Enderton. Exponentiation and some arithmetic exercises.
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@ -34,9 +34,6 @@ For each $m \in \omega$, there exists (by the
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\end{align*}
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\textbf{Addition} ($+$) is the \nameref{ref:binary-operation} on $\omega$ such
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that for any $m$ and $n$ in $\omega$, $$m + n = A_m(n).$$
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Thus when written as a \nameref{ref:relation},
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$$+ = \{\pair{\pair{m, n}, p} \mid
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m \in \omega \land n \in \omega \land p = A_m(n)\}.$$
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\begin{definition}
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@ -185,6 +182,25 @@ Relation $R$ is an \textbf{equivalence relation} on set $A$ if and only if
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\end{definition}
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\section{\defined{Exponentiation}}%
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\hyperlabel{ref:exponentiation}
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For each $m \in \omega$, there exists (by the
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\nameref{sub:recursion-theorem-natural-numbers}) a unique
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\nameref{ref:function} $E_m \colon \omega \rightarrow \omega$ for which
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\begin{align*}
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E_m(0) & = 1, \\
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E_m(n^+) & = E_m(n) \cdot m & \text{for } n \text{ in } \omega.
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\end{align*}
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\textbf{Exponentiation} is the \nameref{ref:binary-operation} on $\omega$
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such that for any $m$ and $n$ in $\omega$, $$m^n = E_m(n).$$
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\begin{definition}
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\lean*{Init/Prelude}{Pow.pow}
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\end{definition}
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\section{\defined{Extensionality Axiom}}%
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\hyperlabel{ref:extensionality-axiom}
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@ -7512,25 +7528,25 @@ Show that each natural number is either even or odd, but never both.
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\end{proof}
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\subsection{\sorry{Exercise 4.15}}%
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\subsection{\verified{Exercise 4.15}}%
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\hyperlabel{sub:exercise-4.15}
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Complete the proof of \nameref{sub:theorem-4k-1}.
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\begin{proof}
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TODO
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Refer to \nameref{sub:theorem-4k-1}.
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\end{proof}
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\subsection{\sorry{Exercise 4.16}}%
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\subsection{\verified{Exercise 4.16}}%
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\hyperlabel{sub:exercise-4.16}
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Complete the proof of \nameref{sub:theorem-4k-5}.
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\begin{proof}
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TODO
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Refer to \nameref{sub:theorem-4k-5}.
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\end{proof}
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