p = f' ⟨↑x, hxm⟩ := by
rw [show f' fin_p = n by simp, ← ny]
have hp₂ := Fin.val_eq_of_eq (hf'_inj hp₁)
exact (lt_self_iff_false ↑x).mp $ calc ↑x
_ < nat_p := x.isLt
_ = ↑fin_p := by simp
_ = ↑x := hp₂⟩
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 hg_inj : Function.Injective g := by
intro x₁ x₂ hg
simp only [Fin.mk.injEq] at hg
rw [if_neg (Nat.ne_of_lt x₁.isLt), if_neg (Nat.ne_of_lt x₂.isLt)] at hg
let x₁m : Fin m := ⟨↑x₁, calc ↑x₁
_ < nat_p := x₁.isLt
_ < m := hnat_p_lt_m⟩
let x₂m : Fin m := ⟨↑x₂, calc ↑x₂
_ < nat_p := x₂.isLt
_ < m := hnat_p_lt_m⟩
by_cases hx₁ : x₁m = fin_t
· by_cases hx₂ : x₂m = fin_t
· rw [Fin.ext_iff] at hx₁ hx₂ ⊢
rw [show x₁.1 = x₁m.1 from rfl, show x₂.1 = x₂m.1 from rfl, hx₁, hx₂]
· rw [if_pos hx₁, if_neg hx₂, ← Fin.ext_iff] at hg
have := hf_inj hg
rw [Fin.ext_iff] at this
exact absurd this.symm (Nat.ne_of_lt x₂.isLt)
· by_cases hx₂ : x₂m = fin_t
· rw [if_neg hx₁, if_pos hx₂, ← Fin.ext_iff] at hg
have := hf_inj hg
rw [Fin.ext_iff] at this
exact absurd this (Nat.ne_of_lt x₁.isLt)
· rw [if_neg hx₁, if_neg hx₂, ← Fin.ext_iff] at hg
have := hf_inj hg
simp only [Fin.mk.injEq] at this
exact Fin.ext_iff.mpr this
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)
have ng_surj : ¬ Function.Surjective g := ih nat_p (calc nat_p
_ < m := hnat_p_lt_m
_ ≤ n := Nat.lt_succ.mp hm) g hg_inj
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, ha⟩ : ∃ a, a ∉ Set.range g := by
unfold Function.Surjective at ng_surj
unfold Set.range
simp only [not_forall, not_exists] at ng_surj
have ⟨a, ha₁⟩ := ng_surj
simp only [Fin.mk.injEq] at ha₁
refine ⟨a, ?_⟩
intro ha₂
simp only [Fin.mk.injEq, Set.mem_setOf_eq] at ha₂
have ⟨y, hy⟩ := ha₂
exact absurd hy (ha₁ y)
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
-- By construction, if `g` isn't surjective then neither is `f'`.
have hf'a : ↑a ∉ Set.range f' := by
-- 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 ?_)
-- It suffices to prove that `f'` and `g` agree on all values found in
-- `g`'s domain. The only input that complicates things is `p`, which is
-- found in the domains of `f'` and `f`. So long as we can prove
-- `f' p ≠ a`, then we can be sure `a` appears nowhere in `ran f'`.
suffices ∀ x : Fin m, (ht : x < fin_p) → f' x = g ⟨x, ht⟩ by
unfold Set.range
simp only [Set.mem_setOf_eq, not_exists]
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
intro x
by_cases hp : x = fin_p
· intro nx
rw [if_pos hp, Fin.ext_iff] at nx
simp only [
Fin.coe_ofNat_eq_mod,
Fin.coe_eq_castSucc,
Fin.coe_castSucc
] at nx
rw [Nat.mod_succ_eq_iff_lt.mpr (show n < n + 1 by simp)] at nx
exact absurd nx (Nat.ne_of_lt a.isLt).symm
· show f' x ≠ ↑↑a
rw [show ¬x = fin_p ↔ x ≠ fin_p from Iff.rfl, ← Fin.val_ne_iff] at hp
-- Apply our `suffice` hypothesis.
