Enderton. Heterogeneous relations and additional exercises.
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@ -4491,7 +4491,7 @@ Show that for any sets $B$ and $C$,
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\end{proof}
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\subsection{\pending{Exercise 3.24}}%
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\subsection{\verified{Exercise 3.24}}%
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\label{sub:exercise-3.24}
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Show that for a function $F$,
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@ -4499,6 +4499,9 @@ Show that for a function $F$,
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Relation}
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{Enderton.Set.Chapter\_3.exercise\_3\_24}
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Let $F$ be a function.
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By definition of the \nameref{ref:inverse} of a set,
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\begin{align*}
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@ -4506,13 +4509,13 @@ Show that for a function $F$,
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& = \{x \mid (\exists y \in A) yF^{-1}x\} \\
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& = \{x \mid (\exists y \in A) xFy\} \\
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& = \{x \mid (\exists y \in A) \left< x, y \right> \in F\} \\
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& = \{x \mid \exists x \in F, F(x) \in A\} \\
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& = \{x \mid x \in \dom{F} \land F(x) \in A\} \\
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& = \{x \in \dom{F} \mid F(x) \in A\}.
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\end{align*}
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\end{proof}
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\subsection{\pending{Exercise 3.25}}%
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\subsection{\verified{Exercise 3.25}}%
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\label{sub:exercise-3.25}
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\begin{enumerate}[(a)]
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@ -4525,18 +4528,45 @@ Show that for a function $F$,
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\begin{proof}
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\statementpadding
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\lean*{Bookshelf/Enderton/Set/Relation}
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{Enderton.Set.Chapter\_3.exercise\_3\_25\_b}
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\lean{Bookshelf/Enderton/Set/Relation}
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{Enderton.Set.Chapter\_3.exercise\_3\_25\_a}
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\paragraph{(b)}%
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\label{par:exercise-3.25-b}
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Let $G$ be a function.
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Let $\left< x, y \right> \in G \circ G^{-1}$.
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By definition of the \nameref{ref:composition} of sets, there exists some
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set $t$ such that $x(G^{-1})t$ and $tGy$.
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By definition of the \nameref{ref:inverse} of a set,
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$$x(G^{-1})t \iff tGx.$$
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By hypothesis, $G$ is single-valued.
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Thus $x = y$.
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That is, $G \circ G^{-1}$ is the identity function on $\ran{G}$.
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Let $G$ be an arbitrary function.
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We show that $G \circ G^{-1} \subseteq I_{\ran{G}}$ and that
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$I_{\ran{G}} \subseteq G \circ G^{-1}$.
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\subparagraph{($\subseteq$)}%
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Let $\left< x, y \right> \in G \circ G^{-1}$.
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By definition of the \nameref{ref:composition} of sets, there exists some
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set $t$ such that $x(G^{-1})t$ and $tGy$.
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By definition of the \nameref{ref:inverse} of a set,
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$$x(G^{-1})t \iff tGx.$$
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The right hand side of the above biconditional indicates $x \in \ran{G}$.
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Since $G$ is single-valued, $tGy \land tGx$ implies $x = y$.
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Thus $\left< x, y \right> \in I_{\ran{G}}$.
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\subparagraph{($\supseteq$)}%
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Let $\left< x, x \right> \in I_{\ran{G}}$ where $x \in \ran{G}$.
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By definition of the \nameref{ref:range} of a function, there exists some
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$t$ such that $\left< t, x \right> \in G$.
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By definition of the \nameref{ref:inverse} of a set, it follows
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$\left< x, t \right> \in G^{-1}$.
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Thus $\left< x, x \right> \in G \circ G^{-1}$.
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\subparagraph{Conclusion}%
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Since $G \circ G^{-1}$ is a subset of $I_{\ran{G}}$ and vice versa, it
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follows that these two sets are equal.
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\paragraph{(a)}%
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@ -4544,7 +4574,7 @@ Show that for a function $F$,
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\end{proof}
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\subsection{\pending{Exercise 3.26}}%
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\subsection{\verified{Exercise 3.26}}%
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\label{sub:exercise-3.26}
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Prove the second halves of parts (a) and (b) of Theorem 3K.
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@ -4556,7 +4586,7 @@ Prove the second halves of parts (a) and (b) of Theorem 3K.
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\end{proof}
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\subsection{\pending{Exercise 3.27}}%
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\subsection{\verified{Exercise 3.27}}%
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\label{sub:exercise-3.27}
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Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
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@ -4564,6 +4594,9 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Relation}
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{Enderton.Set.Chapter\_3.exercise\_3\_27}
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Let $F$ and $G$ be arbitrary sets.
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We show that each side of our desired equality is a subset of the other.
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@ -4597,7 +4630,7 @@ Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
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\end{proof}
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\subsection{\pending{Exercise 3.28}}%
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\subsection{\verified{Exercise 3.28}}%
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\label{sub:exercise-3.28}
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Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the
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@ -4607,9 +4640,14 @@ Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Relation}
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{Enderton.Set.Chapter\_3.exercise\_3\_28}
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By construction, $\dom{G} = \powerset{A}$.
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Likewise, $\ran{G} \subseteq \powerset{B}$ by definition of the
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\nameref{ref:image} of sets.
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Thus $G$ maps $\powerset{A}$ into $\powerset{B}$.
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Let $y \in \ran{G}$.
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Then there exists an $X_1 \in \powerset{A}$ such that $\img{f}{X_1} = y$.
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To prove $G$ is one-to-one into $\powerset{B}$, assume there exists an
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@ -229,12 +229,16 @@ theorem theorem_3d {A : Set (Set (Set α))} (h : OrderedPair x y ∈ A)
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have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq
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exact ⟨this (by simp), this (by simp)⟩
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section Relation
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open Set.Relation
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/-- #### Exercise 3.6
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Show that a set `A` is a relation **iff** `A ⊆ dom A × ran A`.
