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Enderton (Set). Additions to exercise set 6. (#8)

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305
Bookshelf/Enderton/Set.tex
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@ -421,6 +421,17 @@
\lean{Init/Prelude}
{Mul.mul}
\section{\defined{Natural Map}}%
\hyperlabel{ref:natural-map}
Let $R$ be an \nameref{ref:equivalence-relation} on $A$.
Then the \textbf{natural map} (or \textbf{canonical map})
$\phi \colon A \rightarrow A / R$ is defined as $$\phi(x) = [x]_R$$ for
$x \in A$.
\lean*{Init/Core}
{Quotient.lift}
\section{\defined{Natural Number}}%
\hyperlabel{ref:natural-number}
@ -3072,6 +3083,9 @@
\code{Bookshelf/Enderton/Set/Relation}
{Set.Relation.one\_to\_one\_comp\_is\_one\_to\_one}
\lean{Mathlib/Data/Set/Function}
{Set.InjOn.comp}
\begin{proof}
Let $F \colon B \rightarrow C$ and $G \colon A \rightarrow B$ be
one-to-one \nameref{ref:function}s from sets $A$, $B$, and $C$.
@ -3116,35 +3130,23 @@
\end{align*}
\end{proof}
\subsection{\verified{Theorem 3J}}%
\hyperlabel{sub:theorem-3j}
\subsection{\verified{Theorem 3J (a)}}%
\hyperlabel{sub:theorem-3j-a}
\begin{theorem}[3J]
\begin{theorem}[3J(a)]
Assume that $F \colon A \rightarrow B$, and that $A$ is nonempty.
\begin{enumerate}[(a)]
\item There exists a function $G \colon B \rightarrow A$
(a "left inverse") such that $G \circ F$ is the identity function $I_A$
on $A$ iff $F$ is one-to-one.
\item There exists a function $H \colon B \rightarrow A$
(a "right inverse") such that $F \circ H$ is the identity function $I_B$
on $B$ iff $F$ maps $A$ \textit{onto} $B$.
\end{enumerate}
There exists a function $G \colon B \rightarrow A$ (a "left inverse") such
that $G \circ F$ is the identity function $I_A$ on $A$ iff $F$ is
one-to-one.
\end{theorem}
\code{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.theorem\_3j\_a}
\code{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.theorem\_3j\_b}
\begin{proof}
Let $F$ be a \nameref{ref:function} from nonempty set $A$ to set $B$.
\paragraph{(a)}%
We prove there exists a function $G \colon B \rightarrow A$ such that
$G \circ F = I_A$ if and only if $F$ is one-to-one.
\subparagraph{($\Rightarrow$)}%
Let $G \colon B \rightarrow A$ such that $G \circ F = I_A$.
@ -3172,10 +3174,24 @@
Thus $(G \circ F)(x) = G(F(x)) = F^{-1}(F(x)) = x$ by
\nameref{sub:theorem-3g}.
\paragraph{(b)}%
\end{proof}
We prove there exists a function $H \colon B \rightarrow A$ such that
$F \circ H = I_A$ if and only if $F$ maps $A$ onto $B$.
\subsection{\unverified{Theorem 3J (b)}}%
\hyperlabel{sub:theorem-3j-b}
\begin{theorem}[3J(b)]
Assume that $F \colon A \rightarrow B$, and that $A$ is nonempty.
There exists a function $H \colon B \rightarrow A$ (a "right inverse") such
that $F \circ H$ is the identity function $I_B$ on $B$ iff $F$ maps $A$
\textit{onto} $B$.
\end{theorem}
\code{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.theorem\_3j\_b}
\begin{proof}
Let $F$ be a \nameref{ref:function} from nonempty set $A$ to set $B$.
\subparagraph{($\Rightarrow$)}%
@ -3215,9 +3231,9 @@
\begin{proof}
By definition, a one-to-one correspondence $f$ between sets $A$ and $B$ must
be both one-to-one and onto.
By \nameref{sub:theorem-3j}, $f$ is one-to-one if and only if it has a left
inverse.
The same theorem states that $f$ is onto $B$ if and only if it has a right
By \nameref{sub:theorem-3j-a}, $f$ is one-to-one if and only if it has a
left inverse.
By \nameref{sub:theorem-3j-b}, $f$ is onto $B$ if and only if it has a right
inverse.
\end{proof}
@ -9099,7 +9115,7 @@
\hyperlabel{sub:lemma-6f}
\begin{lemma}[6F]
If $C$ is a proper subset of a natural number $n$, then $C \approx m$ for
If $C$ is a proper subset of a natural number $n$, then $C \equin m$ for
some $m$ less than $n$.
\end{lemma}
@ -9199,6 +9215,214 @@
Hence $S'$ is a finite set.
\end{proof}
\subsection{\verified{Subset Size}}%
\hyperlabel{sub:subset-size}
\begin{lemma}
Let $A$ be a finite set and $B \subseteq A$.
Then there exist natural numbers $m, n \in \omega$ such that
$B \equin m$, $A \equin n$, and $m \leq n$.
\end{lemma}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.subset\_size}
\begin{proof}
Let $A$ be a \nameref{ref:finite-set} and $B$ be a subset of $A$.
By \nameref{sub:corollary-6g}, $B$ must be finite.
By definition of a finite set, there exists natural numbers
$m, n \in \omega$ such that $B \equin m$ and $A \equin n$.
By \nameref{sub:trichotomy-law-natural-numbers}, it suffices to prove that
$m > n$ is not possible for then either $m < n$ or $m = n$.
For the sake of contradiction, assume $m > n$.
By definition of \nameref{ref:equinumerous}, there exists a one-to-one
correspondence between $B$ and $m$.
\nameref{sub:theorem-6a} indicates there then exists a one-to-one
correspondence $f$ between $m$ and $B$.
Likewise, there exists a one-to-one correspondence $g$ between $A$ and
$n$.
Define $h \colon A \rightarrow B$ as $h(x) = f(g(x))$ for all $x \in A$.
Since $n \subset m$ by \nameref{sub:corollary-4m}, $h$ is well-defined.
By \nameref{sub:one-to-one-composition}, $h$ must be one-to-one.
Thus $h$ is a one-to-one correspondence between $A$ and $\ran{h}$, i.e.
$A \equin \ran{h}$.
But $n < m$ meaning $\ran{h} \subset B$ which in turn is a proper subset
of $A$ by hypothesis.
\nameref{sub:corollary-6c} states no finite set is equinumerous to a
proper subset of itself, a contradiction.
