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a/semester4/fmfp/formal-methods-functional-programming-summary.tex b/semester4/fmfp/formal-methods-functional-programming-summary.tex index 9b9aadf..9be707c 100644 --- a/semester4/fmfp/formal-methods-functional-programming-summary.tex +++ b/semester4/fmfp/formal-methods-functional-programming-summary.tex @@ -75,6 +75,8 @@ \input{parts/01_induction-proofs/00_intro.tex} \input{parts/01_induction-proofs/01_mathematical-induction.tex} \input{parts/01_induction-proofs/02_structural-induction.tex} +\input{parts/01_induction-proofs/03_induction-on-trees.tex} +\input{parts/01_induction-proofs/04_other-induction.tex} % \input{parts/01_induction-proofs/} \newsection diff --git a/semester4/fmfp/parts/01_induction-proofs/01_mathematical-induction.tex b/semester4/fmfp/parts/01_induction-proofs/01_mathematical-induction.tex index c8bc8be..0f10068 100644 --- a/semester4/fmfp/parts/01_induction-proofs/01_mathematical-induction.tex +++ b/semester4/fmfp/parts/01_induction-proofs/01_mathematical-induction.tex @@ -1,24 +1,34 @@ \subsection{Mathematical Induction} -{\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} +To prove something of the form $\forall n \in \N. P$ (with $n$ free in $P$), we can use one of these two schemes \subsubsection{Weak Mathematical Induction} -To prove $\forall n \in \N. P$ (with $n$ free in $P$), we do the following: +If $P$ is not defined yet (e.g. we just have a description of what we need to prove), give a full definition of $P$, which is our induction hypothesis. +Then state that we are proving $\forall n \in \N. P$ using \textit{weak} mathematical induction. -\shade{blue}{Base case} We show that $P[n \mapsto 0]$ is correct +\shade{blue}{Base case} We show that $P[n \mapsto 0]$ holds. -\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. - -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. +\shade{green}{Step case} We proceed by stating the following: +``Let $m \geq 0$ be arbitrary. We assume $P[n \mapsto m]$ holds, and we prove $P[n \mapsto m + 1]$.'' +Finally, do the actual proof. It is also possible to state $P[n \mapsto m]$ simply as $P(m)$ The same, but expressed as a Natural Deduction rule: \[ \begin{prooftree} \hypo{\Gamma \vdash P(0)} \hypo{\Gamma, P(n) \vdash P(n + 1)} - \infer2{\Gamma \vdash \forall n. P(n)} + \infer2[$n$ not free in $\Gamma$]{\Gamma \vdash \forall n. P(n)} \end{prooftree} \] - \subsubsection{Strong Mathematical Induction} +If $P$ is not yet defined, then define it. It is our induction hypothesis. +Then state: ``We prove $\forall k. P(k)$ by strong (mathematical) induction. +Let $k$ be arbitary and let us assume $P(j)$ for all $j < k$''. Finally, do case distinction on the different values $k$ can take. + +The same, but expressed as a Natural Deduction rule: +\[ + \begin{prooftree} + \hypo{\Gamma, \forall m < n. P(m) \vdash P(n)} + \infer1[$n$ not free in $\Gamma$, $m$ not free in $P(n)$]{\Gamma \vdash \forall n . P(n)} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/01_induction-proofs/03_induction-on-trees.tex b/semester4/fmfp/parts/01_induction-proofs/03_induction-on-trees.tex index e69de29..9c034af 100644 --- a/semester4/fmfp/parts/01_induction-proofs/03_induction-on-trees.tex +++ b/semester4/fmfp/parts/01_induction-proofs/03_induction-on-trees.tex @@ -0,0 +1 @@ +\subsection{Induction on Trees}