have hx_lt_fin_p : x < fin_p := by
refine Or.elim (Nat.lt_or_eq_of_lt $ calc ↑x
_ < m := x.isLt
_ = nat_p + 1 := hnat_p) id ?_
intro hxp
exact absurd hxp hp
rw [this x hx_lt_fin_p]
have ha₁ : ¬∃ y, g y = a := ha
simp only [not_exists] at ha₁
have ha₂ : g ⟨↑x, _⟩ ≠ a :=
ha₁ ⟨↑x, by rwa [Fin.lt_iff_val_lt_val] at hx_lt_fin_p⟩
norm_cast at ha₂ ⊢
intro nx
exact absurd (Fin.castSucc_injective n nx) ha₂
intro t ht
rw [Fin.ext_iff]
simp only [Fin.coe_ofNat_eq_mod]
generalize (
if t = fin_p then ↑n
else if t = fin_t then f fin_p
else f t
) = y
exact (Nat.mod_succ_eq_iff_lt.mpr y.isLt).symm
-- Likewise, if `f'` isn't surjective then neither is `f`.
have hfa : ↑a ∉ Set.range f := by
suffices Set.range f = Set.range f' by rw [this]; exact hf'a
unfold Set.range
ext x
apply Iff.intro
· intro ⟨y, hy⟩
simp only [Set.mem_setOf_eq]
by_cases hx₁ : x = n
· refine ⟨fin_p, ?_⟩
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 hx₁.symm
· by_cases hx₂ : x = ⟨f fin_p, (f fin_p).isLt⟩
· refine ⟨fin_t, ?_⟩
by_cases ht : fin_t = fin_p
· rw [if_pos ht, hx₂]
rw [ht] at hfin_t
exact hfin_t.symm
· rw [if_neg ht, if_pos rfl, hx₂]
· refine ⟨y, ?_⟩
have hy₁ : y ≠ fin_p := by
by_contra ny
rw [ny] at hy
exact absurd hy.symm hx₂
have hy₂ : y ≠ fin_t := by
by_contra ny
rw [ny, hfin_t] at hy
exact absurd hy.symm hx₁
rw [if_neg hy₁, if_neg hy₂]
exact hy
· intro ⟨y, hy⟩
dsimp only at hy
by_cases hy₁ : y = fin_p
· rw [if_pos hy₁] at hy
have := hf_surj ⟨n, show n < n + 1 by simp⟩
rwa [← hy]
· rw [if_neg hy₁] at hy
by_cases hy₂ : y = fin_t
· rw [if_pos hy₂] at hy
exact ⟨fin_p, hy⟩
· rw [if_neg hy₂] at hy
exact ⟨y, hy⟩
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]
simp only [Fin.coe_eq_castSucc, Set.mem_setOf_eq] at hfa
exact absurd (hf_surj $ Fin.castSucc a) hfa
theorem pigeonhole_principle (m n : ℕ) (h : m < n)
: ∀ f : Fin m → Fin n, ¬ Function.Bijective f := by
intro f nf
have := pigeonhole_principle_aux n m h f nf.left
exact absurd nf.right this
/--
No natural number is equinumerous to a proper subset of itself.
-/
theorem pigeonhole_principle {n : ℕ}
: ∀ M, M ⊂ Set.Iio n → ∀ f, ¬ Set.BijOn f M (Set.Iio n) := by
intro M hM f nf
have := pigeonhole_principle_aux n M hM f ⟨nf.left, nf.right.left⟩
exact absurd nf.right.right this
/-- #### Corollary 6C
No finite set is equinumerous to a proper subset of itself.