\end{proof}
\subsection{\pending{Finite Domain and Range Size}}%
\hyperlabel{sub:finite-domain-range-size}
\begin{lemma}
Let $A$ and $B$ be finite sets and $f \colon A \rightarrow B$ be a function.
Then there exist natural numbers $m, n \in \omega$ such that
$\dom{f} \equin m$, $\ran{f} \equin n$, and $m \geq n$.
\end{lemma}
\begin{note}
This proof avoids the \nameref{ref:axiom-of-choice-1} because $A$ and $B$
are finite.
In particular, we are able to choose a "smallest" element of each preimage
set.
Contrast this to \nameref{sub:theorem-3j-b}.
\end{note}
\begin{proof}
Let $A$ and $B$ be \nameref{ref:finite-set}s and $f \colon A \rightarrow B$
be a function.
By definition of finite sets, there exists \nameref{ref:natural-number}s
$m, p \in \omega$ such that $A \equin m$ and $B \equin p$.
By definition of the \nameref{ref:domain} of a function, $\dom{f} = A$.
Thus $\dom{f} \equin m$.
By \nameref{sub:theorem-6a}, there exists a one-to-one correspondence $g$
between $m$ and $\dom{f} = A$.
For all $y \in \ran{f}$, consider $\img{f^{-1}}{\{y\}}$.
Let $$A_y = \{ x \in m \mid f(g(x)) = y \}.$$
Since $g$ is a one-to-one correspondence, it follows that
$A_y \equin \img{f^{-1}}{\{y\}}$.
Since $A_y$ is a nonempty subset of natural numbers, the
\nameref{sub:well-ordering-natural-numbers} implies there exists a least
element, say $q_y$.
Define $C = \{q_y \mid y \in \ran{f}\}$.
Thus $h \colon C \rightarrow \ran{f}$ given by $h(x) = f(g(x))$ is a
one-to-one correspondence between $C$ and $\ran{f}$ by construction.
That is, $C \equin \ran{f}$.
By \nameref{sub:lemma-6f}, there exists some $n \leq m$ such that
$C \equin n$.
By \nameref{sub:theorem-6a}, $n \equin \ran{f}$.
\end{proof}
\subsection{\pending{Set Difference Size}}%
\hyperlabel{sub:set-difference-size}
\begin{lemma}
Let $A \equin m$ for some natural number $m$ and $B \subseteq A$.
Then there exists some $n \in \omega$ such that $B \equin n$ and
$A - B \equin m - n$.
\end{lemma}
\code{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.sdiff\_size}
TODO: Update IH and add additional lemmas to fully prove this out.
\begin{proof}
Let
\begin{equation}
\hyperlabel{sub:set-difference-size-ih}
S = \{m \in \omega \mid
\forall A \equin m, \forall B \subseteq A, \exists n
\in \omega (B \equin n \land A - B \equin m - n)\}.
\end{equation}
We prove that (i) $0 \in S$ and (ii) if $n \in S$ then $n^+ \in S$.
Afterward we prove (iii) the lemma statement.
\paragraph{(i)}%
\hyperlabel{par:set-difference-size-i}
Let $A \equin 0$ and $B \subseteq A$.
Then it follows $A = B = \emptyset$.
Therefore $B \equin 0$ and $$A - B = \emptyset \equin 0 = 0 - 0$$ as
expected.
\paragraph{(ii)}%
\hyperlabel{par:set-difference-size-ii}
Suppose $m \in S$ and consider $m^+$.
Let $A \equin m^+$ and let $B \subseteq A$.
By definition of \nameref{ref:equinumerous}, there exists a one-to-one
correspondence $f$ between $A$ and $m^+$.
Since $f$ is one-to-one and onto, there exists a unique value $a \in A$
such that $f(a) = m$.
Then $B - \{a\} \subseteq A - \{a\}$ and $f$ is a one-to-one
correspondence between $A - \{a\}$ and $m$.
By \ihref{sub:set-difference-size-ih}, there exists some $n \in \omega$
such that $B - \{a\} \equin n$ and
\begin{equation}
\hyperlabel{par:set-difference-size-ii-eq1}
(A - \{a\}) - (B - \{a\}) \equin m - n.
\end{equation}
There are two cases to consider:
\subparagraph{Case 1}%
Assume $a \in B$.
Then $B \equin n^+$.
Furthermore, by definition of the set difference,
\begin{align}
(A - \{a\}) & - (B - \{a\}) \nonumber \\
& = \{x \mid
x \in A - \{a\} \land x \not\in B - \{a\}\} \nonumber \\
& = \{x \mid
(x \in A \land x \neq a) \land
\neg(x \in B \land x \neq a)\} \nonumber \\
& = \{x \mid
(x \in A \land x \neq a) \land
(x \not\in B \lor x = a)\} \nonumber \\
& = \{x \mid
(x \in A \land x \neq a \land x \not\in B) \lor
(x \in A \land x \neq a \land x = a)\} \nonumber \\
& = \{x \mid
(x \in A \land x \neq a \land x \not\in B) \lor F\} \nonumber \\
& = \{x \mid
(x \in A \land x \neq a \land x \not\in B)\} \nonumber \\
& = \{x \mid x \in A - B \land x \neq a\} \nonumber \\
& = \{x \mid x \in A - B \land x \not\in \{a\}\} \nonumber \\
& = (A - B) - \{a\}.
\label{par:set-difference-size-ii-eq2}
\end{align}
Since $a \in A$ and $a \in B$, $(A - B) - \{a\} = A - B$.
Thus
\begin{align*}
(A - \{a\} - (B - \{a\})
& = (A - B) - \{a\} & \eqref{par:set-difference-size-ii-eq2} \\
& = A - B \\
& \equin m - n & \eqref{par:set-difference-size-ii-eq1} \\
& \equin m^+ - n^+ & \textref{sub:theorem-6a}.
\end{align*}
\subparagraph{Case 2}%
Assume $a \not\in B$.
Then $B - \{a\} = B$ (i.e. $B \approx n$) and
\begin{align*}
(A - \{a\}) - (B - \{a\})
& = (A - \{a\}) - B \\
& \equin m - n & \eqref{par:set-difference-size-ii-eq1}.
\end{align*}
The above implies that there exists a one-to-one correspondence $g$
between $(A - \{a\}) - B$ and $m - n$.
Therefore $g \cup \{\tuple{a, m}\}$ is a one-to-one correspondence
between $A - B$ and $(m - n) \cup \{m\}$.
