Update proofs to pending if there is no custom proof set.
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@ -1114,7 +1114,7 @@ List all the members of $V_4$.
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\section{Algebra of Sets}%
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\hyperlabel{sec:algebra-sets}
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\subsection{\verified{Commutative Laws}}%
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\subsection{\pending{Commutative Laws}}%
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\hyperlabel{sub:commutative-laws}
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For any sets $A$ and $B$,
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@ -1160,7 +1160,7 @@ For any sets $A$ and $B$,
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\end{proof}
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\subsection{\verified{Associative Laws}}%
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\subsection{\pending{Associative Laws}}%
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\hyperlabel{sub:associative-laws}
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For any sets $A$, $B$ and $C$,
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@ -1217,7 +1217,7 @@ For any sets $A$, $B$ and $C$,
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\end{proof}
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\subsection{\verified{Distributive Laws}}%
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\subsection{\pending{Distributive Laws}}%
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\hyperlabel{sub:distributive-laws}
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For any sets $A$, $B$, and $C$,
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@ -1273,7 +1273,7 @@ For any sets $A$, $B$, and $C$,
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\end{proof}
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\subsection{\verified{De Morgan's Laws}}%
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\subsection{\pending{De Morgan's Laws}}%
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\hyperlabel{sub:de-morgans-laws}
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For any sets $A$, $B$, and $C$,
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@ -1331,7 +1331,7 @@ For any sets $A$, $B$, and $C$,
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\end{proof}
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\subsection{\verified{%
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\subsection{\pending{%
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Identities Involving \texorpdfstring{$\emptyset$}{the Empty Set}}}%
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\hyperlabel{sub:identitives-involving-empty-set}
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@ -1401,7 +1401,7 @@ For any set $A$,
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\end{proof}
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\subsection{\verified{Monotonicity}}%
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\subsection{\pending{Monotonicity}}%
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\hyperlabel{sub:monotonicity}
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For any sets $A$, $B$, and $C$,
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@ -1474,7 +1474,7 @@ For any sets $A$, $B$, and $C$,
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\end{proof}
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\subsection{\verified{Anti-monotonicity}}%
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\subsection{\pending{Anti-monotonicity}}%
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\hyperlabel{sub:anti-monotonicity}
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For any sets $A$, $B$, and $C$,
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@ -1644,7 +1644,7 @@ For any set $C$ and $\mathscr{A} \neq \emptyset$,
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\end{proof}
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\subsection{\verified{%
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\subsection{\pending{%
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\texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
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\hyperlabel{sub:intersection-difference-associativity}
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@ -2187,7 +2187,7 @@ Show by example that for some sets $A$, $B$, and $C$, the set $A - (B - C)$ is
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\end{proof}
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\subsection{\verified{Exercise 2.15a}}%
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\subsection{\pending{Exercise 2.15a}}%
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\hyperlabel{sub:exercise-2.15a}
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Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$.
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@ -2224,7 +2224,7 @@ Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$.
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\end{proof}
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\subsection{\verified{Exercise 2.15b}}%
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\subsection{\pending{Exercise 2.15b}}%
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\hyperlabel{sub:exercise-2.15b}
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Show that $A + (B + C) = (A + B) + C$.
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@ -2914,7 +2914,7 @@ If not, then under what conditions does equality hold?
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\end{proof}
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\subsection{\verified{Corollary 3C}}%
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\subsection{\pending{Corollary 3C}}%
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\hyperlabel{sub:corollary-3c}
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\begin{theorem}[3C]
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@ -3730,7 +3730,8 @@ If not, then under what conditions does equality hold?
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Relation}{Set.Relation.modEquiv\_partition}
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\lean{Bookshelf/Enderton/Set/Relation}
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{Set.Relation.modEquiv\_partition}
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Let $\Pi = \{[x]_R \mid x \in A\}$.
