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Enderton. Continue through theorem 3 set.

finite-set-exercises
Joshua Potter 2023-06-25 07:27:35 -06:00
parent d215265a91
commit a6e6251627
3 changed files with 219 additions and 90 deletions
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34
Bookshelf/Enderton/Set.tex
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@ -3037,14 +3037,14 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\statementpadding \statementpadding
\lean*{Bookshelf/Enderton/Set/Chapter\_3} \lean*{Common/Set/Relation}
{Enderton.Set.Chapter\_3.theorem\_3e\_i} {Set.Relation.dom\_inv\_eq\_ran\_self}
\lean*{Bookshelf/Enderton/Set/Chapter\_3} \lean*{Common/Set/Relation}
{Enderton.Set.Chapter\_3.theorem\_3e\_ii} {Set.Relation.ran\_inv\_eq\_dom\_self}
\lean{Bookshelf/Enderton/Set/Chapter\_3} \lean{Common/Set/Relation}
{Enderton.Set.Chapter\_3.theorem\_3e\_iii} {Set.Relation.inv\_inv\_eq\_self}
We prove that (i) $\dom{(F^{-1})} = \ran{F}$, (ii) $\ran{(F^{-1})} = \dom{F}$, We prove that (i) $\dom{(F^{-1})} = \ran{F}$, (ii) $\ran{(F^{-1})} = \dom{F}$,
and (iii) $(F^{-1})^{-1} = F$. and (iii) $(F^{-1})^{-1} = F$.
@ -3081,7 +3081,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\end{proof} \end{proof}
\subsection{\partial{Theorem 3F}}% \subsection{\verified{Theorem 3F}}%
\label{sub:theorem-3f} \label{sub:theorem-3f}
\begin{theorem}[3F] \begin{theorem}[3F]
@ -3095,11 +3095,11 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\statementpadding \statementpadding
\lean*{Bookshelf/Enderton/Set/Chapter\_3} \lean*{Common/Set/Relation}
{Enderton.Set.Chapter\_3.theorem\_3f\_i} {Set.Relation.single\_valued\_inv\_iff\_single\_rooted\_self}
\lean{Bookshelf/Enderton/Set/Chapter\_3} \lean{Common/Set/Relation}
{Enderton.Set.Chapter\_3.theorem\_3f\_ii} {Set.Relation.single\_valued\_self\_iff\_single\_rooted\_inv}
We prove that (i) any set $F$, $F^{-1}$ is a function iff $F$ is We prove that (i) any set $F$, $F^{-1}$ is a function iff $F$ is
single-rooted and (ii) any relation $F$ is a function iff $F^{-1}$ is single-rooted and (ii) any relation $F$ is a function iff $F^{-1}$ is
@ -3151,14 +3151,16 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\end{proof} \end{proof}
\subsection{\partial{Lemma 1}}% \subsection{\verified{Lemma 1}}%
\label{sub:lemma-1} \label{sub:lemma-1}
Assume that $F$ is a one-to-one function. For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
Then $F^{-1}$ is a one-to-one function.
\begin{proof} \begin{proof}
\lean{Common/Set/Relation}
{Set.Relation.one\_to\_one\_self\_iff\_one\_to\_one\_inv}
We prove that (i) $F^{-1}$ is a function and (ii) $F^{-1}$ is single-rooted. We prove that (i) $F^{-1}$ is a function and (ii) $F^{-1}$ is single-rooted.
\paragraph{(i)}% \paragraph{(i)}%
@ -3211,11 +3213,11 @@ Then $F^{-1}$ is a one-to-one function.
For all $x \in \dom{F}$, $\left< x, F(x) \right> \in F$. For all $x \in \dom{F}$, $\left< x, F(x) \right> \in F$.
Then $\left< F(x), x \right> \in F^{-1}$. Then $\left< F(x), x \right> \in F^{-1}$.
Since $F^{-1}$ is single-valued, $F^{-1}(F(x)) = x$ is well-defined. Since $F^{-1}$ is single-valued, $F^{-1}(F(x)) = x$.
For all $y \in \ran{F}$, $\left< y, F^{-1}(y) \right> \in F^{-1}$. For all $y \in \ran{F}$, $\left< y, F^{-1}(y) \right> \in F^{-1}$.
