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\documentclass[a4paper, 10pt, pagesize, smallheadings]{article}
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\usepackage{graphicx}
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%\usepackage[latin1]{inputenc}
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\usepackage{amsmath, amsthm, amssymb}
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\usepackage{typearea}
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\usepackage{algorithm}
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\usepackage{algorithmic}
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\usepackage{fullpage}
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\usepackage{mathtools}
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\usepackage[all]{xy}
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\title{Theory I, Sheet 8 Solution}
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\author{Eugen Sawin}
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\renewcommand{\familydefault}{\sfdefault}
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\newcommand{\Pos}{\mathcal{P}os}
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\include{pythonlisting}
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\pagestyle{empty}
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\begin{document}
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\maketitle
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\section*{Exercise 8.1}
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Let $s, t \in T(\Sigma, X)$ be terms over $X$ and $p, q$ be strings over the natural numbers. We provide proofs by induction.
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\subsection*{1. $pq \in \Pos(s) \implies p \in \Pos(s)$ }
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$(IH): pq \in \Pos(s) \implies p \in \Pos(s)$\\\\
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\emph{Base case:}\\
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From $s \in X$ or $s \in \Sigma^{(0)}$ follows $\Pos(s) = \{\epsilon\}$.\\
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Trivially it follows that $pq \in \Pos(s) \iff pq = \epsilon$ and $pq \in \Pos(s) \implies p \in \Pos(s)$ with $pq \notin \Pos(s) \lor pq = p = \epsilon$. \\
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\noindent \emph{Inductive case:}\\
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Let $s = f(t_1, ...,t_n) \in T(\Sigma, X)$ it follows $\Pos(s) = \{\epsilon\} \cup \bigcup_{i = 1}^{n}\{ip \mid p \in \Pos(t_i)\}$.\\
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\begin{align*}
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pq \in \Pos(s) &\implies p \in \Pos(s)\\
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\iff pq \in \{\epsilon\} \cup \bigcup_{i = 1}^{n}\{ip \mid p \in \Pos(t_i)\} &\implies p \in \{\epsilon\} \cup \bigcup_{i = 1}^{n}\{ip \mid p \in \Pos(t_i)\}\\
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\end{align*}
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\section*{Exercise 8.2}
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\subsection*{1. $t = \neg (x \land (\top \lor y)), \sigma = \{x \mapsto \bot, y \mapsto \top \land x\}, \tau = \{z \mapsto \top, x \mapsto \top\}$}
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\begin{align*}
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\sigma(t) &= \neg (\bot \land (\top \lor ( \top \land x)))\\
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\tau \sigma(t) &= \tau(\neg (\bot \land (\top \lor ( \top \land x)))) = \neg (\bot \land (\top \lor (\top \land \top)))\\
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\end{align*}
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\subsection*{2. $t = \neg(\top \land (\bot \lor y)), s = \neg(x \land (\bot \lor (x \land \bot)))$}
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\[\sigma(t) = \sigma(s) \text{ with } \sigma = \{x \mapsto \top, y \mapsto \top \land \bot \} \]
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\subsection*{3.}
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Substitution composition is \emph{not} commutative, we give a counterexample.
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Let $t = x$, $\sigma = \{x \mapsto \bot\}$ and $\tau = \{x \mapsto \top\}$, it follows:
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\[\tau\sigma(t) = \bot\]
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\[\sigma\tau(t) = \top\]
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We see that $\tau\sigma(t) \neq \sigma\tau(t)$.
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\end{document}
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