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Enderton. Heterogeneous relations and additional exercises.

finite-set-exercises
Joshua Potter 2023-07-07 06:15:17 -06:00
parent 1c6fec389f
commit 04fe6c66db
3 changed files with 438 additions and 195 deletions
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58
Bookshelf/Enderton/Set.tex
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@ -4491,7 +4491,7 @@ Show that for any sets $B$ and $C$,
\end{proof} \end{proof}
\subsection{\pending{Exercise 3.24}}% \subsection{\verified{Exercise 3.24}}%
\label{sub:exercise-3.24} \label{sub:exercise-3.24}
Show that for a function $F$, Show that for a function $F$,
@ -4499,6 +4499,9 @@ Show that for a function $F$,
\begin{proof} \begin{proof}
\lean{Bookshelf/Enderton/Set/Relation}
{Enderton.Set.Chapter\_3.exercise\_3\_24}
Let $F$ be a function. Let $F$ be a function.
By definition of the \nameref{ref:inverse} of a set, By definition of the \nameref{ref:inverse} of a set,
\begin{align*} \begin{align*}
@ -4506,13 +4509,13 @@ Show that for a function $F$,
& = \{x \mid (\exists y \in A) yF^{-1}x\} \\ & = \{x \mid (\exists y \in A) yF^{-1}x\} \\
& = \{x \mid (\exists y \in A) xFy\} \\ & = \{x \mid (\exists y \in A) xFy\} \\
& = \{x \mid (\exists y \in A) \left< x, y \right> \in F\} \\ & = \{x \mid (\exists y \in A) \left< x, y \right> \in F\} \\
& = \{x \mid \exists x \in F, F(x) \in A\} \\ & = \{x \mid x \in \dom{F} \land F(x) \in A\} \\
& = \{x \in \dom{F} \mid F(x) \in A\}. & = \{x \in \dom{F} \mid F(x) \in A\}.
\end{align*} \end{align*}
\end{proof} \end{proof}
\subsection{\pending{Exercise 3.25}}% \subsection{\verified{Exercise 3.25}}%
\label{sub:exercise-3.25} \label{sub:exercise-3.25}
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
@ -4525,18 +4528,45 @@ Show that for a function $F$,
\begin{proof} \begin{proof}
\statementpadding
\lean*{Bookshelf/Enderton/Set/Relation}
{Enderton.Set.Chapter\_3.exercise\_3\_25\_b}
\lean{Bookshelf/Enderton/Set/Relation}
{Enderton.Set.Chapter\_3.exercise\_3\_25\_a}
\paragraph{(b)}% \paragraph{(b)}%
\label{par:exercise-3.25-b} \label{par:exercise-3.25-b}
Let $G$ be a function. Let $G$ be an arbitrary function.
We show that $G \circ G^{-1} \subseteq I_{\ran{G}}$ and that
$I_{\ran{G}} \subseteq G \circ G^{-1}$.
\subparagraph{($\subseteq$)}%
Let $\left< x, y \right> \in G \circ G^{-1}$. Let $\left< x, y \right> \in G \circ G^{-1}$.
By definition of the \nameref{ref:composition} of sets, there exists some By definition of the \nameref{ref:composition} of sets, there exists some
set $t$ such that $x(G^{-1})t$ and $tGy$. set $t$ such that $x(G^{-1})t$ and $tGy$.
By definition of the \nameref{ref:inverse} of a set, By definition of the \nameref{ref:inverse} of a set,
$$x(G^{-1})t \iff tGx.$$ $$x(G^{-1})t \iff tGx.$$
By hypothesis, $G$ is single-valued. The right hand side of the above biconditional indicates $x \in \ran{G}$.
Thus $x = y$. Since $G$ is single-valued, $tGy \land tGx$ implies $x = y$.
That is, $G \circ G^{-1}$ is the identity function on $\ran{G}$. Thus $\left< x, y \right> \in I_{\ran{G}}$.
\subparagraph{($\supseteq$)}%
Let $\left< x, x \right> \in I_{\ran{G}}$ where $x \in \ran{G}$.
By definition of the \nameref{ref:range} of a function, there exists some
$t$ such that $\left< t, x \right> \in G$.
By definition of the \nameref{ref:inverse} of a set, it follows
$\left< x, t \right> \in G^{-1}$.
Thus $\left< x, x \right> \in G \circ G^{-1}$.
\subparagraph{Conclusion}%
Since $G \circ G^{-1}$ is a subset of $I_{\ran{G}}$ and vice versa, it
follows that these two sets are equal.
\paragraph{(a)}% \paragraph{(a)}%
@ -4544,7 +4574,7 @@ Show that for a function $F$,
\end{proof} \end{proof}
\subsection{\pending{Exercise 3.26}}% \subsection{\verified{Exercise 3.26}}%
\label{sub:exercise-3.26} \label{sub:exercise-3.26}
Prove the second halves of parts (a) and (b) of Theorem 3K. Prove the second halves of parts (a) and (b) of Theorem 3K.
@ -4556,7 +4586,7 @@ Prove the second halves of parts (a) and (b) of Theorem 3K.
\end{proof} \end{proof}
\subsection{\pending{Exercise 3.27}}% \subsection{\verified{Exercise 3.27}}%
\label{sub:exercise-3.27} \label{sub:exercise-3.27}
Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$. Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
@ -4564,6 +4594,9 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
\begin{proof} \begin{proof}
\lean{Bookshelf/Enderton/Set/Relation}
{Enderton.Set.Chapter\_3.exercise\_3\_27}
Let $F$ and $G$ be arbitrary sets. Let $F$ and $G$ be arbitrary sets.
We show that each side of our desired equality is a subset of the other. We show that each side of our desired equality is a subset of the other.
@ -4597,7 +4630,7 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
\end{proof} \end{proof}
\subsection{\pending{Exercise 3.28}}% \subsection{\verified{Exercise 3.28}}%
\label{sub:exercise-3.28} \label{sub:exercise-3.28}
Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the
@ -4607,9 +4640,14 @@ Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$.
\begin{proof} \begin{proof}
\lean{Bookshelf/Enderton/Set/Relation}
{Enderton.Set.Chapter\_3.exercise\_3\_28}
By construction, $\dom{G} = \powerset{A}$. By construction, $\dom{G} = \powerset{A}$.
Likewise, $\ran{G} \subseteq \powerset{B}$ by definition of the Likewise, $\ran{G} \subseteq \powerset{B}$ by definition of the
\nameref{ref:image} of sets. \nameref{ref:image} of sets.
Thus $G$ maps $\powerset{A}$ into $\powerset{B}$.
Let $y \in \ran{G}$. Let $y \in \ran{G}$.
Then there exists an $X_1 \in \powerset{A}$ such that $\img{f}{X_1} = y$. Then there exists an $X_1 \in \powerset{A}$ such that $\img{f}{X_1} = y$.
To prove $G$ is one-to-one into $\powerset{B}$, assume there exists an To prove $G$ is one-to-one into $\powerset{B}$, assume there exists an

435
Bookshelf/Enderton/Set/Chapter_3.lean
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@ -229,12 +229,16 @@ theorem theorem_3d {A : Set (Set (Set α))} (h : OrderedPair x y ∈ A)
have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq
exact ⟨this (by simp), this (by simp)⟩ exact ⟨this (by simp), this (by simp)⟩
section Relation
open Set.Relation
/-- #### Exercise 3.6 /-- #### Exercise 3.6
Show that a set `A` is a relation **iff** `A ⊆ dom A × ran A`. Show that a set `A` is a relation **iff** `A ⊆ dom A × ran A`.
-/ -/
theorem exercise_3_6 {A : Set.Relation α} theorem exercise_3_6 {A : Set.HRelation α β}
: A ⊆ Set.prod (A.dom) (A.ran) := by : A ⊆ Set.prod (dom A) (ran A) := by
show ∀ t, t ∈ A → t ∈ Set.prod (Prod.fst '' A) (Prod.snd '' A) show ∀ t, t ∈ A → t ∈ Set.prod (Prod.fst '' A) (Prod.snd '' A)
intro (a, b) ht intro (a, b) ht
unfold Set.prod unfold Set.prod
@ -301,8 +305,8 @@ theorem exercise_3_7 {R : Set.Relation α}
-- `t = y` then `t ∈ ran R`. -- `t = y` then `t ∈ ran R`.
have hxy_mem : t = x ∨ t = y → t ∈ Set.Relation.fld R := by have hxy_mem : t = x ∨ t = y → t ∈ Set.Relation.fld R := by
intro ht intro ht
have hz : R ⊆ Set.prod (R.dom) (R.ran) := exercise_3_6 have hz : R ⊆ Set.prod (dom R) (ran R) := exercise_3_6
have : (x, y) ∈ Set.prod (R.dom) (R.ran) := hz p have : (x, y) ∈ Set.prod (dom R) (ran R) := hz p
unfold Set.prod at this unfold Set.prod at this
simp at this simp at this
apply Or.elim ht apply Or.elim ht
@ -326,10 +330,6 @@ theorem exercise_3_7 {R : Set.Relation α}
simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
exact hxy_mem this exact hxy_mem this
section Relation
open Set.Relation
/-- #### Exercise 3.8 (i) /-- #### Exercise 3.8 (i)
Show that for any set `𝓐`: Show that for any set `𝓐`:
@ -337,7 +337,7 @@ Show that for any set `𝓐`:
dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 } dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
``` ```
-/ -/
theorem exercise_3_8_i {A : Set (Set.Relation α)} theorem exercise_3_8_i {A : Set (Set.HRelation α β)}
: dom (⋃₀ A) = ⋃₀ { dom R | R ∈ A } := by : dom (⋃₀ A) = ⋃₀ { dom R | R ∈ A } := by
ext x ext x
unfold dom Prod.fst unfold dom Prod.fst
@ -363,7 +363,7 @@ Show that for any set `𝓐`:
ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 } ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
``` ```
-/ -/
theorem exercise_3_8_ii {A : Set (Set.Relation α)} theorem exercise_3_8_ii {A : Set (Set.HRelation α β)}
: ran (⋃₀ A) = ⋃₀ { ran R | R ∈ A } := by : ran (⋃₀ A) = ⋃₀ { ran R | R ∈ A } := by
ext x ext x
unfold ran Prod.snd unfold ran Prod.snd
@ -389,7 +389,7 @@ operation in the preceding problem.
dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 } dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
``` ```
-/ -/
theorem exercise_3_9_i {A : Set (Set.Relation α)} theorem exercise_3_9_i {A : Set (Set.HRelation α β)}
: dom (⋂₀ A) ⊆ ⋂₀ { dom R | R ∈ A } := by : dom (⋂₀ A) ⊆ ⋂₀ { dom R | R ∈ A } := by
show ∀ x, x ∈ dom (⋂₀ A) → x ∈ ⋂₀ { dom R | R ∈ A } show ∀ x, x ∈ dom (⋂₀ A) → x ∈ ⋂₀ { dom R | R ∈ A }
unfold dom Prod.fst unfold dom Prod.fst
@ -415,7 +415,7 @@ operation in the preceding problem.
ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 } ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
``` ```
-/ -/
theorem exercise_3_9_ii {A : Set (Set.Relation α)} theorem exercise_3_9_ii {A : Set (Set.HRelation α β)}
: ran (⋂₀ A) ⊆ ⋂₀ { ran R | R ∈ A } := by : ran (⋂₀ A) ⊆ ⋂₀ { ran R | R ∈ A } := by
show ∀ x, x ∈ ran (⋂₀ A) → x ∈ ⋂₀ { ran R | R ∈ A } show ∀ x, x ∈ ran (⋂₀ A) → x ∈ ⋂₀ { ran R | R ∈ A }
unfold ran Prod.snd unfold ran Prod.snd
@ -437,9 +437,9 @@ theorem exercise_3_9_ii {A : Set (Set.Relation α)}
Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`. Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
-/ -/
theorem theorem_3g_i {F : Set.Relation α} theorem theorem_3g_i {F : Set.HRelation α β}
(hF : F.isOneToOne) (hx : x ∈ dom F) (hF : isOneToOne F) (hx : x ∈ dom F)
: ∃! y, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by : ∃! y, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
simp only [mem_self_comm_mem_inv, and_self] simp only [mem_self_comm_mem_inv, and_self]
have ⟨y, hy⟩ := dom_exists hx have ⟨y, hy⟩ := dom_exists hx
refine ⟨y, hy, ?_⟩ refine ⟨y, hy, ?_⟩
@ -451,9 +451,9 @@ theorem theorem_3g_i {F : Set.Relation α}
Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`. Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`.
-/ -/
theorem theorem_3g_ii {F : Set.Relation α} theorem theorem_3g_ii {F : Set.HRelation α β}
(hF : F.isOneToOne) (hy : y ∈ F.ran) (hF : isOneToOne F) (hy : y ∈ ran F)
: ∃! x, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by : ∃! x, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
simp only [mem_self_comm_mem_inv, and_self] simp only [mem_self_comm_mem_inv, and_self]
have ⟨x, hx⟩ := ran_exists hy have ⟨x, hx⟩ := ran_exists hy
refine ⟨x, hx, ?_⟩ refine ⟨x, hx, ?_⟩
@ -468,13 +468,13 @@ Assume that `F` and `G` are functions. Then
dom (F ∘ G) = {x ∈ dom G | G(x) ∈ dom F}. dom (F ∘ G) = {x ∈ dom G | G(x) ∈ dom F}.
``` ```
-/ -/
theorem theorem_3h_dom {F G : Set.Relation α} theorem theorem_3h_dom {F : Set.HRelation β γ} {G : Set.HRelation α β}
(_ : F.isSingleValued) (hG : G.isSingleValued) (_ : isSingleValued F) (hG : isSingleValued G)
: dom (F.comp G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F} := by : dom (comp F G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F} := by
let rhs := {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F } let rhs := {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F }
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
apply And.intro apply And.intro
· show ∀ t, t ∈ dom (F.comp G) → t ∈ rhs · show ∀ t, t ∈ dom (comp F G) → t ∈ rhs
intro t ht intro t ht
simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq]
have ⟨z, hz⟩ := dom_exists ht have ⟨z, hz⟩ := dom_exists ht
@ -488,7 +488,7 @@ theorem theorem_3h_dom {F G : Set.Relation α}
refine ⟨a, ⟨ha.left, z, ha.right⟩, ?_⟩ refine ⟨a, ⟨ha.left, z, ha.right⟩, ?_⟩
intro y₁ hy₁ intro y₁ hy₁
exact fun _ _ => single_valued_eq_unique hG hy₁ ha.left exact fun _ _ => single_valued_eq_unique hG hy₁ ha.left
· show ∀ t, t ∈ rhs → t ∈ dom (F.comp G) · show ∀ t, t ∈ rhs → t ∈ dom (comp F G)
intro t ht intro t ht
simp only [Set.mem_setOf_eq] at ht simp only [Set.mem_setOf_eq] at ht
unfold dom unfold dom
@ -507,15 +507,15 @@ Assume that `F : A → B`, and that `A` is nonempty. There exists a function
`G : B → A` (a "left inverse") such that `G ∘ F` is the identity function on `A` `G : B → A` (a "left inverse") such that `G ∘ F` is the identity function on `A`
**iff** `F` is one-to-one. **iff** `F` is one-to-one.
-/ -/
theorem theorem_3j_a {F : Set.Relation α} {A B : Set α} theorem theorem_3j_a {F : Set.HRelation α β} {A : Set α} {B : Set β}
(hF : F.isSingleValued ∧ F.mapsInto A B) (hA : Set.Nonempty A) (hF : isSingleValued F ∧ mapsInto F A B) (hA : Set.Nonempty A)
: (∃ G : Set.Relation α, : (∃ G : Set.HRelation β α,
G.isSingleValued ∧ G.mapsInto B A ∧ isSingleValued G ∧ mapsInto G B A ∧
(∀ p ∈ G.comp F, p.1 = p.2)) ↔ F.isOneToOne := by (∀ p ∈ comp G F, p.1 = p.2)) ↔ isOneToOne F := by
apply Iff.intro apply Iff.intro
· intro ⟨G, ⟨hG₁, hG₂, hI⟩⟩ · intro ⟨G, ⟨hG₁, hG₂, hI⟩⟩
refine ⟨hF.left, ?_⟩ refine ⟨hF.left, ?_⟩
show F.isSingleRooted show isSingleRooted F
intro y hy intro y hy
have ⟨x, hx⟩ := ran_exists hy have ⟨x, hx⟩ := ran_exists hy
sorry sorry
@ -528,11 +528,11 @@ 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 `H : B → A` (a "right inverse") such that `F ∘ H` is the identity function on
`B` **iff** `F` maps `A` onto `B`. `B` **iff** `F` maps `A` onto `B`.
-/ -/
theorem theorem_3j_b {F : Set.Relation α} {A B : Set α} theorem theorem_3j_b {F : Set.HRelation α β} {A : Set α} {B : Set β}
(hF : F.isSingleValued ∧ F.mapsInto A B) (hA : Set.Nonempty A) (hF : isSingleValued F ∧ mapsInto F A B) (hA : Set.Nonempty A)
: (∃ H : Set.Relation α, : (∃ H : Set.HRelation β α,
H.isSingleValued ∧ H.mapsInto B A ∧ isSingleValued H ∧ mapsInto H B A ∧
(∀ p ∈ F.comp H, p.1 = p.2)) ↔ F.mapsOnto A B := by (∀ p ∈ comp F H, p.1 = p.2)) ↔ mapsOnto F A B := by
sorry sorry
/-- #### Theorem 3K (a) /-- #### Theorem 3K (a)
@ -543,11 +543,11 @@ The image of a union is the union of the images:
F⟦⋃ 𝓐⟧ = ⋃ {F⟦A⟧ | A ∈ 𝓐} F⟦⋃ 𝓐⟧ = ⋃ {F⟦A⟧ | A ∈ 𝓐}
``` ```
-/ -/
theorem theorem_3k_a {F : Set.Relation α} {𝓐 : Set (Set α)} theorem theorem_3k_a {F : Set.HRelation α β} {𝓐 : Set (Set α)}
: F.image (⋃₀ 𝓐) = ⋃₀ { F.image A | A ∈ 𝓐 } := by : image F (⋃₀ 𝓐) = ⋃₀ { image F A | A ∈ 𝓐 } := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
apply And.intro apply And.intro
· show ∀ v, v ∈ F.image (⋃₀ 𝓐) → v ∈ ⋃₀ { F.image A | A ∈ 𝓐 } · show ∀ v, v ∈ image F (⋃₀ 𝓐) → v ∈ ⋃₀ { image F A | A ∈ 𝓐 }
intro v hv intro v hv
unfold image at hv unfold image at hv
simp only [Set.mem_sUnion, Set.mem_setOf_eq] at hv simp only [Set.mem_sUnion, Set.mem_setOf_eq] at hv
@ -555,11 +555,11 @@ theorem theorem_3k_a {F : Set.Relation α} {𝓐 : Set (Set α)}
have ⟨A, hA⟩ := hu.left have ⟨A, hA⟩ := hu.left
simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and] simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and]
refine ⟨A, hA.left, ?_⟩ refine ⟨A, hA.left, ?_⟩
show v ∈ F.image A show v ∈ image F A
unfold image unfold image
simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq]
exact ⟨u, hA.right, hu.right⟩ exact ⟨u, hA.right, hu.right⟩
· show ∀ v, v ∈ ⋃₀ {x | ∃ A, A ∈ 𝓐 ∧ F.image A = x} → v ∈ F.image (⋃₀ 𝓐) · show ∀ v, v ∈ ⋃₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋃₀ 𝓐)
intro v hv intro v hv
simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and] at hv simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and] at hv
have ⟨A, hA⟩ := hv have ⟨A, hA⟩ := hv
@ -580,9 +580,9 @@ F⟦⋂ 𝓐⟧ ⊆ ⋂ {F⟦A⟧ | A ∈ 𝓐}
Equality holds if `F` is single-rooted. Equality holds if `F` is single-rooted.
