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a/semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex +++ b/semester4/fmfp/parts/01_formal-reasoning/05_correctness/03_induction.tex @@ -1,9 +1,9 @@ \subsubsection{Induction} -To prove recursive formulas, or more precisely formulated, a formula $P$ (with free variable $n$) for all $n \in \N$. - -A proof by cases is not possible, as there are infinitely many cases (one for each input). +To prove recursive formulas, or more precisely formulated, a formula $P$ (with free variable $n$) for all $n \in \N$, +we can't really do a proof by cases, as there are infinitely many cases (one for each input). Thus: We can use induction to prove recursive formulas or functions. + \paragraph{The schema} To prove $\forall n \in \N. P$ (with $n$ free in $P$), we do the following: @@ -13,3 +13,13 @@ To prove $\forall n \in \N. P$ (with $n$ free in $P$), we do the following: For \bi{well-founded} domains, we have to adjust the induction hypothesis slightly: We assume $\forall l \in \N. l < m \rightarrow P[n \mapsto l]$ and then prove $P[n \mapsto m]$ under our assumption. + + +\paragraph{Induction over Lists} +To prove $P$ for all $xs$ in \texttt{[T]}, we do the following: + +\bi{Base case}: We prove that $P[xs \mapsto []]$ is correct + +\bi{Step case}: We prove that $\forall y :: T, ys :: [T]. P[xs \mapsto ys] \rightarrow P[xs \mapsto y : ys]$, or in other words: +We fix arbitrary $y :: T$ and $ys :: [T]$, which both are not free in $P$. +We then apply our induction hypothesis $P[xs \mapsto ys]$ to prove $P[xs \mapsto y : ys]$