Then $\left< F^{-1}(y), y \right> \in F$. Then $\left< F^{-1}(y), y \right> \in F$.
Since $F$ is single-valued, $F(F^{-1}(y)) = y$ is also well-defined. Since $F$ is single-valued, $F(F^{-1}(y)) = y$.
\end{proof} \end{proof}

84
Bookshelf/Enderton/Set/Chapter_3.lean
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@ -432,86 +432,22 @@ theorem exercise_6_9_ii {A : Set (Set.Relation α)}
intro _ y hy R hR intro _ y hy R hR
exact ⟨y, hy R hR⟩ exact ⟨y, hy R hR⟩
/-- ### Theorem 3E (i) /-- ### Theorem 3G (i)
For a set `F`, `dom F⁻¹ = ran F`. Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
-/ -/
theorem theorem_3e_i {F : Set.Relation α} theorem theorem_3g_i {F : Set.Relation α}
: Set.Relation.dom (F.inv) = Set.Relation.ran F := by (hF : Set.Relation.isOneToOne F) (hx : x ∈ F.dom)
ext x : ∃! y, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
simp only [
Prod.exists,
Set.mem_image,
Set.mem_setOf_eq,
Prod.mk.injEq,
exists_and_right,
exists_eq_right
]
apply Iff.intro
· intro ⟨y, a, _, h⟩
rw [← h.right.left]
exact ⟨a, h.left⟩
· intro ⟨y, hy⟩
exact ⟨y, y, x, hy, rfl, rfl⟩
/-- ### Theorem 3E (ii)
For a set `F`, `ran F⁻¹ = dom F`.
-/
theorem theorem_3e_ii {F : Set.Relation α}
: Set.Relation.ran (F.inv) = Set.Relation.dom F := by
ext x
unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
simp only [
Prod.exists,
Set.mem_image,
Set.mem_setOf_eq,
Prod.mk.injEq,
exists_eq_right,
exists_and_right
]
apply Iff.intro
· intro ⟨a, y, b, h⟩
rw [← h.right.right]
exact ⟨b, h.left⟩
· intro ⟨y, hy⟩
exact ⟨y, x, y, hy, rfl, rfl⟩
/-- ### Theorem 3E (iii)
For a set `F`, `(F⁻¹)⁻¹ = F`.
-/
theorem theorem_3e_iii {F : Set.Relation α}
: F.inv.inv = F := by
unfold Set.Relation.inv
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
ext x
apply Iff.intro
· intro hx
have ⟨a₁, b₁, ⟨⟨a₂, b₂, h₁⟩, h₂⟩⟩ := hx
rw [← h₂, ← h₁.right.right, ← h₁.right.left]
exact h₁.left
· intro hx
have (p, q) := x
refine ⟨q, p, ⟨?_, ?_⟩⟩
· exact ⟨p, q, hx, rfl, rfl⟩
· rfl
/-- ### Theorem 3F (i)
For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted.
-/
theorem theorem_3f_i {F : Set.Relation α}
: Set.Relation.isSingleValued F.inv ↔ Set.Relation.isSingleRooted F := by
sorry sorry
/-- ### Theorem 3F (ii) /-- ### Theorem 3G (ii)
For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted. Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`.
-/ -/
theorem theorem_3f_ii {F : Set.Relation α} theorem theorem_3g_ii {F : Set.Relation α}
: Set.Relation.isSingleValued F ↔ Set.Relation.isSingleRooted F.inv := by (hF : Set.Relation.isOneToOne F) (hy : x ∈ F.ran)
: ∃! x, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
sorry sorry
end Enderton.Set.Chapter_3 end Enderton.Set.Chapter_3

191
Common/Set/Relation.lean
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@ -22,16 +22,52 @@ abbrev Relation (α : Type _) := Set (α × α)
namespace Relation namespace Relation
/-! ## Domain and Range -/
/-- /--
The domain of a `Relation`. The domain of a `Relation`.
-/ -/
def dom (R : Relation α) : Set α := Prod.fst '' R def dom (R : Relation α) : Set α := Prod.fst '' R
/--
The first component of any pair in a `Relation` must be a member of the
`Relation`'s domain.