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-/
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theorem exercise_3_6 {A : Set.Relation α}
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: A ⊆ Set.prod (A.dom) (A.ran) := by
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theorem exercise_3_6 {A : Set.HRelation α β}
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: A ⊆ Set.prod (dom A) (ran A) := by
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show ∀ t, t ∈ A → t ∈ Set.prod (Prod.fst '' A) (Prod.snd '' A)
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intro (a, b) ht
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unfold Set.prod
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@ -301,8 +305,8 @@ theorem exercise_3_7 {R : Set.Relation α}
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-- `t = y` then `t ∈ ran R`.
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have hxy_mem : t = x ∨ t = y → t ∈ Set.Relation.fld R := by
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intro ht
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have hz : R ⊆ Set.prod (R.dom) (R.ran) := exercise_3_6
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have : (x, y) ∈ Set.prod (R.dom) (R.ran) := hz p
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have hz : R ⊆ Set.prod (dom R) (ran R) := exercise_3_6
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have : (x, y) ∈ Set.prod (dom R) (ran R) := hz p
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unfold Set.prod at this
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simp at this
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apply Or.elim ht
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@ -326,10 +330,6 @@ theorem exercise_3_7 {R : Set.Relation α}
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simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
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exact hxy_mem this
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section Relation
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open Set.Relation
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/-- #### Exercise 3.8 (i)
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Show that for any set `𝓐`:
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@ -337,7 +337,7 @@ Show that for any set `𝓐`:
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dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_8_i {A : Set (Set.Relation α)}
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theorem exercise_3_8_i {A : Set (Set.HRelation α β)}
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: dom (⋃₀ A) = ⋃₀ { dom R | R ∈ A } := by
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ext x
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unfold dom Prod.fst
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@ -363,7 +363,7 @@ Show that for any set `𝓐`:
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ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_8_ii {A : Set (Set.Relation α)}
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theorem exercise_3_8_ii {A : Set (Set.HRelation α β)}
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: ran (⋃₀ A) = ⋃₀ { ran R | R ∈ A } := by
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ext x
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unfold ran Prod.snd
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@ -389,7 +389,7 @@ operation in the preceding problem.
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dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_9_i {A : Set (Set.Relation α)}
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theorem exercise_3_9_i {A : Set (Set.HRelation α β)}
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: dom (⋂₀ A) ⊆ ⋂₀ { dom R | R ∈ A } := by
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show ∀ x, x ∈ dom (⋂₀ A) → x ∈ ⋂₀ { dom R | R ∈ A }
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unfold dom Prod.fst
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@ -415,7 +415,7 @@ operation in the preceding problem.
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ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
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```
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-/
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theorem exercise_3_9_ii {A : Set (Set.Relation α)}
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theorem exercise_3_9_ii {A : Set (Set.HRelation α β)}
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: ran (⋂₀ A) ⊆ ⋂₀ { ran R | R ∈ A } := by
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show ∀ x, x ∈ ran (⋂₀ A) → x ∈ ⋂₀ { ran R | R ∈ A }
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unfold ran Prod.snd
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@ -437,9 +437,9 @@ theorem exercise_3_9_ii {A : Set (Set.Relation α)}
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Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
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-/
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theorem theorem_3g_i {F : Set.Relation α}
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(hF : F.isOneToOne) (hx : x ∈ dom F)
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: ∃! y, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
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theorem theorem_3g_i {F : Set.HRelation α β}
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(hF : isOneToOne F) (hx : x ∈ dom F)
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: ∃! y, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
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simp only [mem_self_comm_mem_inv, and_self]
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have ⟨y, hy⟩ := dom_exists hx
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refine ⟨y, hy, ?_⟩
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@ -451,9 +451,9 @@ theorem theorem_3g_i {F : Set.Relation α}
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Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`.
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-/
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theorem theorem_3g_ii {F : Set.Relation α}
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(hF : F.isOneToOne) (hy : y ∈ F.ran)
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: ∃! x, (x, y) ∈ F ∧ (y, x) ∈ F.inv := by
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theorem theorem_3g_ii {F : Set.HRelation α β}
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(hF : isOneToOne F) (hy : y ∈ ran F)
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: ∃! x, (x, y) ∈ F ∧ (y, x) ∈ inv F := by
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simp only [mem_self_comm_mem_inv, and_self]
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have ⟨x, hx⟩ := ran_exists hy
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refine ⟨x, hx, ?_⟩
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@ -468,13 +468,13 @@ Assume that `F` and `G` are functions. Then
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dom (F ∘ G) = {x ∈ dom G | G(x) ∈ dom F}.
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```
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-/
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theorem theorem_3h_dom {F G : Set.Relation α}
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(_ : F.isSingleValued) (hG : G.isSingleValued)
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: dom (F.comp G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F} := by
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theorem theorem_3h_dom {F : Set.HRelation β γ} {G : Set.HRelation α β}
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(_ : isSingleValued F) (hG : isSingleValued G)
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: dom (comp F G) = {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F} := by
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let rhs := {x ∈ dom G | ∃! y, (x, y) ∈ G ∧ y ∈ dom F }
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rw [Set.Subset.antisymm_iff]
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apply And.intro
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· show ∀ t, t ∈ dom (F.comp G) → t ∈ rhs
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· show ∀ t, t ∈ dom (comp F G) → t ∈ rhs
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intro t ht
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simp only [Set.mem_setOf_eq]
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have ⟨z, hz⟩ := dom_exists ht
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@ -488,7 +488,7 @@ theorem theorem_3h_dom {F G : Set.Relation α}
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refine ⟨a, ⟨ha.left, z, ha.right⟩, ?_⟩
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intro y₁ hy₁
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exact fun _ _ => single_valued_eq_unique hG hy₁ ha.left
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· show ∀ t, t ∈ rhs → t ∈ dom (F.comp G)
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· show ∀ t, t ∈ rhs → t ∈ dom (comp F G)
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intro t ht
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simp only [Set.mem_setOf_eq] at ht
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unfold dom
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@ -507,15 +507,15 @@ Assume that `F : A → B`, and that `A` is nonempty. There exists a function
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`G : B → A` (a "left inverse") such that `G ∘ F` is the identity function on `A`
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**iff** `F` is one-to-one.
