OnProgrammingforCorrectness/11_.tex at master · kammitama5/OnProgrammingforCorrectness · GitHub

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\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{mathtools}
\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
\title{The Hoare Triple}
\author{Krystal Maughan }
\date{April 28th 2017}

\begin{document}

\maketitle

\section{The Hoare Triple : Simultaneous Assignment}
Problems in Simultaneous Assignment
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\section{Homework 2.3.5.1}
Evaluate
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wp("$i : = i + 1; j : = 2i$" , $2i = j$)
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TRUE
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wp("$j : 2i; i : = i + 1$", $2i = j$)
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FALSE
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wp("$i, j : = i + 1, 2i$", $2i = j$)
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FALSE
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\section{Homework 2.3.5.2}
Prove the following code segment correct.
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It swaps the values of variables x and y.
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$\left\{Q: x = \hat{x} \land (y = \hat{y}\right\}$
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$S : x, y : = y , x$
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$\left\{R: x = \hat{y} \land (y = \hat{x}\right\}$
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$Q \Rightarrow wp("S", R)$
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$<$ Instantiate S and R $>$
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$Q \Rightarrow wp("x, y : = y, x", (x = \hat{y}) \land (y = \hat{x}))$
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$<$ Definition of $wp(: =,) >$
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$Q \Rightarrow {(( x = \hat{y}) \land (y = \hat{x})}_{(y),(x)}^{x, y}$
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$<$ Instantiate $Q$, definition of $R_{(y),(x)}^{x,y} >$
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$((x = \hat{x} \land (y = \hat{y})) \Rightarrow ((y = \hat{y}) \land (x = \hat{x}))$
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$<$ $\land$-commutivity; $\Rightarrow$-simplification $>$
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T
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\section{Homework 2.3.5.3}
Evaluate
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wp($i : = 2i + j; j : = i + 2j + 4$", $i = j$)
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$<$ definition of $: = $ $>$
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wp("$i : = 2i + j$", $i = i + 2j + 4$)
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$<$ definition of $: = $ $>$
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wp("$i : = 2i + j$", $i = i + 2j + 4$)
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simplification $\rightarrow$ $j = -2$
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Evaluate
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wp("$j : = i + 2j + 4; i : = 2i + j$", $i = j$)
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$<$ definition of $: = $ $>$
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wp("$j : = i + 2j + 4$", $2i + j = j$)
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$<$ definition of $: = $ $>$
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$2i + (i + 2j + 4) = i + 2j + 4$
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simplification $\Rightarrow i = 0$
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Evaluate
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wp("$i, j: = 2i + j, i + 2j + 4$", $i = j$)
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$<$ definition of $: = $ $>$
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$2i + j = i + 2j + 4$
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simplification $\Rightarrow i = j + 4$
\end{document}