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We show that (i) there are no empty sets in $\Pi$, (ii) no two different sets
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@ -4345,7 +4346,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
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\end{answer}
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\subsection{\verified{Exercise 3.11}}%
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\subsection{\pending{Exercise 3.11}}%
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\hyperlabel{sub:exercise-3.11}
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Prove the following version (for functions) of the extensionality principle:
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@ -6527,7 +6528,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\section{Arithmetic}%
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\hyperlabel{sec:arithmetic}
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\subsection{\verified{Theorem 4I}}
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\subsection{\pending{Theorem 4I}}
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\hyperlabel{sub:theorem-4i}
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\begin{theorem}[4I]
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@ -6563,7 +6564,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4J}}
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\subsection{\pending{Theorem 4J}}
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\hyperlabel{sub:theorem-4j}
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\begin{theorem}[4J]
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@ -6597,7 +6598,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Left Additive Identity}}%
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\subsection{\pending{Left Additive Identity}}%
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\hyperlabel{sub:left-additive-identity}
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\begin{lemma}
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@ -6639,7 +6640,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Lemma 3}}%
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\subsection{\pending{Lemma 3}}%
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\hyperlabel{sub:lemma-3}
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\hyperlabel{sub:succ-add-eq-add-succ}
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@ -6683,7 +6684,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4K-1}}%
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\subsection{\pending{Theorem 4K-1}}%
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\label{sub:theorem-4k-1}
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\begin{theorem}[4K-1]
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@ -6739,7 +6740,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4K-2}}%
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\subsection{\pending{Theorem 4K-2}}%
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\label{sub:theorem-4k-2}
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\begin{theorem}[4K-2]
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@ -6791,7 +6792,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Zero Multiplicand}}%
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\subsection{\pending{Zero Multiplicand}}%
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\hyperlabel{sub:zero-multiplicand}
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\begin{lemma}
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@ -6841,7 +6842,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Successor Distribution}}%
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\subsection{\pending{Successor Distribution}}%
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\hyperlabel{sub:successor-distribution}
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\begin{lemma}
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@ -6897,7 +6898,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4K-3}}
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\subsection{\pending{Theorem 4K-3}}
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\hyperlabel{sub:theorem-4k-3}
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\begin{theorem}[4K-3]
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@ -6958,7 +6959,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Successor Identity}}%
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\subsection{\pending{Successor Identity}}%
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\hyperlabel{sub:successor-identity}
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\begin{lemma}
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@ -7011,7 +7012,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Right Multiplicative Identity}}%
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\subsection{\pending{Right Multiplicative Identity}}%
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\hyperlabel{sub:right-multiplicative-identity}
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\begin{lemma}
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@ -7062,7 +7063,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4K-5}}
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\subsection{\pending{Theorem 4K-5}}
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\hyperlabel{sub:theorem-4k-5}
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\begin{theorem}[4K-5]
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@ -7122,7 +7123,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4K-4}}%
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\subsection{\pending{Theorem 4K-4}}%
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\hyperlabel{sub:theorem-4k-4}
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\begin{theorem}[4K-4]
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@ -7180,7 +7181,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\section{Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}%
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\hyperlabel{sec:ordering-natural-numbers}
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\subsection{\verified{Ordering on Successor}}%
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\subsection{\pending{Ordering on Successor}}%
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\hyperlabel{sub:ordering-successor}
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\begin{lemma}
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@ -7226,7 +7227,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Lemma 4L(a)}}%
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\subsection{\pending{Lemma 4L(a)}}%
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\hyperlabel{sub:lemma-4l-a}
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\begin{lemma}[4L(a)]
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@ -7294,7 +7295,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Lemma 4L(b)}}%
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\subsection{\pending{Lemma 4L(b)}}%
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\hyperlabel{sub:lemma-4l-b}
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\begin{lemma}[4L(b)]
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@ -7577,7 +7578,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Theorem 4N}}%
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\subsection{\pending{Theorem 4N}}%
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\hyperlabel{sub:theorem-4n}
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\begin{theorem}[4N]
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@ -7695,7 +7696,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
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\end{proof}
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\subsection{\verified{Corollary 4P}}%
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\subsection{\pending{Corollary 4P}}%
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\hyperlabel{sub:corollary-4p}
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\begin{corollary}[4P]
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@ -8320,7 +8321,7 @@ Complete the proof of \nameref{sub:theorem-4k-5}.
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\end{proof}
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\subsection{\verified{Exercise 4.17}}%
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\subsection{\pending{Exercise 4.17}}%
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\hyperlabel{sub:exercise-4.17}
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Prove that $m^{n+p} = m^n \cdot m^p$.
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