-/ -/
theorem theorem_3k_b_i {F : Set.Relation α} {𝓐 : Set (Set α)} theorem theorem_3k_b_i {F : Set.HRelation α β} {𝓐 : Set (Set α)}
: F.image (⋂₀ 𝓐) ⊆ ⋂₀ { F.image A | A ∈ 𝓐} := by : image F (⋂₀ 𝓐) ⊆ ⋂₀ { image F A | A ∈ 𝓐} := by
show ∀ v, v ∈ F.image (⋂₀ 𝓐) → v ∈ ⋂₀ { F.image A | A ∈ 𝓐} show ∀ v, v ∈ image F (⋂₀ 𝓐) → v ∈ ⋂₀ { image F A | A ∈ 𝓐}
intro v hv intro v hv
unfold image at hv unfold image at hv
simp only [Set.mem_sInter, Set.mem_setOf_eq] at hv simp only [Set.mem_sInter, Set.mem_setOf_eq] at hv
@ -599,9 +599,9 @@ theorem theorem_3k_b_i {F : Set.Relation α} {𝓐 : Set (Set α)}
simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq]
exact ⟨u, hu.left A hA, hu.right⟩ exact ⟨u, hu.left A hA, hu.right⟩
theorem theorem_3k_b_ii {F : Set.Relation α} {𝓐 : Set (Set α)} theorem theorem_3k_b_ii {F : Set.HRelation α β} {𝓐 : Set (Set α)}
(hF : F.isSingleRooted) (h𝓐 : Set.Nonempty 𝓐) (hF : isSingleRooted F) (h𝓐 : Set.Nonempty 𝓐)
: F.image (⋂₀ 𝓐) = ⋂₀ { F.image A | A ∈ 𝓐} := by : image F (⋂₀ 𝓐) = ⋂₀ { image F A | A ∈ 𝓐} := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
refine ⟨theorem_3k_b_i, ?_⟩ refine ⟨theorem_3k_b_i, ?_⟩
show ∀ v, v ∈ ⋂₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋂₀ 𝓐) show ∀ v, v ∈ ⋂₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋂₀ 𝓐)
@ -640,9 +640,9 @@ F⟦A⟧ - F⟦B⟧ ⊆ F⟦A - B⟧.
Equality holds if `F` is single-rooted. Equality holds if `F` is single-rooted.
-/ -/
theorem theorem_3k_c_i {F : Set.Relation α} {A B : Set α} theorem theorem_3k_c_i {F : Set.HRelation α β} {A B : Set α}
: F.image A \ F.image B ⊆ F.image (A \ B) := by : image F A \ image F B ⊆ image F (A \ B) := by
show ∀ v, v ∈ F.image A \ F.image B → v ∈ F.image (A \ B) show ∀ v, v ∈ image F A \ image F B → v ∈ image F (A \ B)
intro v hv intro v hv
have hv' : v ∈ image F A ∧ v ∉ image F B := hv have hv' : v ∈ image F A ∧ v ∉ image F B := hv
conv at hv' => arg 1; unfold image; simp only [Set.mem_setOf_eq, eq_iff_iff] conv at hv' => arg 1; unfold image; simp only [Set.mem_setOf_eq, eq_iff_iff]
@ -660,9 +660,9 @@ theorem theorem_3k_c_i {F : Set.Relation α} {A B : Set α}
simp only [Set.mem_diff, Set.mem_setOf_eq] simp only [Set.mem_diff, Set.mem_setOf_eq]
exact ⟨u, ⟨hu.left, hu'⟩, hu.right⟩ exact ⟨u, ⟨hu.left, hu'⟩, hu.right⟩
theorem theorem_3k_c_ii {F : Set.Relation α} {A B : Set α} theorem theorem_3k_c_ii {F : Set.HRelation α β} {A B : Set α}
(hF : F.isSingleRooted) (hF : isSingleRooted F)
: F.image A \ F.image B = F.image (A \ B) := by : image F A \ image F B = image F (A \ B) := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
refine ⟨theorem_3k_c_i, ?_⟩ refine ⟨theorem_3k_c_i, ?_⟩
show ∀ v, v ∈ image F (A \ B) → v ∈ image F A \ image F B show ∀ v, v ∈ image F (A \ B) → v ∈ image F A \ image F B
@ -670,11 +670,11 @@ theorem theorem_3k_c_ii {F : Set.Relation α} {A B : Set α}
unfold image at hv unfold image at hv
simp only [Set.mem_diff, Set.mem_setOf_eq] at hv simp only [Set.mem_diff, Set.mem_setOf_eq] at hv
have ⟨u, hu⟩ := hv have ⟨u, hu⟩ := hv
have hv₁ : v ∈ F.image A := by have hv₁ : v ∈ image F A := by
unfold image unfold image
simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq]
exact ⟨u, hu.left.left, hu.right⟩ exact ⟨u, hu.left.left, hu.right⟩
have hv₂ : v ∉ F.image B := by have hv₂ : v ∉ image F B := by
intro nv intro nv
unfold image at nv unfold image at nv
simp only [Set.mem_setOf_eq] at nv simp only [Set.mem_setOf_eq] at nv
@ -695,20 +695,20 @@ G⁻¹⟦A - B⟧ = G⁻¹⟦A⟧ - G⁻¹⟦B⟧.
``` ```
-/ -/
theorem corollary_3l_i {G : Set.Relation α} {𝓐 : Set (Set α)} theorem corollary_3l_i {G : Set.HRelation β α} {𝓐 : Set (Set α)}
: G.inv.image (⋃₀ 𝓐) = ⋃₀ {G.inv.image A | A ∈ 𝓐} := theorem_3k_a : image (inv G) (⋃₀ 𝓐) = ⋃₀ {image (inv G) A | A ∈ 𝓐} := theorem_3k_a
theorem corollary_3l_ii {G : Set.Relation α} {𝓐 : Set (Set α)} theorem corollary_3l_ii {G : Set.HRelation β α} {𝓐 : Set (Set α)}
(hG : G.isSingleValued) (h𝓐 : Set.Nonempty 𝓐) (hG : isSingleValued G) (h𝓐 : Set.Nonempty 𝓐)
: G.inv.image (⋂₀ 𝓐) = ⋂₀ {G.inv.image A | A ∈ 𝓐} := by : image (inv G) (⋂₀ 𝓐) = ⋂₀ {image (inv G) A | A ∈ 𝓐} := by
have hG' : G.inv.isSingleRooted := have hG' : isSingleRooted (inv G) :=
single_valued_self_iff_single_rooted_inv.mp hG single_valued_self_iff_single_rooted_inv.mp hG
exact theorem_3k_b_ii hG' h𝓐 exact theorem_3k_b_ii hG' h𝓐
theorem corollary_3l_iii {G : Set.Relation α} {A B : Set α} theorem corollary_3l_iii {G : Set.HRelation β α} {A B : Set α}
(hG : G.isSingleValued) (hG : isSingleValued G)
: G.inv.image (A \ B) = G.inv.image A \ G.inv.image B := by : image (inv G) (A \ B) = image (inv G) A \ image (inv G) B := by
have hG' : G.inv.isSingleRooted := have hG' : isSingleRooted (inv G) :=
single_valued_self_iff_single_rooted_inv.mp hG single_valued_self_iff_single_rooted_inv.mp hG
exact (theorem_3k_c_ii hG').symm exact (theorem_3k_c_ii hG').symm
@ -719,10 +719,10 @@ Assume that `f` and `g` are functions and show that
f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f) f(x) = g(x). f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f) f(x) = g(x).