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-/
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theorem theorem_3j_a {F : Set.Relation α} {A B : Set α}
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(hF : F.isSingleValued ∧ F.mapsInto A B) (hA : Set.Nonempty A)
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: (∃ G : Set.Relation α,
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G.isSingleValued ∧ G.mapsInto B A ∧
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(∀ p ∈ G.comp F, p.1 = p.2)) ↔ F.isOneToOne := by
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theorem theorem_3j_a {F : Set.HRelation α β} {A : Set α} {B : Set β}
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(hF : isSingleValued F ∧ mapsInto F A B) (hA : Set.Nonempty A)
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: (∃ G : Set.HRelation β α,
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isSingleValued G ∧ mapsInto G B A ∧
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(∀ p ∈ comp G F, p.1 = p.2)) ↔ isOneToOne F := by
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apply Iff.intro
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· intro ⟨G, ⟨hG₁, hG₂, hI⟩⟩
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refine ⟨hF.left, ?_⟩
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show F.isSingleRooted
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show isSingleRooted F
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intro y hy
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have ⟨x, hx⟩ := ran_exists hy
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sorry
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@ -528,11 +528,11 @@ Assume that `F : A → B`, and that `A` is nonempty. There exists a function
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`H : B → A` (a "right inverse") such that `F ∘ H` is the identity function on
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`B` **iff** `F` maps `A` onto `B`.
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-/
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theorem theorem_3j_b {F : Set.Relation α} {A B : Set α}
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(hF : F.isSingleValued ∧ F.mapsInto A B) (hA : Set.Nonempty A)
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: (∃ H : Set.Relation α,
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H.isSingleValued ∧ H.mapsInto B A ∧
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(∀ p ∈ F.comp H, p.1 = p.2)) ↔ F.mapsOnto A B := by
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theorem theorem_3j_b {F : Set.HRelation α β} {A : Set α} {B : Set β}
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(hF : isSingleValued F ∧ mapsInto F A B) (hA : Set.Nonempty A)
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: (∃ H : Set.HRelation β α,
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isSingleValued H ∧ mapsInto H B A ∧
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(∀ p ∈ comp F H, p.1 = p.2)) ↔ mapsOnto F A B := by
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sorry
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/-- #### Theorem 3K (a)
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@ -543,11 +543,11 @@ The image of a union is the union of the images:
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F⟦⋃ 𝓐⟧ = ⋃ {F⟦A⟧ | A ∈ 𝓐}
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```
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-/
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theorem theorem_3k_a {F : Set.Relation α} {𝓐 : Set (Set α)}
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: F.image (⋃₀ 𝓐) = ⋃₀ { F.image A | A ∈ 𝓐 } := by
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theorem theorem_3k_a {F : Set.HRelation α β} {𝓐 : Set (Set α)}
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: image F (⋃₀ 𝓐) = ⋃₀ { image F A | A ∈ 𝓐 } := by
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rw [Set.Subset.antisymm_iff]
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apply And.intro
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· show ∀ v, v ∈ F.image (⋃₀ 𝓐) → v ∈ ⋃₀ { F.image A | A ∈ 𝓐 }
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· show ∀ v, v ∈ image F (⋃₀ 𝓐) → v ∈ ⋃₀ { image F A | A ∈ 𝓐 }
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intro v hv
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unfold image at hv
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simp only [Set.mem_sUnion, Set.mem_setOf_eq] at hv
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@ -555,11 +555,11 @@ theorem theorem_3k_a {F : Set.Relation α} {𝓐 : Set (Set α)}
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have ⟨A, hA⟩ := hu.left
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simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and]
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refine ⟨A, hA.left, ?_⟩
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show v ∈ F.image A
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show v ∈ image F A
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unfold image
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simp only [Set.mem_setOf_eq]
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exact ⟨u, hA.right, hu.right⟩
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· show ∀ v, v ∈ ⋃₀ {x | ∃ A, A ∈ 𝓐 ∧ F.image A = x} → v ∈ F.image (⋃₀ 𝓐)
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· show ∀ v, v ∈ ⋃₀ {x | ∃ A, A ∈ 𝓐 ∧ image F A = x} → v ∈ image F (⋃₀ 𝓐)
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intro v hv
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simp only [Set.mem_sUnion, Set.mem_setOf_eq, exists_exists_and_eq_and] at hv
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have ⟨A, hA⟩ := hv
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@ -580,9 +580,9 @@ F⟦⋂ 𝓐⟧ ⊆ ⋂ {F⟦A⟧ | A ∈ 𝓐}
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|||
Equality holds if `F` is single-rooted.
|
||||
-/
|
||||
|
||||
theorem theorem_3k_b_i {F : Set.Relation α} {𝓐 : Set (Set α)}
|
||||
: F.image (⋂₀ 𝓐) ⊆ ⋂₀ { F.image A | A ∈ 𝓐} := by
|
||||
show ∀ v, v ∈ F.image (⋂₀ 𝓐) → v ∈ ⋂₀ { F.image A | A ∈ 𝓐}
|
||||
theorem theorem_3k_b_i {F : Set.HRelation α β} {𝓐 : Set (Set α)}
|
||||
: image F (⋂₀ 𝓐) ⊆ ⋂₀ { image F A | A ∈ 𝓐} := by
|
||||
show ∀ v, v ∈ image F (⋂₀ 𝓐) → v ∈ ⋂₀ { image F A | A ∈ 𝓐}
|
||||
intro v hv
|
||||
unfold image 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]
|
||||
exact ⟨u, hu.left A hA, hu.right⟩
|
||||
|
||||
theorem theorem_3k_b_ii {F : Set.Relation α} {𝓐 : Set (Set α)}
|
||||
(hF : F.isSingleRooted) (h𝓐 : Set.Nonempty 𝓐)
|
||||
: F.image (⋂₀ 𝓐) = ⋂₀ { F.image A | A ∈ 𝓐} := by
|
||||
theorem theorem_3k_b_ii {F : Set.HRelation α β} {𝓐 : Set (Set α)}
|
||||
(hF : isSingleRooted F) (h𝓐 : Set.Nonempty 𝓐)
|
||||
: image F (⋂₀ 𝓐) = ⋂₀ { image F A | A ∈ 𝓐} := by
|
||||
rw [Set.Subset.antisymm_iff]
|
||||
refine ⟨theorem_3k_b_i, ?_⟩
|
||||
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.
