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10 lines
512 B
TeX
10 lines
512 B
TeX
\subsection{Other types of induction}
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\subsubsection{Induction over Lists}
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To prove $P$ for all $xs$ in \texttt{[T]}, we do the following:
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\shade{blue}{Base case} We prove that $P[xs \mapsto []]$ is correct
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\shade{green}{Step case} We prove that $\forall y :: T, ys :: [T]. P[xs \mapsto ys] \rightarrow P[xs \mapsto y : ys]$, or in other words:
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We fix arbitrary $y :: T$ and $ys :: [T]$, which both are not free in $P$.
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We then apply our induction hypothesis $P[xs \mapsto ys]$ to prove $P[xs \mapsto y : ys]$
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