``` ```
-/ -/
theorem exercise_3_12 {f g : Set.Relation α} theorem exercise_3_12 {f g : Set.HRelation α β}
(hf : f.isSingleValued) (_ : g.isSingleValued) (hf : isSingleValued f) (_ : isSingleValued g)
: f ⊆ g ↔ dom f ⊆ dom g ∧ : f ⊆ g ↔ dom f ⊆ dom g ∧
(∀ x ∈ dom f, ∃! y : α, (x, y) ∈ f ∧ (x, y) ∈ g) := by (∀ x ∈ dom f, ∃! y : β, (x, y) ∈ f ∧ (x, y) ∈ g) := by
apply Iff.intro apply Iff.intro
· intro h · intro h
apply And.intro apply And.intro
@ -747,8 +747,8 @@ theorem exercise_3_12 {f g : Set.Relation α}
Assume that `f` and `g` are functions with `f ⊆ g` and `dom g ⊆ dom f`. Show Assume that `f` and `g` are functions with `f ⊆ g` and `dom g ⊆ dom f`. Show
that `f = g`. that `f = g`.
-/ -/
theorem exercise_3_13 {f g : Set.Relation α} theorem exercise_3_13 {f g : Set.HRelation α β}
(hf : f.isSingleValued) (hg : g.isSingleValued) (hf : isSingleValued f) (hg : isSingleValued g)
(h : f ⊆ g) (h₁ : dom g ⊆ dom f) (h : f ⊆ g) (h₁ : dom g ⊆ dom f)
: f = g := by : f = g := by
have h₂ := (exercise_3_12 hf hg).mp h have h₂ := (exercise_3_12 hf hg).mp h
@ -770,9 +770,9 @@ theorem exercise_3_13 {f g : Set.Relation α}
Assume that `f` and `g` are functions. Show that `f ∩ g` is a function. Assume that `f` and `g` are functions. Show that `f ∩ g` is a function.
-/ -/
theorem exercise_3_14_a {f g : Set.Relation α} theorem exercise_3_14_a {f g : Set.HRelation α β}
(hf : f.isSingleValued) (_ : g.isSingleValued) (hf : isSingleValued f) (_ : isSingleValued g)
: (f ∩ g).isSingleValued := : isSingleValued (f ∩ g) :=
single_valued_subset hf (Set.inter_subset_left f g) single_valued_subset hf (Set.inter_subset_left f g)
/-- #### Exercise 3.14 (b) /-- #### Exercise 3.14 (b)
@ -780,9 +780,9 @@ theorem exercise_3_14_a {f g : Set.Relation α}
Assume that `f` and `g` are functions. Show that `f ∪ g` is a function **iff** Assume that `f` and `g` are functions. Show that `f ∪ g` is a function **iff**
`f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`. `f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`.
-/ -/
theorem exercise_3_14_b {f g : Set.Relation α} theorem exercise_3_14_b {f g : Set.HRelation α β}
(hf : f.isSingleValued) (hg : g.isSingleValued) (hf : isSingleValued f) (hg : isSingleValued g)
: (f ∪ g).isSingleValued ↔ : isSingleValued (f ∪ g) ↔
(∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by (∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by
apply Iff.intro apply Iff.intro
· intro h x hx · intro h x hx
@ -860,16 +860,16 @@ theorem exercise_3_14_b {f g : Set.Relation α}
Let `𝓐` be a set of functions such that for any `f` and `g` in `𝓐`, either Let `𝓐` be a set of functions such that for any `f` and `g` in `𝓐`, either
`f ⊆ g` or `g ⊆ f`. Show that `⋃ 𝓐` is a function. `f ⊆ g` or `g ⊆ f`. Show that `⋃ 𝓐` is a function.
-/ -/
theorem exercise_3_15 {𝓐 : Set (Set.Relation α)} theorem exercise_3_15 {𝓐 : Set (Set.HRelation α β)}
(h𝓐 : ∀ F ∈ 𝓐, F.isSingleValued) (h𝓐 : ∀ F ∈ 𝓐, isSingleValued F)
(h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G ∨ G ⊆ F) (h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G ∨ G ⊆ F)
: isSingleValued (⋃₀ 𝓐) := by : isSingleValued (⋃₀ 𝓐) := by
intro x hx intro x hx
have ⟨y₁, hy₁⟩ := dom_exists hx have ⟨y₁, hy₁⟩ := dom_exists hx
refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩ refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩
intro y₂ hy₂ intro y₂ hy₂
have ⟨f, hf⟩ : ∃ f : Set.Relation α, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁ have ⟨f, hf⟩ : ∃ f : Set.HRelation α β, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁
have ⟨g, hg⟩ : ∃ g : Set.Relation α, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right have ⟨g, hg⟩ : ∃ g : Set.HRelation α β, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right
apply Or.elim (h f g hf.left hg.left) apply Or.elim (h f g hf.left hg.left)
· intro hf' · intro hf'
have := hf' hf.right have := hf' hf.right
@ -884,9 +884,9 @@ Show that the composition of two single-rooted sets is again single-rooted.
Conclude that the composition of two one-to-one functions is again one-to-one. Conclude that the composition of two one-to-one functions is again one-to-one.
-/ -/
theorem exercise_3_17_i {F G : Set.Relation α} theorem exercise_3_17_i {F : Set.HRelation β γ} {G : Set.HRelation α β}
(hF : F.isSingleRooted) (hG : G.isSingleRooted) (hF : isSingleRooted F) (hG : isSingleRooted G)
: (F.comp G).isSingleRooted := by : isSingleRooted (comp F G):= by
intro v hv intro v hv
have ⟨u₁, hu₁⟩ := ran_exists hv have ⟨u₁, hu₁⟩ := ran_exists hv
@ -907,9 +907,9 @@ theorem exercise_3_17_i {F G : Set.Relation α}
rw [ht] at ht₁ rw [ht] at ht₁
exact single_rooted_eq_unique hG ht₂.left ht₁.left exact single_rooted_eq_unique hG ht₂.left ht₁.left
theorem exercise_3_17_ii {F G : Set.Relation α} theorem exercise_3_17_ii {F : Set.HRelation β γ} {G : Set.HRelation α β}
(hF : F.isOneToOne) (hG : G.isOneToOne) (hF : isOneToOne F) (hG : isOneToOne G)
: (F.comp G).isOneToOne := And.intro : isOneToOne (comp F G) := And.intro
(single_valued_comp_is_single_valued hF.left hG.left) (single_valued_comp_is_single_valued hF.left hG.left)
(exercise_3_17_i hF.right hG.right) (exercise_3_17_i hF.right hG.right)
@ -929,7 +929,7 @@ variable {R : Set.Relation ℕ}
variable (hR : R = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)}) variable (hR : R = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)})
theorem exercise_3_18_i theorem exercise_3_18_i
: R.comp R = {(0, 2), (0, 3), (1, 3)} := by : comp R R = {(0, 2), (0, 3), (1, 3)} := by
rw [hR] rw [hR]
unfold comp unfold comp
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, or_self, Prod.mk.injEq] simp only [Set.mem_singleton_iff, Set.mem_insert_iff, or_self, Prod.mk.injEq]
@ -970,7 +970,7 @@ theorem exercise_3_18_i
exact ⟨rfl, h.right⟩ exact ⟨rfl, h.right⟩
theorem exercise_3_18_ii theorem exercise_3_18_ii
: R.restriction {1} = {(1, 2), (1, 3)} := by : restriction R {1} = {(1, 2), (1, 3)} := by
rw [hR] rw [hR]
unfold restriction unfold restriction
ext p ext p
@ -1006,7 +1006,7 @@ theorem exercise_3_18_ii
rw [hab.left, hab.right] rw [hab.left, hab.right]
theorem exercise_3_18_iii theorem exercise_3_18_iii
: R.inv.restriction {1} = {(1, 0)} := by : restriction (inv R) {1} = {(1, 0)} := by
rw [hR] rw [hR]
unfold inv restriction unfold inv restriction
ext p ext p
@ -1031,14 +1031,14 @@ theorem exercise_3_18_iii
simp simp
theorem exercise_3_18_iv theorem exercise_3_18_iv
: R.image {1} = {2, 3} := by : image R {1} = {2, 3} := by
rw [hR] rw [hR]
unfold image unfold image
ext y ext y
simp simp
theorem exercise_3_18_v theorem exercise_3_18_v
: R.inv.image {1} = {0} := by : image (inv R) {1} = {0} := by
rw [hR] rw [hR]
unfold inv image unfold inv image
ext y ext y
@ -1067,12 +1067,12 @@ theorem exercise_3_19_i
simp simp
theorem exercise_3_19_ii theorem exercise_3_19_ii
: A.image ∅ = ∅ := by : image A ∅ = ∅ := by
unfold image unfold image
simp simp
theorem exercise_3_19_iii theorem exercise_3_19_iii
: A.image {∅} = {{∅, {∅}}} := by : image A {∅} = {{∅, {∅}}} := by
unfold image unfold image
rw [hA] rw [hA]
ext x ext x
@ -1098,7 +1098,7 @@ theorem exercise_3_19_iii
simp simp
theorem exercise_3_19_iv theorem exercise_3_19_iv
: A.image {∅, {∅}} = {{∅, {∅}}, ∅} := by : image A {∅, {∅}} = {{∅, {∅}}, ∅} := by
unfold image unfold image
rw [hA] rw [hA]
ext x ext x
@ -1131,7 +1131,7 @@ theorem exercise_3_19_iv
· intro hx₁; right; exact hx₁ · intro hx₁; right; exact hx₁
theorem exercise_3_19_v theorem exercise_3_19_v
: A.inv = {({∅, {∅}}, ∅), (∅, {∅})} := by : inv A = {({∅, {∅}}, ∅), (∅, {∅})} := by
unfold inv unfold inv
rw [hA] rw [hA]
ext x ext x
@ -1154,7 +1154,7 @@ theorem exercise_3_19_v
· intro hx₁; right; rw [← hx₁] · intro hx₁; right; rw [← hx₁]
theorem exercise_3_19_vi theorem exercise_3_19_vi
: A.comp A = {({∅}, {∅, {∅}})} := by : comp A A = {({∅}, {∅, {∅}})} := by
unfold comp unfold comp
rw [hA] rw [hA]
ext x ext x
@ -1179,13 +1179,13 @@ theorem exercise_3_19_vi
· left ; rw [hb]; simp · left ; rw [hb]; simp
theorem exercise_3_19_vii theorem exercise_3_19_vii
: A.restriction ∅ = ∅ := by : restriction A ∅ = ∅ := by
unfold restriction unfold restriction
rw [hA] rw [hA]
simp simp
theorem exercise_3_19_viii theorem exercise_3_19_viii
: A.restriction {∅} = {(∅, {∅, {∅}})} := by : restriction A {∅} = {(∅, {∅, {∅}})} := by
unfold restriction unfold restriction
rw [hA] rw [hA]
ext x ext x
@ -1206,7 +1206,7 @@ theorem exercise_3_19_viii
simp simp
theorem exercise_3_19_ix theorem exercise_3_19_ix
: A.restriction {∅, {∅}} = A := by : restriction A {∅, {∅}} = A := by
unfold restriction unfold restriction
rw [hA] rw [hA]
ext x ext x
@ -1300,9 +1300,9 @@ end Exercise_3_19
Show that `F ↾ A = F ∩ (A × ran F)`. Show that `F ↾ A = F ∩ (A × ran F)`.