|
||||
-/
|
||||
|
||||
theorem theorem_3k_c_i {F : Set.Relation α} {A B : Set α}
|
||||
: F.image A \ F.image B ⊆ F.image (A \ B) := by
|
||||
show ∀ v, v ∈ F.image A \ F.image B → v ∈ F.image (A \ B)
|
||||
theorem theorem_3k_c_i {F : Set.HRelation α β} {A B : Set α}
|
||||
: image F A \ image F B ⊆ image F (A \ B) := by
|
||||
show ∀ v, v ∈ image F A \ image F B → v ∈ image F (A \ B)
|
||||
intro v 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]
|
||||
|
|
@ -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]
|
||||
exact ⟨u, ⟨hu.left, hu'⟩, hu.right⟩
|
||||
|
||||
theorem theorem_3k_c_ii {F : Set.Relation α} {A B : Set α}
|
||||
(hF : F.isSingleRooted)
|
||||
: F.image A \ F.image B = F.image (A \ B) := by
|
||||
theorem theorem_3k_c_ii {F : Set.HRelation α β} {A B : Set α}
|
||||
(hF : isSingleRooted F)
|
||||
: image F A \ image F B = image F (A \ B) := by
|
||||
rw [Set.Subset.antisymm_iff]
|
||||
refine ⟨theorem_3k_c_i, ?_⟩
|
||||
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
|
||||
simp only [Set.mem_diff, Set.mem_setOf_eq] at hv
|
||||
have ⟨u, hu⟩ := hv
|
||||
have hv₁ : v ∈ F.image A := by
|
||||
have hv₁ : v ∈ image F A := by
|
||||
unfold image
|
||||
simp only [Set.mem_setOf_eq]
|
||||
exact ⟨u, hu.left.left, hu.right⟩
|
||||
have hv₂ : v ∉ F.image B := by
|
||||
have hv₂ : v ∉ image F B := by
|
||||
intro nv
|
||||
unfold image 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)}
|
||||
: G.inv.image (⋃₀ 𝓐) = ⋃₀ {G.inv.image A | A ∈ 𝓐} := theorem_3k_a
|
||||
theorem corollary_3l_i {G : Set.HRelation β α} {𝓐 : Set (Set α)}
|
||||
: 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)}
|
||||
(hG : G.isSingleValued) (h𝓐 : Set.Nonempty 𝓐)
|
||||
: G.inv.image (⋂₀ 𝓐) = ⋂₀ {G.inv.image A | A ∈ 𝓐} := by
|
||||
have hG' : G.inv.isSingleRooted :=
|
||||
theorem corollary_3l_ii {G : Set.HRelation β α} {𝓐 : Set (Set α)}
|
||||
(hG : isSingleValued G) (h𝓐 : Set.Nonempty 𝓐)
|
||||
: image (inv G) (⋂₀ 𝓐) = ⋂₀ {image (inv G) A | A ∈ 𝓐} := by
|
||||
have hG' : isSingleRooted (inv G) :=
|
||||
single_valued_self_iff_single_rooted_inv.mp hG
|
||||
exact theorem_3k_b_ii hG' h𝓐
|
||||
|
||||
theorem corollary_3l_iii {G : Set.Relation α} {A B : Set α}
|
||||
(hG : G.isSingleValued)
|
||||
: G.inv.image (A \ B) = G.inv.image A \ G.inv.image B := by
|
||||
have hG' : G.inv.isSingleRooted :=
|
||||
theorem corollary_3l_iii {G : Set.HRelation β α} {A B : Set α}
|
||||
(hG : isSingleValued G)
|
||||
: image (inv G) (A \ B) = image (inv G) A \ image (inv G) B := by
|
||||
have hG' : isSingleRooted (inv G) :=
|
||||
single_valued_self_iff_single_rooted_inv.mp hG
|
||||
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).
|
||||
```
|
||||
-/
|
||||
theorem exercise_3_12 {f g : Set.Relation α}
|
||||
(hf : f.isSingleValued) (_ : g.isSingleValued)
|
||||
theorem exercise_3_12 {f g : Set.HRelation α β}
|
||||
(hf : isSingleValued f) (_ : isSingleValued 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
|
||||
· intro h
|
||||
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
|
||||
that `f = g`.
|
||||
-/
|
||||
theorem exercise_3_13 {f g : Set.Relation α}
|
||||
(hf : f.isSingleValued) (hg : g.isSingleValued)
|
||||
theorem exercise_3_13 {f g : Set.HRelation α β}
|
||||
(hf : isSingleValued f) (hg : isSingleValued g)
|
||||
(h : f ⊆ g) (h₁ : dom g ⊆ dom f)
|
||||
: f = g := by
|
||||
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.
|
||||
-/
|
||||
theorem exercise_3_14_a {f g : Set.Relation α}
|
||||
(hf : f.isSingleValued) (_ : g.isSingleValued)
|
||||
: (f ∩ g).isSingleValued :=
|
||||
theorem exercise_3_14_a {f g : Set.HRelation α β}
|
||||
(hf : isSingleValued f) (_ : isSingleValued g)
|
||||
: isSingleValued (f ∩ g) :=
|
||||
single_valued_subset hf (Set.inter_subset_left f g)
|
||||
|
||||
/-- #### 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**
|
||||
`f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`.
|
||||
-/
|
||||
theorem exercise_3_14_b {f g : Set.Relation α}
|
||||
(hf : f.isSingleValued) (hg : g.isSingleValued)
|
||||
: (f ∪ g).isSingleValued ↔
|
||||
theorem exercise_3_14_b {f g : Set.HRelation α β}
|
||||
(hf : isSingleValued f) (hg : isSingleValued g)
|
||||
: isSingleValued (f ∪ g) ↔
|
||||
(∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by
|
||||
apply Iff.intro
|
||||
· 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
|
||||
`f ⊆ g` or `g ⊆ f`. Show that `⋃ 𝓐` is a function.
