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\documentclass[uebung]{../../../lecture} |
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\title{Übungsblatt Nr. 5} |
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\author{Christian Merten, Samuel Weidemaier} |
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\usepackage[]{enumerate} |
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\usepackage{listings} |
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\begin{document} |
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\begin{aufgabe} Schleifeninvarianz |
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\begin{enumerate}[(a)] |
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\item $v = (a,b,i)$\\ |
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$B(v)= i < n$ \\ |
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$H(v)= (b, a+b, i+1)$ |
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\item Hier $F$ Fibonacci Folge.\\ |
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$INV(v^{j}) \equiv F(j) = a \land F(j+1) = b \land i - 1 < n$ |
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Verifikation der Schleifeninvariante: |
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\begin{enumerate}[1.] |
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\item |
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Für $v^{0} = (a^{0}, b^{0}, i^{0}) = (0, 1, 0)$ gilt: |
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\[ |
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INV(0,1,0) \equiv F(0) = 0 \land F(1) = 1 \land 0 - 1 < n |
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.\] |
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\item Es gelte nun $INV(v^{j}) \land B(v^{j})$. |
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Wende $H$ auf $v^{j}$ an: |
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\begin{itemize} |
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\item |
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$a_{j+1} = F(j+1) = b_j \land b_{j+1} = a_j + b_{j} = F(j) + F(j+1) = F(j+2)$ |
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\item |
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$i_{j+1} - 1 = i_{j} + 1 - 1 = i_j < n$ |
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\end{itemize} |
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\item |
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Am Schleifenende gilt $\neg (i < n)$, also $i \ge n \land i - 1 < n \implies i = n$. |
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\begin{align*} |
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&INV(a, b, n-1) \land \neg B(v^{n-1}) \\ |
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\iff & F(n-1) = a \land F(n) = b \land n - 1 < n \\ |
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\iff &Q(v^{n}) |
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.\end{align*} |
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\end{enumerate} |
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\end{enumerate} |
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\end{aufgabe} |
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\begin{aufgabe} |
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siehe \textit{readSortedArray.ccp} |
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\end{aufgabe} |
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\begin{aufgabe} |
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siehe \textit{perfectShuffle.cpp} |
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\end{aufgabe} |
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\end{document} |