-/
theorem mem_pair_imp_fst_mem_dom {R : Relation α} (h : (x, y) ∈ R)
: x ∈ dom R := by
unfold dom Prod.fst
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right]
exact ⟨y, h⟩
/--
If `x ∈ dom R`, there exists some `y` such that `⟨x, y⟩ ∈ R`.
-/
theorem dom_exists {R : Relation α} (hx : x ∈ R.dom)
: ∃ y, (x, y) ∈ R := by
unfold dom at hx
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] at hx
exact hx
/-- /--
The range of a `Relation`. The range of a `Relation`.
-/ -/
def ran (R : Relation α) : Set α := Prod.snd '' R def ran (R : Relation α) : Set α := Prod.snd '' R
theorem mem_pair_imp_snd_mem_ran {R : Relation α} (h : (x, y) ∈ R)
: y ∈ ran R := by
unfold ran Prod.snd
simp only [mem_image, Prod.exists, exists_eq_right]
exact ⟨x, h⟩
/--
If `x ∈ ran R`, there exists some `t` such that `⟨t, x⟩ ∈ R`.
-/
theorem ran_exists {R : Relation α} (hx : x ∈ R.ran)
: ∃ t, (t, x) ∈ R := by
unfold ran at hx
simp only [mem_image, Prod.exists, exists_eq_right] at hx
exact hx
/-- /--
The field of a `Relation`. The field of a `Relation`.
-/ -/
@ -42,24 +78,113 @@ The inverse of a `Relation`.
-/ -/
def inv (R : Relation α) : Relation α := { (p.2, p.1) | p ∈ R } def inv (R : Relation α) : Relation α := { (p.2, p.1) | p ∈ R }
/--
`(x, y)` is a member of relation `R` **iff** `(y, x)` is a member of `R⁻¹`.
-/
@[simp]
theorem mem_self_comm_mem_inv {R : Relation α}
: (y, x) ∈ R.inv ↔ (x, y) ∈ R := by
unfold inv
simp only [Prod.exists, mem_setOf_eq, Prod.mk.injEq]
apply Iff.intro
· intro ⟨x', y', hxy⟩
rw [← hxy.right.left, ← hxy.right.right]
exact hxy.left
· intro hxy
exact ⟨x, y, hxy, rfl, rfl⟩
/--
The inverse of the inverse of a `Relation` is the `Relation`.
-/
@[simp]
theorem inv_inv_eq_self (R : Relation α)
: R.inv.inv = R := by
unfold Set.Relation.inv
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
ext x
apply Iff.intro
· intro hx
have ⟨a₁, b₁, ⟨⟨a₂, b₂, h₁⟩, h₂⟩⟩ := hx
rw [← h₂, ← h₁.right.right, ← h₁.right.left]
exact h₁.left
· intro hx
have (p, q) := x
refine ⟨q, p, ⟨?_, ?_⟩⟩
· exact ⟨p, q, hx, rfl, rfl⟩
· rfl
/--
For a set `F`, `dom F⁻¹ = ran F`.
-/
@[simp]
theorem dom_inv_eq_ran_self {F : Set.Relation α}
: Set.Relation.dom (F.inv) = Set.Relation.ran F := by
ext x
unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
simp only [
Prod.exists,
Set.mem_image,
Set.mem_setOf_eq,
Prod.mk.injEq,
exists_and_right,
exists_eq_right
]
apply Iff.intro
· intro ⟨y, a, _, h⟩
rw [← h.right.left]
exact ⟨a, h.left⟩
· intro ⟨y, hy⟩
exact ⟨y, y, x, hy, rfl, rfl⟩
/--
For a set `F`, `ran F⁻¹ = dom F`.
-/
@[simp]
theorem ran_inv_eq_dom_self {F : Set.Relation α}
: Set.Relation.ran (F.inv) = Set.Relation.dom F := by
ext x
unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
simp only [
Prod.exists,
Set.mem_image,
Set.mem_setOf_eq,
Prod.mk.injEq,
exists_eq_right,
exists_and_right
]
apply Iff.intro
· intro ⟨a, y, b, h⟩
rw [← h.right.right]
exact ⟨b, h.left⟩
· intro ⟨y, hy⟩
exact ⟨y, x, y, hy, rfl, rfl⟩
/-! ## Composition -/
/-- /--
The composition of two `Relation`s. The composition of two `Relation`s.