|
||||
-/
|
||||
theorem exercise_3_15 {𝓐 : Set (Set.Relation α)}
|
||||
(h𝓐 : ∀ F ∈ 𝓐, F.isSingleValued)
|
||||
theorem exercise_3_15 {𝓐 : Set (Set.HRelation α β)}
|
||||
(h𝓐 : ∀ F ∈ 𝓐, isSingleValued F)
|
||||
(h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G ∨ G ⊆ F)
|
||||
: isSingleValued (⋃₀ 𝓐) := by
|
||||
intro x hx
|
||||
have ⟨y₁, hy₁⟩ := dom_exists hx
|
||||
refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩
|
||||
intro y₂ hy₂
|
||||
have ⟨f, hf⟩ : ∃ f : Set.Relation α, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁
|
||||
have ⟨g, hg⟩ : ∃ g : Set.Relation α, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right
|
||||
have ⟨f, hf⟩ : ∃ f : Set.HRelation α β, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁
|
||||
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)
|
||||
· intro hf'
|
||||
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.
|
||||
-/
|
||||
|
||||
theorem exercise_3_17_i {F G : Set.Relation α}
|
||||
(hF : F.isSingleRooted) (hG : G.isSingleRooted)
|
||||
: (F.comp G).isSingleRooted := by
|
||||
theorem exercise_3_17_i {F : Set.HRelation β γ} {G : Set.HRelation α β}
|
||||
(hF : isSingleRooted F) (hG : isSingleRooted G)
|
||||
: isSingleRooted (comp F G):= by
|
||||
intro v hv
|
||||
|
||||
have ⟨u₁, hu₁⟩ := ran_exists hv
|
||||
|
|
@ -907,11 +907,11 @@ theorem exercise_3_17_i {F G : Set.Relation α}
|
|||
rw [ht] at ht₁
|
||||
exact single_rooted_eq_unique hG ht₂.left ht₁.left
|
||||
|
||||
theorem exercise_3_17_ii {F G : Set.Relation α}
|
||||
(hF : F.isOneToOne) (hG : G.isOneToOne)
|
||||
: (F.comp G).isOneToOne := And.intro
|
||||
(single_valued_comp_is_single_valued hF.left hG.left)
|
||||
(exercise_3_17_i hF.right hG.right)
|
||||
theorem exercise_3_17_ii {F : Set.HRelation β γ} {G : Set.HRelation α β}
|
||||
(hF : isOneToOne F) (hG : isOneToOne G)
|
||||
: isOneToOne (comp F G) := And.intro
|
||||
(single_valued_comp_is_single_valued hF.left hG.left)
|
||||
(exercise_3_17_i hF.right hG.right)
|
||||
|
||||
/-! #### Exercise 3.18
|
||||
|
||||
|
|
@ -929,7 +929,7 @@ variable {R : Set.Relation ℕ}
|
|||
variable (hR : R = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)})
|
||||
|
||||
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]
|
||||
unfold comp
|
||||
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⟩
|
||||
|
||||
theorem exercise_3_18_ii
|
||||
: R.restriction {1} = {(1, 2), (1, 3)} := by
|
||||
: restriction R {1} = {(1, 2), (1, 3)} := by
|
||||
rw [hR]
|
||||
unfold restriction
|
||||
ext p
|
||||
|
|
@ -1006,7 +1006,7 @@ theorem exercise_3_18_ii
|
|||
rw [hab.left, hab.right]
|
||||
|
||||
theorem exercise_3_18_iii
|
||||
: R.inv.restriction {1} = {(1, 0)} := by
|
||||
: restriction (inv R) {1} = {(1, 0)} := by
|
||||
rw [hR]
|
||||
unfold inv restriction
|
||||
ext p
|
||||
|
|
@ -1031,14 +1031,14 @@ theorem exercise_3_18_iii
|
|||
simp
|
||||
|
||||
theorem exercise_3_18_iv
|
||||
: R.image {1} = {2, 3} := by
|
||||
: image R {1} = {2, 3} := by
|
||||
rw [hR]
|
||||
unfold image
|
||||
ext y
|
||||
simp
|
||||
|
||||
theorem exercise_3_18_v
|
||||
: R.inv.image {1} = {0} := by
|
||||
: image (inv R) {1} = {0} := by
|
||||
rw [hR]
|
||||
unfold inv image
|
||||
ext y
|
||||
|
|
@ -1067,12 +1067,12 @@ theorem exercise_3_19_i
|
|||
simp
|
||||
|
||||
theorem exercise_3_19_ii
|
||||
: A.image ∅ = ∅ := by
|
||||
: image A ∅ = ∅ := by
|
||||
unfold image
|
||||
simp
|
||||
|
||||
theorem exercise_3_19_iii
|
||||
: A.image {∅} = {{∅, {∅}}} := by
|
||||
: image A {∅} = {{∅, {∅}}} := by
|
||||
unfold image
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1098,7 +1098,7 @@ theorem exercise_3_19_iii
|
|||
simp
|
||||
|
||||
theorem exercise_3_19_iv
|
||||
: A.image {∅, {∅}} = {{∅, {∅}}, ∅} := by
|
||||
: image A {∅, {∅}} = {{∅, {∅}}, ∅} := by
|
||||
unfold image
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1131,7 +1131,7 @@ theorem exercise_3_19_iv
|
|||
· intro hx₁; right; exact hx₁
|
||||
|
||||
theorem exercise_3_19_v
|
||||
: A.inv = {({∅, {∅}}, ∅), (∅, {∅})} := by
|
||||
: inv A = {({∅, {∅}}, ∅), (∅, {∅})} := by
|
||||
unfold inv
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1154,7 +1154,7 @@ theorem exercise_3_19_v
|
|||
· intro hx₁; right; rw [← hx₁]
|
||||
|
||||
theorem exercise_3_19_vi
|
||||
: A.comp A = {({∅}, {∅, {∅}})} := by
|
||||
: comp A A = {({∅}, {∅, {∅}})} := by
|
||||
unfold comp
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1179,13 +1179,13 @@ theorem exercise_3_19_vi
|
|||
· left ; rw [hb]; simp
|
||||
|
||||
theorem exercise_3_19_vii
|
||||
: A.restriction ∅ = ∅ := by
|
||||
: restriction A ∅ = ∅ := by
|
||||
unfold restriction
|
||||
rw [hA]
|
||||
simp
|
||||
|
||||
theorem exercise_3_19_viii
|
||||
: A.restriction {∅} = {(∅, {∅, {∅}})} := by
|
||||
: restriction A {∅} = {(∅, {∅, {∅}})} := by
|
||||
unfold restriction
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1206,7 +1206,7 @@ theorem exercise_3_19_viii
|
|||
simp
|
||||
|
||||
theorem exercise_3_19_ix
|
||||
: A.restriction {∅, {∅}} = A := by
|
||||
: restriction A {∅, {∅}} = A := by
|
||||
unfold restriction
|
||||
rw [hA]
|
||||
ext x
|
||||
|
|
@ -1300,9 +1300,9 @@ end Exercise_3_19
|
|||
|
||||
Show that `F ↾ A = F ∩ (A × ran F)`.