By \nameref{sub:theorem-6a},
$$A - B \equin (m - n) \cup \{m\} \equin m^+ - n.$$
\subparagraph{Subconclusion}%
The above two cases are exhaustive and both conclude the existence of
some $n \in \omega$ such that $B \equin n$ and $A - B \equin m^+ - n$.
Hence $m^+ \in S$.
\paragraph{(iii)}%
By \nameref{par:set-difference-size-i} and
\nameref{par:set-difference-size-ii}, $S \subseteq \omega$ is an
\nameref{ref:inductive-set}.
Thus \nameref{sub:theorem-4b} implies $S = \omega$.
Hence, for all $A \equin m$ for some $m \in \omega$, if $B \subseteq A$,
then there exists some $n \in \omega$ such that $B \equin n$ and
$A - B \equin m - n$.
\end{proof}
\subsection{\sorry{Theorem 6H}}%
\hyperlabel{sub:theorem-6h}
@ -9601,8 +9825,33 @@
\paragraph{($\Leftarrow$)}%
Suppose $\ran{f} = A$.
TODO: by induction?
If $A = \emptyset$, then $f$ is trivially one-to-one.
Assume now $A \neq \emptyset$ and $\ran{f} = A$.
Let $x_1, x_2 \in A$ such that $f(x_1) = f(x_2) = y$ for some $y \in A$.
We must prove that $x_1 = x_2$.
Let $B = \img{f^{-1}}{\{y\}}$.
Then $x_1, x_2 \in B$.
Since $B \subseteq A$, \nameref{sub:subset-size} indicates that there
exist natural numbers $m_1, n_1 \in \omega$ such that $B \equin m_1$,
$A \equin n_1$, and $m_1 \leq n_1$.
Define $f' \colon (A - B) \rightarrow (A - \{y\})$ as the
\nameref{ref:restriction} of $f$ to $A - B$.
Since $\ran{f} = A$, it follows $\ran{f'} = A - \{y\}$.
Since \nameref{sub:corollary-6g} implies $A - B$ and $A - \{y\}$ are
finite sets, \nameref{sub:finite-domain-range-size} implies the
existence of natural numbers $m_2, n_2 \in \omega$ such that
$\dom{f'} \equin m_2$, $\ran{f'} \equin n_2$, and $m_2 \geq n_2$.
Thus, by \nameref{sub:set-difference-size},
\begin{align*}
m_2 & \equin \dom{f'} \equin A - B \equin n_1 - m_1 \\
n_2 & \equin \ran{f'} \equin A - \{y\} \equin n_1 - 1.
\end{align*}
By \nameref{sub:corollary-6e}, $m_2 = n_1 - m_1$ and $n_2 = n_1 - 1$.
Since $m_2 \geq n_2$, $n_1 - m_1 \geq n_1 - 1$.
But $1 \leq m_1 \leq n_1$ meaning $m_1 = 1$.
Hence $B$ is a singleton.
Therefore $x_1 = x_2$, i.e. $f$ is one-to-one.
\end{proof}

2
Bookshelf/Enderton/Set/Chapter_3.lean
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@ -264,7 +264,7 @@ theorem theorem_3j_a {F : Set.HRelation α β}
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`.
`B` only if `F` maps `A` onto `B`.
-/
theorem theorem_3j_b {F : Set.HRelation α β} (hF : mapsInto F A B)
: (∃ H, mapsInto H B A ∧ comp F H = { p | p.1 ∈ B ∧ p.1 = p.2 }) →

387
Bookshelf/Enderton/Set/Chapter_6.lean
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@ -3,10 +3,10 @@ import Common.Logic.Basic
import Common.Nat.Basic
import Common.Set.Basic
import Common.Set.Equinumerous
import Common.Set.Function
import Common.Set.Intervals
import Mathlib.Data.Finset.Card
import Mathlib.Data.Set.Finite
import Mathlib.Tactic.LibrarySearch
/-! # Enderton.Set.Chapter_6
@ -83,11 +83,8 @@ lemma pigeonhole_principle_aux (n : ℕ)
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)
· rw [hM', Set.SurjOn_emptyset_Iio_iff_eq_zero] at hf_surj
simp at hf_surj
by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n
-- Trivial case. `f` must not be onto if this is the case.
@ -95,17 +92,18 @@ lemma pigeonhole_principle_aux (n : ℕ)
exact absurd ⟨t, ht⟩ h
-- Continue under the assumption `n ∈ ran f`.
simp only [not_not] at h
have ⟨t, ht₁, ht₂⟩ := h
have ⟨t, ht₁, ht₂⟩ := not_not.mp h
-- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`.
-- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`, i.e. a maximum member.
have ⟨p, hp₁, hp₂⟩ : ∃ p ∈ M, ∀ x, x ∈ M → p ≥ x := by
refine subset_finite_max_nat (show Set.Finite M from ?_) ?_ ?_
· have := Set.finite_lt_nat (n + 1)
· show Set.Finite M
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]
· show Set.Nonempty M
exact Set.nmem_singleton_empty.mp hM'
· show M ⊆ M
exact Eq.subset rfl
-- `g` is a variant of `f` in which the largest element of its domain
-- (i.e. `p`) corresponds to value `n`.
@ -675,6 +673,349 @@ theorem corollary_6g {S S' : Set α} (hS : Set.Finite S) (hS' : S' ⊆ S)
· intro h
rwa [h]
/-- #### Subset Size
Let `A` be a finite set and `B ⊂ A`. Then there exist natural numbers `m, n ∈ ω`
such that `B ≈ m`, `A ≈ n`, and `m ≤ n`.