-/ -/
theorem exercise_3_20 {F : Set.Relation α} {A : Set α} theorem exercise_3_20 {F : Set.HRelation α β} {A : Set α}
: F.restriction A = F ∩ (Set.prod A (ran F)) := by : restriction F A = F ∩ (Set.prod A (ran F)) := by
calc F.restriction A calc restriction F A
_ = {p | p ∈ F ∧ p.fst ∈ A} := rfl _ = {p | p ∈ F ∧ p.fst ∈ A} := rfl
_ = {p | p ∈ F ∧ p.fst ∈ A ∧ p.snd ∈ ran F} := by _ = {p | p ∈ F ∧ p.fst ∈ A ∧ p.snd ∈ ran F} := by
ext x ext x
@ -1323,9 +1323,9 @@ Show that the following is correct for any sets.
A ⊆ B → F⟦A⟧ ⊆ F⟦B⟧ A ⊆ B → F⟦A⟧ ⊆ F⟦B⟧
``` ```
-/ -/
theorem exercise_3_22_a {A B : Set α} {F : Set.Relation α} (h : A ⊆ B) theorem exercise_3_22_a {A B : Set α} {F : Set.HRelation α β} (h : A ⊆ B)
: F.image A ⊆ F.image B := by : image F A ⊆ image F B := by
show ∀ x, x ∈ F.image A → x ∈ F.image B show ∀ x, x ∈ image F A → x ∈ image F B
unfold image unfold image
simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq]
intro x hx intro x hx
@ -1340,10 +1340,10 @@ Show that the following is correct for any sets.
(F ∘ G)⟦A⟧ = F⟦G⟦A⟧⟧ (F ∘ G)⟦A⟧ = F⟦G⟦A⟧⟧
``` ```
-/ -/
theorem exercise_3_22_b {A B : Set α} {F : Set.Relation α} theorem exercise_3_22_b {A B : Set α} {F : Set.HRelation α β}
: (F.comp G).image A = F.image (G.image A) := by : image (comp F G) A = image F (image G A) := by
calc (F.comp G).image A calc image (comp F G) A
_ = { v | ∃ u ∈ A, (u, v) ∈ F.comp G } := rfl _ = { v | ∃ u ∈ A, (u, v) ∈ comp F G } := rfl
_ = { v | ∃ u ∈ A, ∃ a, (u, a) ∈ G ∧ (a, v) ∈ F } := rfl _ = { v | ∃ u ∈ A, ∃ a, (u, a) ∈ G ∧ (a, v) ∈ F } := rfl
_ = { v | ∃ a, ∃ u ∈ A, (u, a) ∈ G ∧ (a, v) ∈ F } := by _ = { v | ∃ a, ∃ u ∈ A, (u, a) ∈ G ∧ (a, v) ∈ F } := by
ext p ext p
@ -1362,8 +1362,8 @@ theorem exercise_3_22_b {A B : Set α} {F : Set.Relation α}
· intro ⟨a, ⟨u, hu⟩, ha⟩ · intro ⟨a, ⟨u, hu⟩, ha⟩
exact ⟨a, u, hu.left, hu.right, ha⟩ exact ⟨a, u, hu.left, hu.right, ha⟩
_ = { v | ∃ a ∈ { w | ∃ u ∈ A, (u, w) ∈ G }, (a, v) ∈ F } := rfl _ = { v | ∃ a ∈ { w | ∃ u ∈ A, (u, w) ∈ G }, (a, v) ∈ F } := rfl
_ = { v | ∃ a ∈ G.image A, (a, v) ∈ F } := rfl _ = { v | ∃ a ∈ image G A, (a, v) ∈ F } := rfl
_ = F.image (G.image A) := rfl _ = image F (image G A) := rfl
/-- #### Exercise 3.22 (c) /-- #### Exercise 3.22 (c)
@ -1373,27 +1373,27 @@ Q ↾ (A ∪ B) = (Q ↾ A) ∪ (Q ↾ B)
``` ```
-/ -/
theorem exercise_3_22_c {A B : Set α} {Q : Set.Relation α} theorem exercise_3_22_c {A B : Set α} {Q : Set.Relation α}
: Q.restriction (A ∪ B) = (Q.restriction A) ∪ (Q.restriction B) := by : restriction Q (A ∪ B) = (restriction Q A) ∪ (restriction Q B) := by
calc Q.restriction (A ∪ B) calc restriction Q (A ∪ B)
_ = { p | p ∈ Q ∧ p.1 ∈ A ∪ B } := rfl _ = { p | p ∈ Q ∧ p.1 ∈ A ∪ B } := rfl
_ = { p | p ∈ Q ∧ (p.1 ∈ A ∨ p.1 ∈ B) } := rfl _ = { p | p ∈ Q ∧ (p.1 ∈ A ∨ p.1 ∈ B) } := rfl
_ = { p | (p ∈ Q ∧ p.1 ∈ A) ∨ (p ∈ Q ∧ p.1 ∈ B) } := by _ = { p | (p ∈ Q ∧ p.1 ∈ A) ∨ (p ∈ Q ∧ p.1 ∈ B) } := by
ext p ext p
simp only [Set.sep_or, Set.mem_union, Set.mem_setOf_eq] simp only [Set.sep_or, Set.mem_union, Set.mem_setOf_eq]
_ = { p | p ∈ Q ∧ p.1 ∈ A} ∪ { p | p ∈ Q ∧ p.1 ∈ B } := rfl _ = { p | p ∈ Q ∧ p.1 ∈ A} ∪ { p | p ∈ Q ∧ p.1 ∈ B } := rfl
_ = (Q.restriction A) ∪ (Q.restriction B) := rfl _ = (restriction Q A) ∪ (restriction Q B) := rfl
/-- #### Exercise 3.23 (i) /-- #### Exercise 3.23 (i)
Let `I` be the identity function on the set `A`. Show that for any sets `B` and Let `I` be the identity function on the set `A`. Show that for any sets `B` and
`C`, `B ∘ I = B ↾ A`. `C`, `B ∘ I = B ↾ A`.
-/ -/
theorem exercise_3_23_i {A : Set α} {B : Set.Relation α} {I : Set.Relation α} theorem exercise_3_23_i {A : Set α} {B : Set.HRelation α β} {I : Set.Relation α}
(hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 }) (hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
: B.comp I = B.restriction A := by : comp B I = restriction B A := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
apply And.intro apply And.intro
· show ∀ p, p ∈ B.comp I → p ∈ B.restriction A · show ∀ p, p ∈ comp B I → p ∈ restriction B A
intro (x, y) hp intro (x, y) hp
have ⟨t, ht⟩ := hp have ⟨t, ht⟩ := hp
rw [hI] at ht rw [hI] at ht
@ -1401,7 +1401,7 @@ theorem exercise_3_23_i {A : Set α} {B : Set.Relation α} {I : Set.Relation α}
show (x, y) ∈ B ∧ x ∈ A show (x, y) ∈ B ∧ x ∈ A
rw [← ht.left.right] at ht rw [← ht.left.right] at ht
exact ⟨ht.right, ht.left.left⟩ exact ⟨ht.right, ht.left.left⟩
· show ∀ p, p ∈ B.restriction A → p ∈ B.comp I · show ∀ p, p ∈ restriction B A → p ∈ comp B I
unfold restriction comp unfold restriction comp
rw [hI] rw [hI]
simp only [Set.mem_setOf_eq, and_true] simp only [Set.mem_setOf_eq, and_true]
@ -1415,8 +1415,8 @@ Let `I` be the identity function on the set `A`. Show that for any sets `B` and
-/ -/
theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α} theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α}
(hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 }) (hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
: I.image C = A ∩ C := by : image I C = A ∩ C := by
calc I.image C calc image I C
_ = { v | ∃ u ∈ C, (u, v) ∈ I } := rfl _ = { v | ∃ u ∈ C, (u, v) ∈ I } := rfl
_ = { v | ∃ u ∈ C, u ∈ A ∧ u = v } := by _ = { v | ∃ u ∈ C, u ∈ A ∧ u = v } := by
ext v ext v
@ -1443,6 +1443,205 @@ theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α}
_ = C ∩ A := rfl _ = C ∩ A := rfl
_ = A ∩ C := Set.inter_comm C A _ = A ∩ C := Set.inter_comm C A
/-- #### Exercise 3.24
Show that for a function `F`, `F⁻¹⟦A⟧ = { x ∈ dom F | F(x) ∈ A }`.