|
||||
-/
|
||||
theorem exercise_3_20 {F : Set.Relation α} {A : Set α}
|
||||
: F.restriction A = F ∩ (Set.prod A (ran F)) := by
|
||||
calc F.restriction A
|
||||
theorem exercise_3_20 {F : Set.HRelation α β} {A : Set α}
|
||||
: restriction F A = F ∩ (Set.prod A (ran F)) := by
|
||||
calc restriction F A
|
||||
_ = {p | p ∈ F ∧ p.fst ∈ A} := rfl
|
||||
_ = {p | p ∈ F ∧ p.fst ∈ A ∧ p.snd ∈ ran F} := by
|
||||
ext x
|
||||
|
|
@ -1323,9 +1323,9 @@ Show that the following is correct for any sets.
|
|||
A ⊆ B → F⟦A⟧ ⊆ F⟦B⟧
|
||||
```
|
||||
-/
|
||||
theorem exercise_3_22_a {A B : Set α} {F : Set.Relation α} (h : A ⊆ B)
|
||||
: F.image A ⊆ F.image B := by
|
||||
show ∀ x, x ∈ F.image A → x ∈ F.image B
|
||||
theorem exercise_3_22_a {A B : Set α} {F : Set.HRelation α β} (h : A ⊆ B)
|
||||
: image F A ⊆ image F B := by
|
||||
show ∀ x, x ∈ image F A → x ∈ image F B
|
||||
unfold image
|
||||
simp only [Set.mem_setOf_eq]
|
||||
intro x hx
|
||||
|
|
@ -1340,10 +1340,10 @@ Show that the following is correct for any sets.
|
|||
(F ∘ G)⟦A⟧ = F⟦G⟦A⟧⟧
|
||||
```
|
||||
-/
|
||||
theorem exercise_3_22_b {A B : Set α} {F : Set.Relation α}
|
||||
: (F.comp G).image A = F.image (G.image A) := by
|
||||
calc (F.comp G).image A
|
||||
_ = { v | ∃ u ∈ A, (u, v) ∈ F.comp G } := rfl
|
||||
theorem exercise_3_22_b {A B : Set α} {F : Set.HRelation α β}
|
||||
: image (comp F G) A = image F (image G A) := by
|
||||
calc image (comp F G) A
|
||||
_ = { v | ∃ u ∈ A, (u, v) ∈ comp F G } := rfl
|
||||
_ = { v | ∃ u ∈ A, ∃ a, (u, a) ∈ G ∧ (a, v) ∈ F } := rfl
|
||||
_ = { v | ∃ a, ∃ u ∈ A, (u, a) ∈ G ∧ (a, v) ∈ F } := by
|
||||
ext p
|
||||
|
|
@ -1362,8 +1362,8 @@ theorem exercise_3_22_b {A B : Set α} {F : Set.Relation α}
|
|||
· intro ⟨a, ⟨u, hu⟩, ha⟩
|
||||
exact ⟨a, u, hu.left, hu.right, ha⟩
|
||||
_ = { v | ∃ a ∈ { w | ∃ u ∈ A, (u, w) ∈ G }, (a, v) ∈ F } := rfl
|
||||
_ = { v | ∃ a ∈ G.image A, (a, v) ∈ F } := rfl
|
||||
_ = F.image (G.image A) := rfl
|
||||
_ = { v | ∃ a ∈ image G A, (a, v) ∈ F } := rfl
|
||||
_ = image F (image G A) := rfl
|
||||
|
||||
/-- #### 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}
|
||||
: Q.restriction (A ∪ B) = (Q.restriction A) ∪ (Q.restriction B) := by
|
||||
calc Q.restriction (A ∪ B)
|
||||
: restriction Q (A ∪ B) = (restriction Q A) ∪ (restriction Q B) := by
|
||||
calc restriction Q (A ∪ B)
|
||||
_ = { 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 ∈ Q ∧ p.1 ∈ B) } := by
|
||||
ext p
|
||||
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
|
||||
_ = (Q.restriction A) ∪ (Q.restriction B) := rfl
|
||||
_ = (restriction Q A) ∪ (restriction Q B) := rfl
|
||||
|
||||
/-- #### Exercise 3.23 (i)
|
||||
|
||||
Let `I` be the identity function on the set `A`. Show that for any sets `B` and
|
||||
`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 })
|
||||
: B.comp I = B.restriction A := by
|
||||
: comp B I = restriction B A := by
|
||||
rw [Set.Subset.antisymm_iff]
|
||||
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
|
||||
have ⟨t, ht⟩ := hp
|
||||
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
|
||||
rw [← ht.left.right] at ht
|
||||
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
|
||||
rw [hI]
|
||||
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}
|
||||
(hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
|
||||
: I.image C = A ∩ C := by
|
||||
calc I.image C
|
||||
: image I C = A ∩ C := by
|
||||
calc image I C
|
||||
_ = { v | ∃ u ∈ C, (u, v) ∈ I } := rfl
|
||||
_ = { v | ∃ u ∈ C, u ∈ A ∧ u = v } := by
|
||||
ext v
|
||||
|
|
@ -1443,6 +1443,205 @@ theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α}
|
|||
_ = C ∩ A := rfl
|
||||
_ = 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 Enderton.Set.Chapter_3
|
||||
|
|
@ -16,7 +16,12 @@ Kuratowski's definition of a set.