-/
lemma subset_size [DecidableEq α] [Nonempty α] {A B : Set α}
(hBA : B ⊆ A) (hA : Set.Finite A)
: ∃ m n : ℕ, B ≈ Set.Iio m ∧ A ≈ Set.Iio n ∧ m ≤ n := by
have ⟨n, hn⟩ := Set.finite_iff_equinumerous_nat.mp hA
have ⟨m, hm⟩ := Set.finite_iff_equinumerous_nat.mp (corollary_6g hA hBA)
refine ⟨m, n, hm, hn, ?_⟩
suffices ¬ m > n by
match @trichotomous ℕ LT.lt _ m n with
| Or.inr (Or.inl hr) => -- m = n
rw [hr]
| Or.inr (Or.inr hr) => -- m > n
exact absurd hr this
| Or.inl hr => -- m < n
exact Nat.le_of_lt hr
by_contra nr
have ⟨f, hf⟩ := Set.equinumerous_symm hm
have ⟨g, hg⟩ := hn
let h x := f (g x)
have hh : Set.BijOn h A (h '' A) := by
refine ⟨?_, ?_, Eq.subset rfl⟩
· -- `Set.MapsTo h A (ran h)`
intro x hx
simp only [Set.mem_image]
exact ⟨x, hx, rfl⟩
· -- `Set.InjOn h A`
refine Set.InjOn.comp hf.right.left hg.right.left ?_
intro x hx
exact Nat.lt_trans (hg.left hx) nr
have : h '' A ⊂ A := by
rw [Set.ssubset_def]
apply And.intro
· show ∀ x, x ∈ h '' A → x ∈ A
intro x hx
have ⟨y, hy₁, hy₂⟩ := hx
have h₁ : g y ∈ Set.Iio n := hg.left hy₁
have h₂ : f (g y) ∈ B := hf.left (Nat.lt_trans h₁ nr)
have h₃ : x ∈ B := by rwa [← hy₂]
exact hBA h₃
· rw [Set.subset_def]
simp only [Set.mem_image, not_forall, not_exists, not_and, exists_prop]
refine ⟨f n, hBA (hf.left nr), ?_⟩
intro x hx
by_contra nh
have h₁ : g x < n := hg.left hx
have h₂ : g x ∈ Set.Iio m := Nat.lt_trans h₁ nr
rw [hf.right.left h₂ nr nh] at h₁
simp at h₁
exact absurd ⟨h, hh⟩ (corollary_6c hA this)
/-- #### Finite Domain and Range Size
Let `A` and `B` be finite sets and `f : A → B` be a function. Then there exist
natural numbers `m, n ∈ ω` such that `dom f ≈ m`, `ran f ≈ n`, and `m ≥ n`.
-/
theorem finite_dom_ran_size [Nonempty α] {A B : Set α}
(hA : Set.Finite A) (hB : Set.Finite B) (hf : Set.MapsTo f A B)
: ∃ m n : ℕ, A ≈ Set.Iio m ∧ f '' A ≈ Set.Iio n ∧ m ≥ n := by
have ⟨m, hm⟩ := Set.finite_iff_equinumerous_nat.mp hA
have ⟨p, hp⟩ := Set.finite_iff_equinumerous_nat.mp hB
have ⟨g, hg⟩ := Set.equinumerous_symm hm
let A_y y := { x ∈ Set.Iio m | f (g x) = y }
have hA₁ : ∀ y ∈ B, A_y y ≈ f ⁻¹' {y} := by
sorry
have hA₂ : ∀ y ∈ B, Set.Nonempty (A_y y) := by
sorry
have hA₃ : ∀ y ∈ B, ∃ q : ℕ, ∀ p ∈ A_y y, q ≤ p := by
sorry
let C := { q | ∃ y ∈ B, ∀ p ∈ A_y y, q ≤ p }
let h x := f (g x)
have hh : C ≈ f '' A := by
sorry
sorry
/-- #### Set Difference Size
Let `A ≈ m` for some natural number `m` and `B ⊆ A`. Then there exists some
`n ∈ ω` such that `B ≈ n` and `A - B ≈ m - n`.
-/
lemma sdiff_size_aux [DecidableEq α] [Nonempty α]
: ∀ A : Set α, A ≈ Set.Iio m →
∀ B, B ⊆ A →
∃ n : ℕ, n ≤ m ∧ B ≈ Set.Iio n ∧ A \ B ≈ (Set.Iio m) \ (Set.Iio n) := by
induction m with
| zero =>
intro A hA B hB
refine ⟨0, ?_⟩
simp only [
Nat.zero_eq,
sdiff_self,
Set.bot_eq_empty,
Set.equinumerous_zero_iff_emptyset
] at hA ⊢
have hB' : B = ∅ := Set.subset_eq_empty hB hA
have : A \ B = ∅ := by
rw [hB']
simp only [Set.diff_empty]
exact hA
rw [this]
refine ⟨trivial, hB', Set.equinumerous_emptyset_emptyset⟩
| succ m ih =>
intro A ⟨f, hf⟩ B hB
-- Since `f` is one-to-one and onto, there exists a unique value `a ∈ A`
-- such that `f(a) = m`.
have hfa := hf.right.right
unfold Set.SurjOn at hfa
have ⟨a, ha₁, ha₂⟩ := (Set.subset_def ▸ hfa) m (by simp)
-- `f` is a one-to-one correspondence between `A - {a}` and `m`.
have hBA : B \ {a} ⊆ A \ {a} := Set.diff_subset_diff_left hB
have hfBA : Set.BijOn f (A \ {a}) (Set.Iio m) := by
refine ⟨?_, ?_, ?_⟩
· intro x hx
have := hf.left hx.left
simp only [Set.mem_Iio, gt_iff_lt] at this ⊢
apply Or.elim (Nat.lt_or_eq_of_lt this)
· simp
· intro h
rw [← ha₂] at h
exact absurd (hf.right.left hx.left ha₁ h) hx.right
· intro x₁ hx₁ x₂ hx₂ h
exact hf.right.left hx₁.left hx₂.left h
· have := hf.right.right
unfold Set.SurjOn Set.image at this ⊢
rw [Set.subset_def] at this ⊢
simp only [
Set.mem_Iio,
Set.mem_diff,
Set.mem_singleton_iff,
Set.mem_setOf_eq
] at this ⊢
intro x hx
have ⟨b, hb⟩ := this x (Nat.lt.step hx)
refine ⟨b, ⟨hb.left, ?_⟩, hb.right⟩
by_contra nb
rw [← nb, hb.right] at ha₂