-/
theorem exercise_3_24 {F : Set.HRelation α β} {A : Set β}
(hF : isSingleValued F)
: image (inv F) A = { x ∈ dom F | ∃! y : β, (x, y) ∈ F ∧ y ∈ A } := by
calc image (inv F) A
_ = { x | ∃ y ∈ A, (y, x) ∈ inv F } := rfl
_ = { x | ∃ y ∈ A, (x, y) ∈ F } := by simp only [mem_self_comm_mem_inv]
_ = { x | x ∈ dom F ∧ (∃ y : β, (x, y) ∈ F ∧ y ∈ A) } := by
ext x
simp only [Set.mem_setOf_eq]
apply Iff.intro
· intro ⟨y, hy, hyx⟩
exact ⟨mem_pair_imp_fst_mem_dom hyx, y, hyx, hy⟩
· intro ⟨_, y, hxy, hy⟩
exact ⟨y, hy, hxy⟩
_ = { x ∈ dom F | ∃ y : β, (x, y) ∈ F ∧ y ∈ A } := rfl
_ = { x ∈ dom F | ∃! y : β, (x, y) ∈ F ∧ y ∈ A } := by
ext x
simp only [Set.mem_setOf_eq, and_congr_right_iff]
intro _
apply Iff.intro
· intro ⟨y, hy⟩
refine ⟨y, hy, ?_⟩
intro y₁ hy₁
exact single_valued_eq_unique hF hy₁.left hy.left
· intro ⟨y, hy⟩
exact ⟨y, hy.left⟩
/-- #### Exercise 3.25 (b)
Show that the result of part (a) holds for any function `G`, not necessarily
one-to-one.
-/
theorem exercise_3_25_b {G : Set.HRelation α β} (hG : isSingleValued G)
: comp G (inv G) = { p | p.1 ∈ ran G ∧ p.1 = p.2 } := by
ext p
have (x, y) := p
apply Iff.intro
· unfold comp inv
intro h
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq] at h
have ⟨t, ⟨a, b, ⟨hab, hb, ha⟩⟩, ht⟩ := h
simp only [Set.mem_setOf_eq]
rw [hb, ha] at hab
exact ⟨mem_pair_imp_snd_mem_ran hab, single_valued_eq_unique hG hab ht⟩
· intro h
simp only [Set.mem_setOf_eq] at h
unfold comp inv
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
have ⟨t, ht⟩ := ran_exists h.left
exact ⟨t, ⟨t, x, ht, rfl, rfl⟩, by rwa [← h.right]⟩
/-- #### Exercise 3.25 (a)
Assume that `G` is a one-to-one function. Show that `G ∘ G⁻¹` is the identity
function on `ran G`.
-/
theorem exercise_3_25_a {G : Set.HRelation α β} (hG : isOneToOne G)
: comp G (inv G) = { p | p.1 ∈ ran G ∧ p.1 = p.2 } :=
exercise_3_25_b hG.left
/-- #### Exercise 3.27
Show that `dom (F ∘ G) = G⁻¹⟦dom F⟧` for any sets `F` and `G`. (`F` and `G` need
not be functions.)
-/
theorem exercise_3_27 {F : Set.HRelation β γ} {G : Set.HRelation α β}
: dom (comp F G) = image (inv G) (dom F) := by
rw [Set.Subset.antisymm_iff]
apply And.intro
· show ∀ x, x ∈ dom (comp F G) → x ∈ image (inv G) (dom F)
intro x hx
have ⟨y, hy⟩ := dom_exists hx
unfold comp at hy
simp only [Set.mem_setOf_eq] at hy
have ⟨t, ht⟩ := hy
have htF : t ∈ dom F := mem_pair_imp_fst_mem_dom ht.right
unfold image inv
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
exact ⟨t, htF, x, t, ht.left, rfl, rfl⟩
· show ∀ x, x ∈ image (inv G) (dom F) → x ∈ dom (comp F G)
intro x hx
unfold image at hx
simp only [mem_self_comm_mem_inv, Set.mem_setOf_eq] at hx
have ⟨u, hu⟩ := hx
have ⟨t, ht⟩ := dom_exists hu.left
unfold dom comp
simp only [
Set.mem_image,
Set.mem_setOf_eq,
Prod.exists,
exists_and_right,
exists_eq_right
]
exact ⟨t, u, hu.right, ht⟩
/-- #### Exercise 3.28
Assume that `f` is a one-to-one function from `A` into `B`, and that `G` is the
function with `dom G = 𝒫 A` defined by the equation `G(X) = f⟦X⟧`. Show that `G`
maps `𝒫 A` one-to-one into `𝒫 B`.
-/
theorem exercise_3_28 {A : Set α} {B : Set β}
{f : Set.HRelation α β} {G : Set.HRelation (Set α) (Set β)}
(hf : isOneToOne f ∧ mapsInto f A B)
(hG : G = { p | p.1 ∈ 𝒫 A ∧ p.2 = image f p.1 })
: isOneToOne G ∧ mapsInto G (𝒫 A) (𝒫 B) := by
have dG : dom G = 𝒫 A := by
rw [hG]
ext p
unfold dom Prod.fst
simp
have hG₁ : isSingleValued G := by
intro x hx
have ⟨y, hy⟩ := dom_exists hx
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hy, hy⟩, ?_⟩
intro y₁ hy₁
rw [hG, Set.mem_setOf_eq] at hy
conv at hy₁ => rhs; rw [hG, Set.mem_setOf_eq]
simp only at *
rw [hy.right, hy₁.right.right]
apply And.intro
· show isOneToOne G
refine ⟨hG₁, ?_⟩
intro y hy
have ⟨X₁, hX₁⟩ := ran_exists hy
refine ⟨X₁, ⟨mem_pair_imp_fst_mem_dom hX₁, hX₁⟩, ?_⟩
intro X₂ hX₂
have hX₁' : y = image f X₁ := by
rw [hG] at hX₁
simp only [Set.mem_powerset_iff, Set.mem_setOf_eq] at hX₁
exact hX₁.right
have hX₂' : y = image f X₂ := by
have := hX₂.right
rw [hG] at this
simp only [Set.mem_powerset_iff, Set.mem_setOf_eq] at this
exact this.right
ext t
apply Iff.intro
· intro ht
rw [dG] at hX₂
simp only [Set.mem_powerset_iff] at hX₂
have ht' := hX₂.left ht
rw [← hf.right.right.left] at ht'
have ⟨ft, hft⟩ := dom_exists ht'
have hft' : ft ∈ image f X₂ := ⟨t, ht, hft⟩
rw [← hX₂', hX₁'] at hft'
have ⟨t₁, ht₁⟩ := hft'
rw [single_rooted_eq_unique hf.left.right hft ht₁.right]
exact ht₁.left
· intro ht
have hX₁sub := mem_pair_imp_fst_mem_dom hX₁
rw [dG] at hX₁sub
simp only [Set.mem_powerset_iff] at hX₁sub
have ht' := hX₁sub ht
rw [← hf.right.right.left] at ht'
have ⟨ft, hft⟩ := dom_exists ht'
have hft' : ft ∈ image f X₁ := ⟨t, ht, hft⟩
rw [← hX₁', hX₂'] at hft'
have ⟨t₁, ht₁⟩ := hft'
rw [single_rooted_eq_unique hf.left.right hft ht₁.right]
exact ht₁.left
· show mapsInto G (𝒫 A) (𝒫 B)
refine ⟨hG₁, dG, ?_⟩
show ∀ x, x ∈ ran G → x ∈ 𝒫 B
intro x hx
rw [hG] at hx
unfold ran Prod.snd at hx
simp only [
Set.mem_powerset_iff,
Set.mem_image,
Set.mem_setOf_eq,
Prod.exists,
exists_eq_right
] at hx
have ⟨a, ha⟩ := hx
rw [ha.right]
show ∀ y, y ∈ image f a → y ∈ B
intro y hy
simp only [Set.mem_setOf_eq] at hy
have ⟨b, hb⟩ := hy
have hz := mem_pair_imp_snd_mem_ran hb.right
exact hf.right.right.right hz
end Relation end Relation
end Enderton.Set.Chapter_3 end Enderton.Set.Chapter_3

126
Bookshelf/Enderton/Set/Relation.lean
View File

@ -16,7 +16,12 @@ Kuratowski's definition of a set.
Not to be confused with the Lean-provided `Rel`. Not to be confused with the Lean-provided `Rel`.
-/ -/
abbrev Relation (α : Type _) := Set (α × α) abbrev HRelation (α β : Type ) := Set (α × β)
/--
A homogeneous variant of the `HRelation` type.
-/
abbrev Relation (α : Type _) := HRelation α α
namespace Relation namespace Relation
@ -25,13 +30,13 @@ namespace Relation
/-- /--
The domain of a `Relation`. The domain of a `Relation`.