-/
theorem corollary_6c (S S' : Finset α) (hS : S' ⊂ S)
: ∀ f : S → S', ¬ Function.Bijective f := by
sorry
theorem corollary_6c [DecidableEq α] [Nonempty α] {S S' : Finset α} (h : S' ⊂ S)
: ∀ f, ¬ Set.BijOn f S.toSet S'.toSet := by
have ⟨T, hT₁, hT₂⟩ : ∃ T, Disjoint S' T ∧ S = S' ∪ T := by
refine ⟨S \ S', ?_, ?_⟩
· intro X hX₁ hX₂
show ∀ t, t ∈ X → t ∈ ⊥
intro t ht
have ht₂ := hX₂ ht
simp only [Finset.mem_sdiff] at ht₂
exact absurd (hX₁ ht) ht₂.right
· simp only [
Finset.union_sdiff_self_eq_union,
Finset.right_eq_union_iff_subset
]
exact subset_of_ssubset h
-- `hF : S' ∪ T ≈ S`.
-- `hG : S ≈ n`.
-- `hH : S' ∪ T ≈ n`.
have ⟨F, hF⟩ := Set.equinumerous_refl S.toSet
conv at hF => arg 2; rw [hT₂]
have ⟨n, G, hG⟩ := Set.finite_iff_equinumerous_nat.mp (Finset.finite_toSet S)
have ⟨H, hH⟩ := Set.equinumerous_trans hF hG
-- Restrict `H` to `S'` to yield a bijection between `S'` and `m < 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 $ Finset.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 := Finset.mem_union_right S' ha₁
have hc₂ : x ∈ S' ∪ T := Finset.mem_union_left T hx
rw [hH.right.left hc₁ hc₂ ha₂] at ha₁
have hx₁ : {x} ⊆ S' := Finset.singleton_subset_iff.mpr hx
have hx₂ : {x} ⊆ T := Finset.singleton_subset_iff.mpr ha₁
have hx₃ := hT₁ hx₁ hx₂
simp only [
Finset.bot_eq_empty,
Finset.le_eq_subset,
Finset.singleton_subset_iff,
Finset.not_mem_empty
] at hx₃
· -- `Set.InjOn H S'`
intro x₁ hx₁ x₂ hx₂ h
have hc₁ : x₁ ∈ S' ∪ T := Finset.mem_union_left T hx₁
have hc₂ : x₂ ∈ S' ∪ T := Finset.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 [Finset.mem_coe, Set.mem_setOf_eq]
dsimp only at hr
have := hH.right.right hr.left
simp only [
Finset.coe_union,
Set.mem_image,
Set.mem_union,
Finset.mem_coe
] 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 f nf
have ⟨f₁, hf₁⟩ : ∃ f₁ : α → ℕ, Set.BijOn f₁ S R :=
Set.equinumerous_trans nf hR
have ⟨f₂, hf₂⟩ : ∃ f₃ : ℕ → ℕ, Set.BijOn f₃ R (Set.Iio n) := by
have ⟨k, hk₁⟩ := Set.equinumerous_symm hf₁
exact Set.equinumerous_trans hk₁ hG
refine absurd hf₂ (pigeonhole_principle R ?_ f₂)
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⟩ : Finset.Nonempty T := by
rw [hT₂, Finset.ssubset_def] at h
have : ¬ ∀ x, x ∈ S' ∪ T → x ∈ S' := h.right
simp only [Finset.mem_union, not_forall, exists_prop] at this
have ⟨x, hx⟩ := this
apply Or.elim hx.left
· intro nx
exact absurd nx hx.right
· intro hx
exact ⟨x, hx⟩
have ht₂ : H t ∈ Set.Iio n := hH.left (Finset.mem_union_right S' ht₁)
have ht₃ : H t ∈ R := nr (H t) ht₂
exact absurd ⟨t, ht₁, rfl⟩ ht₃.right
/-- #### Corollary 6D (a)

2
Common/Real/Floor.lean
View File

@ -99,7 +99,7 @@ end Hermite
open BigOperators
/-- ### Hermite's Identity
/-- #### Hermite's Identity
The following decomposes the floor of a multiplication into a sum of floors.