exact absurd ha₂ (Nat.ne_of_lt hx)
-- `(A - {a}) - (B - {a}) ≈ m - n`
have ⟨n, hn₁, hn₂, hn₃⟩ := ih (A \ {a}) ⟨f, hfBA⟩ (B \ {a}) hBA
by_cases hc : a ∈ B
· refine ⟨n.succ, ?_, ?_, ?_⟩
· exact Nat.succ_le_succ hn₁
· -- `B ≈ Set.Iio n.succ`
have ⟨g, hg⟩ := hn₂
let g' x := if x = a then n else g x
refine ⟨g', ⟨?_, ?_, ?_⟩⟩
· -- `Set.MapsTo g' B (Set.Iio n.succ)`
intro x hx
dsimp only
by_cases hx' : x = a
· rw [if_pos hx']
simp
· rw [if_neg hx']
calc g x
_ < n := hg.left ⟨hx, hx'⟩
_ < n + 1 := by simp
· -- `Set.InjOn g' B`
intro x₁ hx₁ x₂ hx₂ h
dsimp only at h
by_cases hc₁ : x₁ = a <;> by_cases hc₂ : x₂ = a
· rw [hc₁, hc₂]
· rw [if_pos hc₁, if_neg hc₂] at h
exact absurd h.symm (Nat.ne_of_lt $ hg.left ⟨hx₂, hc₂⟩)
· rw [if_neg hc₁, if_pos hc₂] at h
exact absurd h (Nat.ne_of_lt $ hg.left ⟨hx₁, hc₁⟩)
· rw [if_neg hc₁, if_neg hc₂] at h
exact hg.right.left ⟨hx₁, hc₁⟩ ⟨hx₂, hc₂⟩ h
· -- `Set.SurjOn g' B (Set.Iio n.succ)`
have := hg.right.right
unfold Set.SurjOn Set.image at this ⊢
rw [Set.subset_def] at this ⊢
simp only [Set.mem_Iio, Set.mem_setOf_eq] at this ⊢
intro x hx
by_cases hc₁ : x = n
· refine ⟨a, hc, ?_⟩
simp only [ite_true]
exact hc₁.symm
· apply Or.elim (Nat.lt_or_eq_of_lt hx)
· intro hx₁
have ⟨b, ⟨hb₁, hb₂⟩, hb₃⟩ := this x hx₁
refine ⟨b, hb₁, ?_⟩
simp only [Set.mem_singleton_iff] at hb₂
rwa [if_neg hb₂]
· intro hx₁
exact absurd hx₁ hc₁
· have hA₁ : (A \ {a}) \ (B \ {a}) = (A \ B) \ {a} :=
Set.diff_mem_diff_mem_eq_diff_diff_mem
have hA₂ : (A \ B) \ {a} = A \ B := by
refine Set.not_mem_diff_eq_self ?_
by_contra na
exact absurd hc na.right
rw [hA₁, hA₂] at hn₃
suffices (Set.Iio m) \ (Set.Iio n) ≈ (Set.Iio m.succ) \ (Set.Iio n.succ)
from Set.equinumerous_trans hn₃ this
-- `m - n ≈ m⁺ - n⁺`
refine ⟨fun x => x + 1, ?_, ?_, ?_⟩
· intro x ⟨hx₁, hx₂⟩
simp at hx₁ hx₂ ⊢
exact ⟨Nat.le_add_of_sub_le hx₂, Nat.add_lt_of_lt_sub hx₁⟩
· intro _ _ _ _ h
simp only [add_left_inj] at h
exact h
· unfold Set.SurjOn Set.image
rw [Set.subset_def]
intro x ⟨hx₁, hx₂⟩
simp only [
Set.Iio_diff_Iio,
gt_iff_lt,
not_lt,
ge_iff_le,
Set.mem_setOf_eq,
Set.mem_Iio
] at hx₁ hx₂ ⊢
have ⟨p, hp⟩ : ∃ p : ℕ, x = p.succ := by
refine Nat.exists_eq_succ_of_ne_zero ?_
have := calc 0
_ < n.succ := by simp
_ ≤ x := hx₂
exact Nat.pos_iff_ne_zero.mp this
refine ⟨p, ⟨?_, ?_⟩, hp.symm⟩
· rw [hp] at hx₂
exact Nat.lt_succ.mp hx₂
· rw [hp] at hx₁
exact Nat.succ_lt_succ_iff.mp hx₁
· have hB : B \ {a} = B := Set.not_mem_diff_eq_self hc
refine ⟨n, ?_, ?_, ?_⟩
· calc n
_ ≤ m := hn₁
_ ≤ m + 1 := by simp
· rwa [← hB]
· rw [hB] at hn₃
have ⟨g, hg⟩ := hn₃
have hAB : A \ B ≈ (Set.Iio m) \ (Set.Iio n) ∪ {m} := by
refine ⟨fun x => if x = a then m else g x, ?_, ?_, ?_⟩
· intro x hx
dsimp only
by_cases hc₁ : x = a
· rw [if_pos hc₁]
simp
· rw [if_neg hc₁]
have := hg.left ⟨⟨hx.left, hc₁⟩, hx.right⟩
simp only [
Set.Iio_diff_Iio,
gt_iff_lt,
not_lt,
ge_iff_le,
Set.union_singleton,
Set.mem_Ico,
lt_self_iff_false,
and_false,
Set.mem_insert_iff
] at this ⊢
right
exact this
· intro x₁ hx₁ x₂ hx₂ h
dsimp only at h
by_cases hc₁ : x₁ = a <;> by_cases hc₂ : x₂ = a
· rw [hc₁, hc₂]
· rw [if_pos hc₁, if_neg hc₂] at h
have := hg.left ⟨⟨hx₂.left, hc₂⟩, hx₂.right⟩
simp at this
exact absurd h.symm (Nat.ne_of_lt this.right)
· rw [if_neg hc₁, if_pos hc₂] at h
have := hg.left ⟨⟨hx₁.left, hc₁⟩, hx₁.right⟩
simp at this
exact absurd h (Nat.ne_of_lt this.right)
· rw [if_neg hc₁, if_neg hc₂] at h
exact hg.right.left ⟨⟨hx₁.left, hc₁⟩, hx₁.right⟩ ⟨⟨hx₂.left, hc₂⟩, hx₂.right⟩ h
· have := hg.right.right
unfold Set.SurjOn Set.image at this ⊢
rw [Set.subset_def] at this ⊢
simp at this ⊢
refine ⟨⟨a, ⟨ha₁, hc⟩, ?_⟩, ?_⟩
· intro ha
simp at ha
· intro x hx₁ hx₂
have ⟨y, hy₁, hy₂⟩ := this x hx₁ hx₂
refine ⟨y, ?_, ?_⟩
· exact ⟨hy₁.left.left, hy₁.right⟩
· rwa [if_neg hy₁.left.right]
suffices (Set.Iio m) \ (Set.Iio n) ∪ {m} ≈ (Set.Iio m.succ) \ (Set.Iio n)
from Set.equinumerous_trans hAB this
refine ⟨fun x => x, ?_, ?_, ?_⟩
· intro x hx
simp at hx ⊢
apply Or.elim hx
· intro hx₁
rw [hx₁]
exact ⟨hn₁, by simp⟩