-/ -/
def comp (F G : Relation α) : Relation α := def comp (F G : Relation α) : Relation α :=
{ p | ∃ t, (p.1, t) ∈ G ∧ (t, p.2) ∈ F} { p | ∃ t, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
/-! ## Restriction -/
/-- /--
The restriction of a `Relation` to a `Set`. The restriction of a `Relation` to a `Set`.
-/ -/
def restriction (R : Relation α) (A : Set α) : Relation α := def restriction (R : Relation α) (A : Set α) : Relation α :=
{ p ∈ R | p.1 ∈ A } { p ∈ R | p.1 ∈ A }
/-! ## Image -/
/-- /--
The image of a `Relation` under a `Set`. The image of a `Relation` under a `Set`.
-/ -/
def image (R : Relation α) (A : Set α) : Set α := def image (R : Relation α) (A : Set α) : Set α :=
{ y | ∃ u ∈ A, (u, y) ∈ R } { y | ∃ u ∈ A, (u, y) ∈ R }
/-! ## Single-Rooted and Single-Valued -/
/-- /--
A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, there A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, there
exists exactly one `x` such that `⟨x, y⟩ ∈ R`. exists exactly one `x` such that `⟨x, y⟩ ∈ R`.
@ -76,6 +201,72 @@ Notice, a `Relation` that is single-valued is a function.
def isSingleValued (R : Relation α) : Prop := def isSingleValued (R : Relation α) : Prop :=
∀ x ∈ R.dom, ∃! y, y ∈ R.ran ∧ (x, y) ∈ R ∀ x ∈ R.dom, ∃! y, y ∈ R.ran ∧ (x, y) ∈ R
/--
For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted.
-/
theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
: isSingleValued F.inv ↔ isSingleRooted F := by
apply Iff.intro
· intro hF
unfold isSingleValued at hF
simp only [
dom_inv_eq_ran_self,
ran_inv_eq_dom_self,
mem_self_comm_mem_inv
] at hF
suffices ∀ x ∈ F.ran, ∃! y, (y, x) ∈ F from hF
intro x hx
have ⟨y, hy⟩ := hF x hx
simp only [
ran_inv_eq_dom_self,
mem_self_comm_mem_inv,
and_imp
] at hy
refine ⟨y, hy.left.right, ?_⟩
intro y₁ hy₁
exact hy.right y₁ (mem_pair_imp_fst_mem_dom hy₁) hy₁
· intro hF
unfold isSingleRooted at hF
unfold isSingleValued
simp only [
dom_inv_eq_ran_self,
ran_inv_eq_dom_self,
mem_self_comm_mem_inv
]
exact hF
/--
For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted.
-/
theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α}
: Set.Relation.isSingleValued F ↔ Set.Relation.isSingleRooted F.inv := by
conv => lhs; rw [← inv_inv_eq_self F]
rw [single_valued_inv_iff_single_rooted_self]
/--
A `Relation` `R` is one-to-one if it is a single-rooted function.
-/
def isOneToOne (R : Relation α) : Prop :=
isSingleValued R ∧ isSingleRooted R
/--
A `Relation` is one-to-one **iff** it's inverse is.
-/
theorem one_to_one_self_iff_one_to_one_inv {R : Relation α}
: isOneToOne R ↔ isOneToOne R.inv := by
unfold isOneToOne isSingleValued isSingleRooted
conv => rhs; simp only [
dom_inv_eq_ran_self,
ran_inv_eq_dom_self,
mem_self_comm_mem_inv,
eq_iff_iff
]
apply Iff.intro <;>
· intro ⟨hx, hy⟩
exact ⟨hy, hx⟩
/-! ## Ordered Pairs -/
/-- /--
Convert a `Relation` into an equivalent representation using `OrderedPair`s. Convert a `Relation` into an equivalent representation using `OrderedPair`s.
-/ -/