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25 lines
1008 B
TeX
25 lines
1008 B
TeX
\subsection{Mathematical Induction}
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{\small NOTE: These types of induction were (primarily) mentioned in the Formal Methods part of the course, but made most sense to be put here}
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\subsubsection{Weak Mathematical Induction}
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To prove $\forall n \in \N. P$ (with $n$ free in $P$), we do the following:
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\shade{blue}{Base case} We show that $P[n \mapsto 0]$ is correct
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\shade{green}{Step case} For an arbitrary $m$ not free in $P$, we show that $P[n \mapsto m + 1]$ is correct under the assumption that $P[n \mapsto m]$ is correct.
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For \bi{well-founded} domains, we have to adjust the induction hypothesis slightly: We assume $\forall l \in \N. l < m \rightarrow P[n \mapsto l]$
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and then prove $P[n \mapsto m]$ under our assumption.
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The same, but expressed as a Natural Deduction rule:
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\[
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\begin{prooftree}
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\hypo{\Gamma \vdash P(0)}
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\hypo{\Gamma, P(n) \vdash P(n + 1)}
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\infer2{\Gamma \vdash \forall n. P(n)}
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\end{prooftree}
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\]
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\subsubsection{Strong Mathematical Induction}
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