· intro ⟨hx₁, hx₂⟩
exact ⟨hx₁, calc x
_ < m := hx₂
_ < m + 1 := by simp⟩
· intro _ _ _ _ h
exact h
· unfold Set.SurjOn Set.image
rw [Set.subset_def]
simp only [
Set.Iio_diff_Iio,
gt_iff_lt,
not_lt,
ge_iff_le,
Set.mem_Ico,
Set.union_singleton,
lt_self_iff_false,
and_false,
Set.mem_insert_iff,
exists_eq_right,
Set.mem_setOf_eq,
and_imp
]
intro x hn hm
apply Or.elim (Nat.lt_or_eq_of_lt hm)
· intro hx
right
exact ⟨hn, hx⟩
· intro hx
left
exact hx
lemma sdiff_size [DecidableEq α] [Nonempty α] {A B : Set α}
(hB : B ⊆ A) (hA : A ≈ Set.Iio m)
: ∃ n : ℕ, n ≤ m ∧ B ≈ Set.Iio n ∧ A \ B ≈ (Set.Iio m) \ (Set.Iio n) :=
sdiff_size_aux A hA B hB
/-- #### Exercise 6.7
Assume that `A` is finite and `f : A → A`. Show that `f` is one-to-one **iff**
@ -702,6 +1043,26 @@ theorem exercise_6_7 [DecidableEq α] [Nonempty α] {A : Set α} {f : α → α}
rw [subset_iff_ssubset_or_eq] at hf₃
exact Or.elim hf₃ (fun h => absurd hf₂ (corollary_6c hA₁ h)) id
· intro hf₁
by_cases hA₃ : A = ∅
· rw [hA₃]
simp
· intro x₁ hx₁ x₂ hx₂ hf₂
let y := f x₁
let B := f ⁻¹' {y}
have hB₁ : x₁ ∈ B := sorry
have hB₂ : x₂ ∈ B := sorry
have hB₃ : B ⊆ A := sorry
have ⟨m₁, n₁, hm₁, hn₁, hmn₁⟩ := subset_size hB₃ hA₁
have hf'₁ : Set.MapsTo f (A \ B) (A \ {y}) := sorry
have hf'₂ : f '' (A \ B) = A \ {y} := sorry
have hf'₃ : Set.Finite (A \ B) := sorry
have hf'₄ : Set.Finite (A \ {y}) := sorry
have ⟨m₂, n₂, hm₂, hn₂, hmn₂⟩ := finite_dom_ran_size hf'₃ hf'₄ hf'₁
have h₁ : A \ B ≈ Set.Iio (n₁ - m₁) := sorry
have h₂ : A \ {y} ≈ Set.Iio (n₁ - 1) := sorry
sorry
/-- #### Exercise 6.8

1
Common/Set.lean
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@ -1,4 +1,5 @@
import Common.Set.Basic
import Common.Set.Equinumerous
import Common.Set.Function
import Common.Set.Intervals
import Common.Set.Peano

56
Common/Set/Basic.lean
View File

@ -193,6 +193,62 @@ theorem diff_ssubset_nonempty {A B : Set α} (h : A ⊂ B)
· intro hx
exact ⟨x, hx⟩
/--
If an element `x` is not a member of a set `A`, then `A - {x} = A`.
-/
theorem not_mem_diff_eq_self {A : Set α} (h : a ∉ A)
: A \ {a} = A := by
ext x
apply Iff.intro
· exact And.left
· intro hx
refine ⟨hx, ?_⟩
simp only [mem_singleton_iff]
by_contra nx
rw [nx] at hx
exact absurd hx h
/--
Given two sets `A` and `B`, `(A - {a}) - (B - {b}) = (A - B) - {a}`.
-/
theorem diff_mem_diff_mem_eq_diff_diff_mem {A B : Set α} {a : α}
: (A \ {a}) \ (B \ {a}) = (A \ B) \ {a} := by
calc (A \ {a}) \ (B \ {a})
_ = { x | x ∈ A \ {a} ∧ x ∉ B \ {a} } := rfl
_ = { x | x ∈ A \ {a} ∧ ¬(x ∈ B \ {a}) } := rfl
_ = { x | (x ∈ A ∧ x ≠ a) ∧ ¬(x ∈ B ∧ x ≠ a) } := rfl
_ = { x | (x ∈ A ∧ x ≠ a) ∧ (x ∉ B ∨ x = a) } := by
ext x
rw [mem_setOf_eq, not_and_de_morgan]
simp
_ = { x | (x ∈ A ∧ x ≠ a ∧ x ∉ B) ∨ (x ∈ A ∧ x ≠ a ∧ x = a) } := by
ext x
simp only [mem_setOf_eq]
rw [and_or_left, and_assoc, and_assoc]
_ = { x | x ∈ A ∧ x ≠ a ∧ x ∉ B } := by simp
_ = { x | x ∈ A ∧ x ∉ B ∧ x ≠ a } := by
ext x
simp only [ne_eq, sep_and, mem_inter_iff, mem_setOf_eq]
apply Iff.intro <;>
· intro ⟨⟨_, hx₂⟩, hx₃, hx₄⟩
exact ⟨⟨hx₃, hx₄⟩, ⟨hx₃, hx₂⟩⟩
_ = { x | x ∈ A ∧ x ∉ B ∧ x ∉ ({a} : Set α) } := rfl
_ = { x | x ∈ A \ B ∧ x ∉ ({a} : Set α) } := by
ext x
simp only [
mem_singleton_iff,
sep_and,
mem_inter_iff,
mem_setOf_eq,
mem_diff,
and_congr_right_iff,
and_iff_right_iff_imp,
and_imp
]
intro hx _ _
exact hx
_ = (A \ B) \ {a} := rfl
/--
For any set `A`, the difference between the sample space and `A` is the
complement of `A`.

73
Common/Set/Equinumerous.lean
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@ -8,6 +8,8 @@ Additional theorems around finite sets.
namespace Set
/-! ### Definitions -/
/--
A set `A` is equinumerous to a set `B` (written `A ≈ B`) if and only if there is
a one-to-one function from `A` onto `B`.
@ -19,6 +21,26 @@ infix:50 " ≈ " => Equinumerous
theorem equinumerous_def (A : Set α) (B : Set β)
: A ≈ B ↔ ∃ F, Set.BijOn F A B := Iff.rfl
/--
A set `A` is not equinumerous to a set `B` (written `A ≉ B`) if and only if
there is no one-to-one function from `A` onto `B`.