|
|||
|
||||
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
|
||||
|
||||
|
|
@ -25,13 +30,13 @@ namespace 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
|
||||
`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
|
||||
unfold dom Prod.fst
|
||||
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`.
|
||||
-/
|
||||
theorem dom_exists {R : Relation α} (hx : x ∈ R.dom)
|
||||
: ∃ y : α, (x, y) ∈ R := by
|
||||
theorem dom_exists {R : HRelation α β} (hx : x ∈ dom R)
|
||||
: ∃ y : β, (x, y) ∈ R := by
|
||||
unfold dom at hx
|
||||
simp only [mem_image, Prod.exists, exists_and_right, exists_eq_right] at hx
|
||||
exact hx
|
||||
|
|
@ -49,9 +54,9 @@ theorem dom_exists {R : Relation α} (hx : x ∈ R.dom)
|
|||
/--
|
||||
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
|
||||
unfold ran Prod.snd
|
||||
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`.
|
||||
-/
|
||||
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
|
||||
unfold ran 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`.
|
||||
-/
|
||||
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`.
|
||||
-/
|
||||
@[simp]
|
||||
theorem mem_self_comm_mem_inv {R : Relation α}
|
||||
: (y, x) ∈ R.inv ↔ (x, y) ∈ R := by
|
||||
theorem mem_self_comm_mem_inv {R : HRelation α β}
|
||||
: (y, x) ∈ inv R ↔ (x, y) ∈ R := by
|
||||
unfold inv
|
||||
simp only [Prod.exists, mem_setOf_eq, Prod.mk.injEq]
|
||||
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`.
|
||||
-/
|
||||
@[simp]
|
||||
theorem inv_inv_eq_self (R : Relation α)
|
||||
: R.inv.inv = R := by
|
||||
theorem inv_inv_eq_self (R : HRelation α β)
|
||||
: inv (inv R) = R := by
|
||||
unfold inv
|
||||
simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
|
||||
ext x
|
||||
|
|
@ -115,8 +120,8 @@ theorem inv_inv_eq_self (R : Relation α)
|
|||
For a set `F`, `dom F⁻¹ = ran F`.
|
||||
-/
|
||||
@[simp]
|
||||
theorem dom_inv_eq_ran_self {F : Relation α}
|
||||
: dom (F.inv) = ran F := by
|
||||
theorem dom_inv_eq_ran_self {F : HRelation α β}
|
||||
: dom (inv F) = ran F := by
|
||||
ext x
|
||||
unfold dom ran inv
|
||||
simp only [
|
||||
|
|
@ -138,8 +143,8 @@ theorem dom_inv_eq_ran_self {F : Relation α}
|
|||
For a set `F`, `ran F⁻¹ = dom F`.
|
||||
-/
|
||||
@[simp]
|
||||
theorem ran_inv_eq_dom_self {F : Relation α}
|
||||
: ran (F.inv) = dom F := by
|
||||
theorem ran_inv_eq_dom_self {F : HRelation α β}
|
||||
: ran (inv F) = dom F := by
|
||||
ext x
|
||||
unfold dom ran inv
|
||||
simp only [
|
||||
|
|
@ -162,7 +167,7 @@ theorem ran_inv_eq_dom_self {F : Relation α}
|
|||
/--
|
||||
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 }
|
||||
|
||||
/-! ## Image -/
|
||||
|
|
@ -170,7 +175,7 @@ def restriction (R : Relation α) (A : Set α) : Relation α :=
|
|||
/--
|
||||
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 }
|
||||
|
||||
/-! ## 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
|
||||
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
|
||||
|
||||
/--
|
||||
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}
|
||||
(hR : R.isSingleRooted)
|
||||
theorem single_rooted_eq_unique {R : HRelation α β} {x₁ x₂ : α} {y : β}
|
||||
(hR : isSingleRooted R)
|
||||
: (x₁, y) ∈ R → (x₂, y) ∈ R → x₁ = x₂ := by
|
||||
intro hx₁ hx₂
|
||||
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.
|
||||
-/
|
||||
def isSingleValued (R : Relation α) : Prop :=
|
||||
def isSingleValued (R : HRelation α β) : Prop :=
|
||||
∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R
|
||||
|
||||
/--
|
||||
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}
|
||||
(hR : R.isSingleValued)
|
||||
theorem single_valued_eq_unique {R : HRelation α β} {x : α} {y₁ y₂ : β}
|
||||
(hR : isSingleValued R)
|
||||
: (x, y₁) ∈ R → (x, y₂) ∈ R → y₁ = y₂ := by
|
||||
intro hy₁ hy₂
|
||||
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.
|
||||
-/
|
||||
theorem single_valued_inv_iff_single_rooted_self {F : Set.Relation α}
|
||||
: F.inv.isSingleValued ↔ F.isSingleRooted := by
|
||||
theorem single_valued_inv_iff_single_rooted_self {F : HRelation α β}
|
||||
: isSingleValued (inv F) ↔ isSingleRooted F := by
|
||||
apply Iff.intro
|
||||
· intro 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,
|
||||
mem_self_comm_mem_inv
|
||||
] 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
|
||||
have ⟨y, hy⟩ := hF x hx
|
||||
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.
|
||||
-/
|
||||
theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α}
|
||||
: F.isSingleValued ↔ F.inv.isSingleRooted := by
|
||||
theorem single_valued_self_iff_single_rooted_inv {F : HRelation α β}
|
||||
: isSingleValued F ↔ isSingleRooted (inv F) := by
|
||||
conv => lhs; rw [← inv_inv_eq_self F]
|
||||
rw [single_valued_inv_iff_single_rooted_self]
|
||||
|
||||
/--
|
||||
The subset of a function must also be a function.
|
||||
-/
|
||||
theorem single_valued_subset {F G : Set.Relation α}
|
||||
(hG : G.isSingleValued) (h : F ⊆ G)
|
||||
: F.isSingleValued := by
|
||||
theorem single_valued_subset {F G : HRelation α β}
|
||||
(hG : isSingleValued G) (h : F ⊆ G)
|
||||
: isSingleValued F := by
|
||||
unfold isSingleValued
|
||||
intro x 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.