-/

29
Common/Set/Basic.lean
View File

@ -1,6 +1,7 @@
import Common.Logic.Basic
import Mathlib.Data.Set.Basic
import Mathlib.Data.Set.Prod
import Mathlib.Tactic.LibrarySearch
/-! # Common.Set.Basic
@ -68,11 +69,15 @@ theorem pair_eq_singleton_mem_imp_eq_all {x y z : α}
/-! ## Subsets -/
/--
Every `Set` is a subset of itself.
There exists no proper subset of `∅`.
-/
theorem subset_self (S : Set α) : S ⊆ S := by
intro _ hs
exact hs
theorem ssubset_empty_iff_false (S : Set α)
: S ⊂ ∅ ↔ False := by
apply Iff.intro
· intro h
rw [ssubset_iff_subset_ne, subset_empty_iff] at h
exact absurd h.left h.right
· simp only [IsEmpty.forall_iff]
/--
If `Set` `A` is a subset of `Set` `B`, then `A ∪ B = B`.
@ -116,7 +121,7 @@ theorem mem_mem_imp_pair_subset {x y : α}
Every `Set` is a member of its own powerset.
-/
theorem self_mem_powerset_self {A : Set α}
: A ∈ 𝒫 A := subset_self A
: A ∈ 𝒫 A := fun _ => mem_preimage.mp
/-! ## Cartesian Product -/
@ -156,6 +161,20 @@ theorem prod_nonempty_nonempty_imp_nonempty_prod {A : Set α} {B : Set β}
/-! ## Difference -/
/--
For any sets `A ⊂ B`, if `x ∈ A` then `A - {x} ⊂ B - {x}`.
-/
theorem diff_ssubset_diff_left {A B : Set α} (h : A ⊂ B)
: x ∈ A → A \ {x} ⊂ B \ {x} := by
intro hx
rw [Set.ssubset_def]
apply And.intro
· exact diff_subset_diff_left (subset_of_ssubset h)
· by_contra nh
have : {x} ⊆ A := singleton_subset_iff.mpr hx
rw [diff_subset_iff, union_diff_cancel this] at nh
exact LT.lt.false (Set.ssubset_of_ssubset_of_subset h nh)
/--
For any set `A`, the difference between the sample space and `A` is the
complement of `A`.

45
Common/Set/Finite.lean Normal file
View File

@ -0,0 +1,45 @@
import Mathlib.Data.Finset.Card
import Mathlib.Data.Set.Finite
/-! # Common.Set.Finite
Additional theorems around finite sets.
-/
namespace Set
/--
For any set `A`, `A ≈ A`.
-/
theorem equinumerous_refl (A : Set α)
: ∃ F, Set.BijOn F A A := by
refine ⟨fun x => x, ?_⟩
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
simp only [imp_self, implies_true, Set.image_id', true_and]
exact Eq.subset rfl
/--
For any sets `A` and `B`, if `A ≈ B`. then `B ≈ A`.
-/
theorem equinumerous_symm [Nonempty α] {A : Set α} {B : Set β}
{F : α → β} (hF : Set.BijOn F A B)
: ∃ G, Set.BijOn G B A := by
refine ⟨Function.invFunOn F A, ?_⟩
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
/--
For any sets `A`, `B`, and `C`, if `A ≈ B` and `B ≈ C`, then `A ≈ C`.
-/
theorem equinumerous_trans {A : Set α} {B : Set β} {C : Set γ}
{F : α → β} (hF : Set.BijOn F A B)
{G : β → γ} (hG : Set.BijOn G B C)
: ∃ H, Set.BijOn H A C := by
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
/--
A set is finite if and only if it is equinumerous to a natural number.
-/
axiom finite_iff_equinumerous_nat {α : Type _} {S : Set α}
: Set.Finite S ↔ ∃ n : ℕ, ∃ f, Set.BijOn f S (Set.Iio n)
end Set