From ab9484b343abd995cba915bb0ba4be8907dfa6ec Mon Sep 17 00:00:00 2001
From: Gregor Kleen <gkleen@yggdrasil.li>
Date: Fri, 13 Nov 2015 23:45:26 +0000
Subject: Shorter lecture names

---
 ws2015/eip/blaetter/03/H3-1.md | 39 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 39 insertions(+)
 create mode 100644 ws2015/eip/blaetter/03/H3-1.md

(limited to 'ws2015/eip/blaetter/03/H3-1.md')

diff --git a/ws2015/eip/blaetter/03/H3-1.md b/ws2015/eip/blaetter/03/H3-1.md
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+---
+header-includes:
+	- \usepackage{bussproofs}
+	- \renewcommand{\implies}{\rightarrow}
+	- \EnableBpAbbreviations
+---
+# Hoare-Tripel II
+
+d)
+
+\begin{prooftree}
+\AXC{$\forall \phi \ldotp \phi \implies \phi$}
+\UIC{$(0 < n + 1 \land n < m) \implies (0 < n + 1 \land n < m)$}
+\RightLabel{Umformung}
+\UIC{$(0 < n + 1 \land n < m) \implies (0 < n + 1 \land n \leq m - 1)$}
+\RightLabel{Umformung}
+\UIC{$(0 < n + 1 \land n < m) \implies (0 < n + 1 \land n + 1 \leq m)$}
+\RightLabel{Zuweisung}
+\UIC{$ \{ 0 < n + 1 \land n < m \} $ \texttt{n = n + 1} $ \{ 0 < n \land n \leq m \} $}
+\end{prooftree}
+
+e) Nimm als Gegenbeispiel:
+
+~~~
+a = -1
+b = 0
+~~~
+
+f) Seit $\phi$ die Nachfolgerfunktion auf $\mathbb{N}$.
+
+\begin{prooftree}
+\AXC{$\forall n \in \mathbb{N} \ldotp \phi(n) > n$}
+\UIC{$1 > 0$}
+\UIC{$((x + 1) + y > x + y)$}
+\UIC{$(z = x + y) \implies ((x + 1) + y > z)$}
+\UIC{$ \{ z = x + y \} $ \texttt{ x = x + 1} $ \{ x + y > z \} $}
+\UIC{$ \{ z = x + y \land z \equiv 0 \mod 2 \} $ \texttt{x = x + 1} $ \{ x + y > z \} $ \quad $ \{ z = x + y \land z \equiv 1 \mod 2 \} $ \texttt{x = x + 1} $ \{ x + y > z \} $}
+\UIC{$ \{ z = x + y \} $ \texttt{if (z \% 2 == 0) x = x + 1; else y = y + 1;} $ \{ x + y > z \} $}
+\end{prooftree}
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