|
||||
-/
|
||||
def isOneToOne (R : Relation α) : Prop :=
|
||||
R.isSingleValued ∧ R.isSingleRooted
|
||||
def isOneToOne (R : HRelation α β) : 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}
|
||||
: R.isOneToOne ↔ R.inv.isOneToOne := by
|
||||
theorem one_to_one_self_iff_one_to_one_inv {R : HRelation α β}
|
||||
: isOneToOne R ↔ isOneToOne (inv R) := by
|
||||
unfold isOneToOne isSingleValued isSingleRooted
|
||||
conv => rhs; simp only [
|
||||
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`.
|
||||
-/
|
||||
def mapsInto (F : Relation α) (A B : Set α) :=
|
||||
F.isSingleValued ∧ dom F = A ∧ ran F ⊆ B
|
||||
def mapsInto (F : HRelation α β) (A : Set α) (B : Set β) :=
|
||||
isSingleValued F ∧ dom F = A ∧ ran F ⊆ B
|
||||
|
||||
/--
|
||||
Indicates `Relation` `F` is a function from `A` to `ran F = B`.
|
||||
-/
|
||||
def mapsOnto (F : Relation α) (A B : Set α) :=
|
||||
F.isSingleValued ∧ dom F = A ∧ ran F = B
|
||||
def mapsOnto (F : HRelation α β) (A : Set α) (B : Set β) :=
|
||||
isSingleValued F ∧ dom F = A ∧ ran F = B
|
||||
|
||||
/-! ## Composition -/
|
||||
|
||||
/--
|
||||
The composition of two `Relation`s.
|
||||
-/
|
||||
def comp (F G : Relation α) : Relation α :=
|
||||
{ p | ∃ t : α, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
|
||||
def comp (F : HRelation β γ) (G : HRelation α β) : HRelation α γ :=
|
||||
{ p | ∃ t : β, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
|
||||
|
||||
/--
|
||||
If `x ∈ dom (F ∘ G)`, then `x ∈ dom G`.
|
||||
-/
|
||||
theorem dom_comp_imp_dom_self {F G : Relation α}
|
||||
: x ∈ dom (F.comp G) → x ∈ dom G := by
|
||||
theorem dom_comp_imp_dom_self {F : HRelation β γ} {G : HRelation α β}
|
||||
: x ∈ dom (comp F G) → x ∈ dom G := by
|
||||
unfold dom comp
|
||||
simp only [
|
||||
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`.
|
||||
-/
|
||||
theorem ran_comp_imp_ran_self {F G : Relation α}
|
||||
: y ∈ ran (F.comp G) → y ∈ ran F := by
|
||||
theorem ran_comp_imp_ran_self {F : HRelation β γ} {G : HRelation α β}
|
||||
: y ∈ ran (comp F G) → y ∈ ran F := by
|
||||
unfold ran comp
|
||||
simp only [
|
||||
mem_image,
|
||||
|
|
@ -362,10 +367,10 @@ theorem ran_comp_imp_ran_self {F G : Relation α}
|
|||
/--
|
||||
Composition of functions is associative.
|
||||
-/
|
||||
theorem comp_assoc {R S T : Relation α}
|
||||
: (R.comp S).comp T = R.comp (S.comp T) := by
|
||||
calc (R.comp S).comp T
|
||||
_ = { p | ∃ t, (p.1, t) ∈ T ∧ (t, p.2) ∈ R.comp S} := rfl
|
||||
theorem comp_assoc {R : HRelation γ δ} {S : HRelation β γ} {T : HRelation α β}
|
||||
: comp (comp R S) T = comp R (comp S T) := by
|
||||
calc comp (comp R S) T
|
||||
_ = { 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, ∃ a, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
|
||||
ext p
|
||||
|
|
@ -391,15 +396,16 @@ theorem comp_assoc {R S T : Relation α}
|
|||
exact ⟨a, ⟨t, h.left⟩, h.right⟩
|
||||
· intro ⟨a, ⟨t, ht⟩, ha⟩
|
||||
exact ⟨a, t, ht, ha⟩
|
||||
_ = { p | ∃ a, (p.1, a) ∈ S.comp T ∧ (a, p.2) ∈ R } := rfl
|
||||
_ = R.comp (S.comp T) := rfl
|
||||
_ = { p | ∃ a, (p.1, a) ∈ comp S T ∧ (a, p.2) ∈ R } := rfl
|
||||
_ = comp R (comp S T) := rfl
|
||||
|
||||
/--
|
||||
The composition of two functions is itself a function.
|
||||
-/
|
||||
theorem single_valued_comp_is_single_valued {F G : Relation α}
|
||||
(hF : F.isSingleValued) (hG : G.isSingleValued)
|
||||
: (F.comp G).isSingleValued := by
|
||||
theorem single_valued_comp_is_single_valued
|
||||
{F : HRelation β γ} {G : HRelation α β}
|
||||
(hF : isSingleValued F) (hG : isSingleValued G)
|
||||
: isSingleValued (comp F G) := by
|
||||
unfold isSingleValued
|
||||
intro x 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`.
|
||||
-/
|
||||
theorem comp_inv_eq_inv_comp_inv {F G : Relation α}
|
||||
: (F.comp G).inv = G.inv.comp F.inv := by
|
||||
calc (F.comp G).inv
|
||||
theorem comp_inv_eq_inv_comp_inv {F : HRelation β γ} {G : HRelation α β}
|
||||
: inv (comp F G) = comp (inv G) (inv F) := by
|
||||
calc inv (comp F G)
|
||||
_ = {p | ∃ t, (p.2, t) ∈ G ∧ (t, p.1) ∈ F} := by
|
||||
rw [Set.Subset.antisymm_iff]
|
||||
apply And.intro
|
||||
|
|
@ -460,7 +466,7 @@ theorem comp_inv_eq_inv_comp_inv {F G : Relation α}
|
|||
simp only [mem_setOf_eq] at *
|
||||
have ⟨t, p, q⟩ := ht
|
||||
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]
|
||||
apply And.intro
|
||||
· intro (a, b) ht
|
||||
|
|
|
|||
Loading…
Reference in New Issue