-/ -/
def dom (R : Relation α) : Set α := Prod.fst '' R def dom (R : HRelation α β) : Set α := Prod.fst '' R
/-- /--
The first component of any pair in a `Relation` must be a member of the The first component of any pair in a `Relation` must be a member of the
`Relation`'s domain. `Relation`'s domain.
-/ -/
theorem mem_pair_imp_fst_mem_dom {R : Relation α} (h : (x, y) ∈ R) theorem mem_pair_imp_fst_mem_dom {R : HRelation α β} (h : (x, y) ∈ R)
: x ∈ dom R := by : x ∈ dom R := by
unfold dom Prod.fst unfold dom Prod.fst
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right]
@ -40,8 +45,8 @@ theorem mem_pair_imp_fst_mem_dom {R : Relation α} (h : (x, y) ∈ R)
/-- /--
If `x ∈ dom R`, there exists some `y` such that `⟨x, y⟩ ∈ R`. If `x ∈ dom R`, there exists some `y` such that `⟨x, y⟩ ∈ R`.
-/ -/
theorem dom_exists {R : Relation α} (hx : x ∈ R.dom) theorem dom_exists {R : HRelation α β} (hx : x ∈ dom R)
: ∃ y : α, (x, y) ∈ R := by : ∃ y : β, (x, y) ∈ R := by
unfold dom at hx unfold dom at hx
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] at hx simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] at hx
exact hx exact hx
@ -49,9 +54,9 @@ theorem dom_exists {R : Relation α} (hx : x ∈ R.dom)
/-- /--
The range of a `Relation`. The range of a `Relation`.
-/ -/
def ran (R : Relation α) : Set α := Prod.snd '' R def ran (R : HRelation α β) : Set β := Prod.snd '' R
theorem mem_pair_imp_snd_mem_ran {R : Relation α} (h : (x, y) ∈ R) theorem mem_pair_imp_snd_mem_ran {R : HRelation α β} (h : (x, y) ∈ R)
: y ∈ ran R := by : y ∈ ran R := by
unfold ran Prod.snd unfold ran Prod.snd
simp only [mem_image, Prod.exists, exists_eq_right] simp only [mem_image, Prod.exists, exists_eq_right]
@ -60,7 +65,7 @@ theorem mem_pair_imp_snd_mem_ran {R : Relation α} (h : (x, y) ∈ R)
/-- /--
If `x ∈ ran R`, there exists some `t` such that `⟨t, x⟩ ∈ R`. If `x ∈ ran R`, there exists some `t` such that `⟨t, x⟩ ∈ R`.
-/ -/
theorem ran_exists {R : Relation α} (hx : x ∈ R.ran) theorem ran_exists {R : HRelation α β} (hx : x ∈ ran R)
: ∃ t : α, (t, x) ∈ R := by : ∃ t : α, (t, x) ∈ R := by
unfold ran at hx unfold ran at hx
simp only [mem_image, Prod.exists, exists_eq_right] at hx simp only [mem_image, Prod.exists, exists_eq_right] at hx
@ -74,14 +79,14 @@ def fld (R : Relation α) : Set α := dom R ∪ ran R
/-- /--
The inverse of a `Relation`. The inverse of a `Relation`.
-/ -/
def inv (R : Relation α) : Relation α := { (p.2, p.1) | p ∈ R } def inv (R : HRelation α β) : HRelation β α := { (p.2, p.1) | p ∈ R }
/-- /--
`(x, y)` is a member of relation `R` **iff** `(y, x)` is a member of `R⁻¹`. `(x, y)` is a member of relation `R` **iff** `(y, x)` is a member of `R⁻¹`.
-/ -/
@[simp] @[simp]
theorem mem_self_comm_mem_inv {R : Relation α} theorem mem_self_comm_mem_inv {R : HRelation α β}
: (y, x) ∈ R.inv ↔ (x, y) ∈ R := by : (y, x) ∈ inv R ↔ (x, y) ∈ R := by
unfold inv unfold inv
simp only [Prod.exists, mem_setOf_eq, Prod.mk.injEq] simp only [Prod.exists, mem_setOf_eq, Prod.mk.injEq]
apply Iff.intro apply Iff.intro
@ -95,8 +100,8 @@ theorem mem_self_comm_mem_inv {R : Relation α}
The inverse of the inverse of a `Relation` is the `Relation`. The inverse of the inverse of a `Relation` is the `Relation`.
-/ -/
@[simp] @[simp]
theorem inv_inv_eq_self (R : Relation α) theorem inv_inv_eq_self (R : HRelation α β)
: R.inv.inv = R := by : inv (inv R) = R := by
unfold inv unfold inv
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq] simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
ext x ext x
@ -115,8 +120,8 @@ theorem inv_inv_eq_self (R : Relation α)
For a set `F`, `dom F⁻¹ = ran F`. For a set `F`, `dom F⁻¹ = ran F`.
-/ -/
@[simp] @[simp]
theorem dom_inv_eq_ran_self {F : Relation α} theorem dom_inv_eq_ran_self {F : HRelation α β}
: dom (F.inv) = ran F := by : dom (inv F) = ran F := by
ext x ext x
unfold dom ran inv unfold dom ran inv
simp only [ simp only [
@ -138,8 +143,8 @@ theorem dom_inv_eq_ran_self {F : Relation α}
For a set `F`, `ran F⁻¹ = dom F`. For a set `F`, `ran F⁻¹ = dom F`.
-/ -/
@[simp] @[simp]
theorem ran_inv_eq_dom_self {F : Relation α} theorem ran_inv_eq_dom_self {F : HRelation α β}
: ran (F.inv) = dom F := by : ran (inv F) = dom F := by
ext x ext x
unfold dom ran inv unfold dom ran inv
simp only [ simp only [
@ -162,7 +167,7 @@ theorem ran_inv_eq_dom_self {F : Relation α}
/-- /--
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 : HRelation α β) (A : Set α) : HRelation α β :=
{ p ∈ R | p.1 ∈ A } { p ∈ R | p.1 ∈ A }
/-! ## Image -/ /-! ## Image -/
@ -170,7 +175,7 @@ def restriction (R : Relation α) (A : Set α) : Relation α :=
/-- /--
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 : HRelation α β) (A : Set α) : Set β :=
{ y | ∃ u ∈ A, (u, y) ∈ R } { y | ∃ u ∈ A, (u, y) ∈ R }
/-! ## Single-Rooted and Single-Valued -/ /-! ## Single-Rooted and Single-Valued -/
@ -179,14 +184,14 @@ def image (R : Relation α) (A : Set α) : Set α :=
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`.
-/ -/
def isSingleRooted (R : Relation α) : Prop := def isSingleRooted (R : HRelation α β) : Prop :=
∀ y ∈ ran R, ∃! x, x ∈ dom R ∧ (x, y) ∈ R ∀ y ∈ ran R, ∃! x, x ∈ dom R ∧ (x, y) ∈ R
/-- /--
A single-rooted `Relation` should map the same output to the same input. A single-rooted `Relation` should map the same output to the same input.
-/ -/
theorem single_rooted_eq_unique {R : Relation α} {x₁ x₂ y : α} theorem single_rooted_eq_unique {R : HRelation α β} {x₁ x₂ : α} {y : β}
(hR : R.isSingleRooted) (hR : isSingleRooted R)
: (x₁, y) ∈ R → (x₂, y) ∈ R → x₁ = x₂ := by : (x₁, y) ∈ R → (x₂, y) ∈ R → x₁ = x₂ := by
intro hx₁ hx₂ intro hx₁ hx₂
unfold isSingleRooted at hR unfold isSingleRooted at hR
@ -203,14 +208,14 @@ exists exactly one `y` such that `⟨x, y⟩ ∈ R`.
Notice, a `Relation` that is single-valued is a function. Notice, a `Relation` that is single-valued is a function.
-/ -/
def isSingleValued (R : Relation α) : Prop := def isSingleValued (R : HRelation α β) : Prop :=
∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R ∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R
/-- /--
A single-valued `Relation` should map the same input to the same output. A single-valued `Relation` should map the same input to the same output.
-/ -/
theorem single_valued_eq_unique {R : Relation α} {x y₁ y₂ : α} theorem single_valued_eq_unique {R : HRelation α β} {x : α} {y₁ y₂ : β}
(hR : R.isSingleValued) (hR : isSingleValued R)
: (x, y₁) ∈ R → (x, y₂) ∈ R → y₁ = y₂ := by : (x, y₁) ∈ R → (x, y₂) ∈ R → y₁ = y₂ := by
intro hy₁ hy₂ intro hy₁ hy₂
unfold isSingleValued at hR unfold isSingleValued at hR
@ -224,8 +229,8 @@ theorem single_valued_eq_unique {R : Relation α} {x y₁ y₂ : α}
/-- /--
For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted. 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} theorem single_valued_inv_iff_single_rooted_self {F : HRelation α β}
: F.inv.isSingleValued ↔ F.isSingleRooted := by : isSingleValued (inv F) ↔ isSingleRooted F := by
apply Iff.intro apply Iff.intro
· intro hF · intro hF
unfold isSingleValued at hF unfold isSingleValued at hF
@ -234,7 +239,7 @@ theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
ran_inv_eq_dom_self, ran_inv_eq_dom_self,
mem_self_comm_mem_inv mem_self_comm_mem_inv
] at hF ] at hF
suffices ∀ x ∈ F.ran, ∃! y, (y, x) ∈ F from hF suffices ∀ x ∈ ran F, ∃! y, (y, x) ∈ F from hF
intro x hx intro x hx
have ⟨y, hy⟩ := hF x hx have ⟨y, hy⟩ := hF x hx
simp only [ simp only [
@ -258,17 +263,17 @@ theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
/-- /--
For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted. 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} theorem single_valued_self_iff_single_rooted_inv {F : HRelation α β}
: F.isSingleValued ↔ F.inv.isSingleRooted := by : isSingleValued F ↔ isSingleRooted (inv F) := by
conv => lhs; rw [← inv_inv_eq_self F] conv => lhs; rw [← inv_inv_eq_self F]
rw [single_valued_inv_iff_single_rooted_self] rw [single_valued_inv_iff_single_rooted_self]
/-- /--
The subset of a function must also be a function. The subset of a function must also be a function.