-/
def NotEquinumerous (A : Set α) (B : Set β) : Prop := ¬ Equinumerous A B
infix:50 " ≉ " => NotEquinumerous
@[simp]
theorem not_equinumerous_def : A ≉ B ↔ ∀ F, ¬ Set.BijOn F A B := by
apply Iff.intro
· intro h
unfold NotEquinumerous Equinumerous at h
simp only [not_exists] at h
exact h
· intro h
unfold NotEquinumerous Equinumerous
simp only [not_exists]
exact h
/--
For any set `A`, `A ≈ A`.
-/
@ -57,30 +79,53 @@ theorem eq_imp_equinumerous {A B : Set α} (h : A = B)
conv at this => right; rw [h]
exact this
/-! ### Finite Sets -/
/--
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, S ≈ Set.Iio n
/-! ### Emptyset -/
/--
A set `A` is not equinumerous to a set `B` (written `A ≉ B`) if and only if
there is no one-to-one function from `A` onto `B`.
Any set equinumerous to the emptyset is the emptyset.
-/
def NotEquinumerous (A : Set α) (B : Set β) : Prop := ¬ Equinumerous A B
infix:50 " ≉ " => NotEquinumerous
@[simp]
theorem not_equinumerous_def : A ≉ B ↔ ∀ F, ¬ Set.BijOn F A B := by
theorem equinumerous_zero_iff_emptyset {S : Set α}
: S ≈ Set.Iio 0 ↔ S = ∅ := by
apply Iff.intro
· intro ⟨f, hf⟩
by_contra nh
rw [← Ne.def, ← Set.nonempty_iff_ne_empty] at nh
have ⟨x, hx⟩ := nh
have := hf.left hx
simp at this
· intro h
unfold NotEquinumerous Equinumerous at h
simp only [not_exists] at h
exact h
· intro h
unfold NotEquinumerous Equinumerous
simp only [not_exists]
exact h
rw [h]
refine ⟨fun _ => ⊥, ?_, ?_, ?_⟩
· intro _ hx
simp at hx
· intro _ hx
simp at hx
· unfold SurjOn
simp only [bot_eq_zero', image_empty]
show ∀ x, x ∈ Set.Iio 0 → x ∈ ∅
intro _ hx
simp at hx
/--
Empty sets are always equinumerous, regardless of their underlying type.
-/
theorem equinumerous_emptyset_emptyset [Bot β]
: (∅ : Set α) ≈ (∅ : Set β) := by
refine ⟨fun _ => ⊥, ?_, ?_, ?_⟩
· intro _ hx
simp at hx
· intro _ hx
simp at hx
· unfold SurjOn
simp
end Set

235
Common/Set/Function.lean Normal file
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@ -0,0 +1,235 @@
import Mathlib.Data.Set.Function
/-! # Common.Set.Function
Additional theorems around functions defined on sets.
-/
namespace Set
/--
Produce a new function that swaps the outputs of the two specified inputs.
-/
def swap [DecidableEq α] (f : α → β) (x₁ x₂ : α) (x : α) : β :=
if x = x₁ then f x₂ else
if x = x₂ then f x₁ else
f x
/--
Swapping the same input yields the original function.
-/
@[simp]
theorem swap_eq_eq_self [DecidableEq α] {f : α → β} {x : α}
: swap f x x = f := by
refine funext ?_
intro y
unfold swap
by_cases hy : y = x
· rw [if_pos hy, hy]
· rw [if_neg hy, if_neg hy]
/--
Swapping a function twice yields the original function.
-/
@[simp]
theorem swap_swap_eq_self [DecidableEq α] {f : α → β} {x₁ x₂ : α}
: swap (swap f x₁ x₂) x₁ x₂ = f := by
refine funext ?_
intro y
by_cases hc₁ : x₂ = x₁
· rw [hc₁]
simp
rw [← Ne.def] at hc₁
unfold swap
by_cases hc₂ : y = x₁ <;>
by_cases hc₃ : y = x₂
· rw [if_pos hc₂, if_neg hc₁, if_pos rfl, ← hc₂]
· rw [if_pos hc₂, if_neg hc₁, if_pos rfl, ← hc₂]
· rw [if_neg hc₂, if_pos hc₃, if_pos rfl, ← hc₃]
· rw [if_neg hc₂, if_neg hc₃, if_neg hc₂, if_neg hc₃]
/--
If `f : A → B`, then a swapped variant of `f` also maps `A` into `B`.
-/
theorem swap_MapsTo_self [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : MapsTo f A B)
: MapsTo (swap f a₁ a₂) A B := by
intro x hx
unfold swap
by_cases hc₁ : x = a₁ <;> by_cases hc₂ : x = a₂
· rw [if_pos hc₁]
exact hf ha₂
· rw [if_pos hc₁]
exact hf ha₂
· rw [if_neg hc₁, if_pos hc₂]
exact hf ha₁
· rw [if_neg hc₁, if_neg hc₂]
exact hf hx
/--
The converse of `swap_MapsTo_self`.
-/
theorem self_MapsTo_swap [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : MapsTo (swap f a₁ a₂) A B)
: MapsTo f A B := by
rw [← @swap_swap_eq_self _ _ _ f a₁ a₂]
exact swap_MapsTo_self ha₁ ha₂ hf
/--
If `f : A → B`, then `f` maps `A` into `B` **iff** a swap of `f` maps `A` into
`B`.
-/
theorem self_iff_swap_MapsTo [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A)
: MapsTo (swap f a₁ a₂) A B ↔ MapsTo f A B :=
⟨self_MapsTo_swap ha₁ ha₂, swap_MapsTo_self ha₁ ha₂⟩
/--
If `f : A → B` is one-to-one, then a swapped variant of `f` is also one-to-one.
