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+\usepackage[]{enumerate} + +\usepackage{listings} + +\begin{document} + +\begin{aufgabe} Schleifeninvarianz + \begin{enumerate}[(a)] + \item $v = (a,b,i)$\\ + $B(v)= i < n$ \\ + $H(v)= (b, a+b, i+1)$ + \item Hier $F$ Fibonacci Folge.\\ + $INV(v^{j}) \equiv F(j) = a \land F(j+1) = b \land i - 1 < n$ + + Verifikation der Schleifeninvariante: + + \begin{enumerate}[1.] + \item + Für $v^{0} = (a^{0}, b^{0}, i^{0}) = (0, 1, 0)$ gilt: + \[ + INV(0,1,0) \equiv F(0) = 0 \land F(1) = 1 \land 0 - 1 < n + .\] + \item Es gelte nun $INV(v^{j}) \land B(v^{j})$. + Wende $H$ auf $v^{j}$ an: + \begin{itemize} + \item + $a_{j+1} = F(j+1) = b_j \land b_{j+1} = a_j + b_{j} = F(j) + F(j+1) = F(j+2)$ + \item + $i_{j+1} - 1 = i_{j} + 1 - 1 = i_j < n$ + \end{itemize} + + \item + Am Schleifenende gilt $\neg (i < n)$, also $i \ge n \land i - 1 < n \implies i = n$. + \begin{align*} + &INV(a, b, n-1) \land \neg B(v^{n-1}) \\ + \iff & F(n-1) = a \land F(n) = b \land n - 1 < n \\ + \iff &Q(v^{n}) + .\end{align*} + \end{enumerate} + + \end{enumerate} + +\end{aufgabe} + +\begin{aufgabe} + siehe \textit{readSortedArray.ccp} +\end{aufgabe} + +\begin{aufgabe} + siehe \textit{perfectShuffle.cpp} +\end{aufgabe} + +\end{document}