-/ -/
theorem single_valued_subset {F G : Set.Relation α} theorem single_valued_subset {F G : HRelation α β}
(hG : G.isSingleValued) (h : F ⊆ G) (hG : isSingleValued G) (h : F ⊆ G)
: F.isSingleValued := by : isSingleValued F := by
unfold isSingleValued unfold isSingleValued
intro x hx intro x hx
have ⟨y, hy⟩ := dom_exists hx have ⟨y, hy⟩ := dom_exists hx
@ -283,14 +288,14 @@ theorem single_valued_subset {F G : Set.Relation α}
/-- /--
A `Relation` `R` is one-to-one if it is a single-rooted function. A `Relation` `R` is one-to-one if it is a single-rooted function.
-/ -/
def isOneToOne (R : Relation α) : Prop := def isOneToOne (R : HRelation α β) : Prop :=
R.isSingleValued ∧ R.isSingleRooted isSingleValued R ∧ isSingleRooted R
/-- /--
A `Relation` is one-to-one **iff** it's inverse is. A `Relation` is one-to-one **iff** it's inverse is.
-/ -/
theorem one_to_one_self_iff_one_to_one_inv {R : Relation α} theorem one_to_one_self_iff_one_to_one_inv {R : HRelation α β}
: R.isOneToOne ↔ R.inv.isOneToOne := by : isOneToOne R ↔ isOneToOne (inv R) := by
unfold isOneToOne isSingleValued isSingleRooted unfold isOneToOne isSingleValued isSingleRooted
conv => rhs; simp only [ conv => rhs; simp only [
dom_inv_eq_ran_self, dom_inv_eq_ran_self,
@ -309,28 +314,28 @@ Indicates `Relation` `F` is a function from `A` to `B`.
This is usually denoted as `F : A → B`. This is usually denoted as `F : A → B`.
-/ -/
def mapsInto (F : Relation α) (A B : Set α) := def mapsInto (F : HRelation α β) (A : Set α) (B : Set β) :=
F.isSingleValued ∧ dom F = A ∧ ran F ⊆ B isSingleValued F ∧ dom F = A ∧ ran F ⊆ B
/-- /--
Indicates `Relation` `F` is a function from `A` to `ran F = B`. Indicates `Relation` `F` is a function from `A` to `ran F = B`.
-/ -/
def mapsOnto (F : Relation α) (A B : Set α) := def mapsOnto (F : HRelation α β) (A : Set α) (B : Set β) :=
F.isSingleValued ∧ dom F = A ∧ ran F = B isSingleValued F ∧ dom F = A ∧ ran F = B
/-! ## Composition -/ /-! ## Composition -/
/-- /--
The composition of two `Relation`s. The composition of two `Relation`s.
-/ -/
def comp (F G : Relation α) : Relation α := def comp (F : HRelation β γ) (G : HRelation α β) : HRelation α γ :=
{ p | ∃ t : α, (p.1, t) ∈ G ∧ (t, p.2) ∈ F} { p | ∃ t : β, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
/-- /--
If `x ∈ dom (F ∘ G)`, then `x ∈ dom G`. If `x ∈ dom (F ∘ G)`, then `x ∈ dom G`.
-/ -/
theorem dom_comp_imp_dom_self {F G : Relation α} theorem dom_comp_imp_dom_self {F : HRelation β γ} {G : HRelation α β}
: x ∈ dom (F.comp G) → x ∈ dom G := by : x ∈ dom (comp F G) → x ∈ dom G := by
unfold dom comp unfold dom comp
simp only [ simp only [
mem_image, mem_image,
@ -346,8 +351,8 @@ theorem dom_comp_imp_dom_self {F G : Relation α}
/-- /--
If `y ∈ ran (F ∘ G)`, then `y ∈ ran F`. If `y ∈ ran (F ∘ G)`, then `y ∈ ran F`.
-/ -/
theorem ran_comp_imp_ran_self {F G : Relation α} theorem ran_comp_imp_ran_self {F : HRelation β γ} {G : HRelation α β}
: y ∈ ran (F.comp G) → y ∈ ran F := by : y ∈ ran (comp F G) → y ∈ ran F := by
unfold ran comp unfold ran comp
simp only [ simp only [
mem_image, mem_image,
@ -362,10 +367,10 @@ theorem ran_comp_imp_ran_self {F G : Relation α}
/-- /--
Composition of functions is associative. Composition of functions is associative.
-/ -/
theorem comp_assoc {R S T : Relation α} theorem comp_assoc {R : HRelation γ δ} {S : HRelation β γ} {T : HRelation α β}
: (R.comp S).comp T = R.comp (S.comp T) := by : comp (comp R S) T = comp R (comp S T) := by
calc (R.comp S).comp T calc comp (comp R S) T
_ = { p | ∃ t, (p.1, t) ∈ T ∧ (t, p.2) ∈ R.comp S} := rfl _ = { p | ∃ t, (p.1, t) ∈ T ∧ (t, p.2) ∈ comp R S} := rfl
_ = { p | ∃ t, (p.1, t) ∈ T ∧ (∃ a, (t, a) ∈ S ∧ (a, p.2) ∈ R) } := rfl _ = { p | ∃ t, (p.1, t) ∈ T ∧ (∃ a, (t, a) ∈ S ∧ (a, p.2) ∈ R) } := rfl
_ = { p | ∃ t, ∃ a, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by _ = { p | ∃ t, ∃ a, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
ext p ext p
@ -391,15 +396,16 @@ theorem comp_assoc {R S T : Relation α}
exact ⟨a, ⟨t, h.left⟩, h.right⟩ exact ⟨a, ⟨t, h.left⟩, h.right⟩
· intro ⟨a, ⟨t, ht⟩, ha⟩ · intro ⟨a, ⟨t, ht⟩, ha⟩
exact ⟨a, t, ht, ha⟩ exact ⟨a, t, ht, ha⟩
_ = { p | ∃ a, (p.1, a) ∈ S.comp T ∧ (a, p.2) ∈ R } := rfl _ = { p | ∃ a, (p.1, a) ∈ comp S T ∧ (a, p.2) ∈ R } := rfl
_ = R.comp (S.comp T) := rfl _ = comp R (comp S T) := rfl
/-- /--
The composition of two functions is itself a function. The composition of two functions is itself a function.
-/ -/
theorem single_valued_comp_is_single_valued {F G : Relation α} theorem single_valued_comp_is_single_valued
(hF : F.isSingleValued) (hG : G.isSingleValued) {F : HRelation β γ} {G : HRelation α β}
: (F.comp G).isSingleValued := by (hF : isSingleValued F) (hG : isSingleValued G)
: isSingleValued (comp F G) := by
unfold isSingleValued unfold isSingleValued
intro x hx intro x hx
have ⟨y, hxy⟩ := dom_exists hx have ⟨y, hxy⟩ := dom_exists hx
@ -433,9 +439,9 @@ theorem single_valued_comp_is_single_valued {F G : Relation α}
/-- /--
For `Relation`s `F` and `G`, `(F ∘ G)⁻¹ = G⁻¹ ∘ F⁻¹`. For `Relation`s `F` and `G`, `(F ∘ G)⁻¹ = G⁻¹ ∘ F⁻¹`.
-/ -/
theorem comp_inv_eq_inv_comp_inv {F G : Relation α} theorem comp_inv_eq_inv_comp_inv {F : HRelation β γ} {G : HRelation α β}
: (F.comp G).inv = G.inv.comp F.inv := by : inv (comp F G) = comp (inv G) (inv F) := by
calc (F.comp G).inv calc inv (comp F G)
_ = {p | ∃ t, (p.2, t) ∈ G ∧ (t, p.1) ∈ F} := by _ = {p | ∃ t, (p.2, t) ∈ G ∧ (t, p.1) ∈ F} := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
apply And.intro apply And.intro
@ -460,7 +466,7 @@ theorem comp_inv_eq_inv_comp_inv {F G : Relation α}
simp only [mem_setOf_eq] at * simp only [mem_setOf_eq] at *
have ⟨t, p, q⟩ := ht have ⟨t, p, q⟩ := ht
exact ⟨t, q, p⟩ exact ⟨t, q, p⟩
_ = {p | ∃ t, (p.1, t) ∈ F.inv ∧ (t, p.2) ∈ G.inv } := by _ = {p | ∃ t, (p.1, t) ∈ inv F ∧ (t, p.2) ∈ inv G } := by
rw [Set.Subset.antisymm_iff] rw [Set.Subset.antisymm_iff]
apply And.intro apply And.intro
· intro (a, b) ht · intro (a, b) ht