-/
theorem swap_InjOn_self [DecidableEq α]
{A : Set α} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : InjOn f A)
: InjOn (swap f a₁ a₂) A := by
intro x₁ hx₁ x₂ hx₂ h
unfold swap at h
by_cases hc₁ : x₁ = a₁ <;>
by_cases hc₂ : x₁ = a₂ <;>
by_cases hc₃ : x₂ = a₁ <;>
by_cases hc₄ : x₂ = a₂
· rw [hc₁, hc₃]
· rw [hc₁, hc₃]
· rw [hc₂, hc₄]
· rw [if_pos hc₁, if_neg hc₃, if_neg hc₄, ← hc₂] at h
exact hf hx₁ hx₂ h
· rw [hc₁, hc₃]
· rw [hc₁, hc₃]
· rw [if_pos hc₁, if_neg hc₃, if_pos hc₄, ← hc₁, ← hc₄] at h
exact hf hx₁ hx₂ h.symm
· rw [if_pos hc₁, if_neg hc₃, if_neg hc₄] at h
exact absurd (hf hx₂ ha₂ h.symm) hc₄
· rw [hc₂, hc₄]
· rw [if_neg hc₁, if_pos hc₂, if_pos hc₃, ← hc₂, ← hc₃] at h
exact hf hx₁ hx₂ h.symm
· rw [hc₂, hc₄]
· rw [if_neg hc₁, if_pos hc₂, if_neg hc₃, if_neg hc₄] at h
exact absurd (hf hx₂ ha₁ h.symm) hc₃
· rw [if_neg hc₁, if_neg hc₂, if_pos hc₃, ← hc₄] at h
exact hf hx₁ hx₂ h
· rw [if_neg hc₁, if_neg hc₂, if_pos hc₃] at h
exact absurd (hf hx₁ ha₂ h) hc₂
· rw [if_neg hc₁, if_neg hc₂, if_neg hc₃, if_pos hc₄] at h
exact absurd (hf hx₁ ha₁ h) hc₁
· rw [if_neg hc₁, if_neg hc₂, if_neg hc₃, if_neg hc₄] at h
exact hf hx₁ hx₂ h
/--
The converse of `swap_InjOn_self`.
-/
theorem self_InjOn_swap [DecidableEq α]
{A : Set α} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : InjOn (swap f a₁ a₂) A)
: InjOn f A := by
rw [← @swap_swap_eq_self _ _ _ f a₁ a₂]
exact swap_InjOn_self ha₁ ha₂ hf
/--
If `f : A → B`, then `f` is one-to-one **iff** a swap of `f` is one-to-one.
-/
theorem self_iff_swap_InjOn [DecidableEq α]
{A : Set α} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A)
: InjOn (swap f a₁ a₂) A ↔ InjOn f A :=
⟨self_InjOn_swap ha₁ ha₂, swap_InjOn_self ha₁ ha₂⟩
/--
If `f : A → B` is onto, then a swapped variant of `f` is also onto.
-/
theorem swap_SurjOn_self [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : SurjOn f A B)
: SurjOn (swap f a₁ a₂) A B := by
show ∀ x, x ∈ B → ∃ a ∈ A, swap f a₁ a₂ a = x
intro x hx
have ⟨a, ha⟩ := hf hx
by_cases hc₁ : a = a₁
· refine ⟨a₂, ha₂, ?_⟩
unfold swap
by_cases hc₂ : a₁ = a₂
· rw [if_pos hc₂.symm, ← hc₂, ← hc₁]
exact ha.right
· rw [← Ne.def] at hc₂
rw [if_neg hc₂.symm, if_pos rfl, ← hc₁]
exact ha.right
· by_cases hc₂ : a = a₂
· unfold swap
refine ⟨a₁, ha₁, ?_⟩
rw [if_pos rfl, ← hc₂]
exact ha.right
· refine ⟨a, ha.left, ?_⟩
unfold swap
rw [if_neg hc₁, if_neg hc₂]
exact ha.right
/--
The converse of `swap_SurjOn_self`.
-/
theorem self_SurjOn_swap [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : SurjOn (swap f a₁ a₂) A B)
: SurjOn f A B := by
rw [← @swap_swap_eq_self _ _ _ f a₁ a₂]
exact swap_SurjOn_self ha₁ ha₂ hf
/--
If `f : A → B`, then `f` is onto **iff** a swap of `f` is onto.
-/
theorem self_iff_swap_SurjOn [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A)
: SurjOn (swap f a₁ a₂) A B ↔ SurjOn f A B :=
⟨self_SurjOn_swap ha₁ ha₂, swap_SurjOn_self ha₁ ha₂⟩
/--
If `f : A → B` is a one-to-one correspondence, then a swapped variant of `f` is
also a one-to-one correspondence.
-/
theorem swap_BijOn_self [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : BijOn f A B)
: BijOn (swap f a₁ a₂) A B := by
have ⟨hf₁, hf₂, hf₃⟩ := hf
exact ⟨
swap_MapsTo_self ha₁ ha₂ hf₁,
swap_InjOn_self ha₁ ha₂ hf₂,
swap_SurjOn_self ha₁ ha₂ hf₃
⟩
/--
The converse of `swap_BijOn_self`.
-/
theorem self_BijOn_swap [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A) (hf : BijOn (swap f a₁ a₂) A B)
: BijOn f A B := by
have ⟨hf₁, hf₂, hf₃⟩ := hf
exact ⟨
self_MapsTo_swap ha₁ ha₂ hf₁,
self_InjOn_swap ha₁ ha₂ hf₂,
self_SurjOn_swap ha₁ ha₂ hf₃
⟩
/--
If `f : A → B`, `f` is a one-to-one correspondence **iff** a swap of `f` is a
one-to-one correspondence.
-/
theorem self_iff_swap_BijOn [DecidableEq α]
{A : Set α} {B : Set β} {f : α → β}
(ha₁ : a₁ ∈ A) (ha₂ : a₂ ∈ A)
: BijOn (swap f a₁ a₂) A B ↔ BijOn f A B :=
⟨self_BijOn_swap ha₁ ha₂, swap_BijOn_self ha₁ ha₂⟩
end Set

22
Common/Set/Intervals.lean
View File

@ -1,4 +1,5 @@
import Common.Logic.Basic
import Mathlib.Data.Set.Function
import Mathlib.Data.Set.Intervals.Basic
namespace Set
@ -8,6 +9,9 @@ namespace Set
Additional theorems around intervals.
-/
/--
If `m < n` then `{0, …, m - 1} ⊂ {0, …, n - 1}`.
-/
theorem Iio_nat_lt_ssubset {m n : ℕ} (h : m < n)
: Iio m ⊂ Iio n := by
rw [ssubset_def]
@ -22,4 +26,22 @@ theorem Iio_nat_lt_ssubset {m n : ℕ} (h : m < n)
simp only [not_forall, not_lt, exists_prop]
exact ⟨m, h, by simp⟩
/--
It is never the case that the emptyset is surjective
-/
theorem SurjOn_emptyset_Iio_iff_eq_zero {n : ℕ} {f : α → ℕ}
: SurjOn f ∅ (Set.Iio n) ↔ n = 0 := by
apply Iff.intro
· intro h
unfold SurjOn at h
rw [subset_def] at h
simp only [mem_Iio, image_empty, mem_empty_iff_false] at h
by_contra nh
exact h 0 (Nat.pos_of_ne_zero nh)
· intro hn
unfold SurjOn
rw [hn, subset_def]
intro x hx
exact absurd hx (Nat.not_lt_zero x)
end Set