OnProgrammingforCorrectness/4_.tex at master · kammitama5/OnProgrammingforCorrectness · GitHub

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\documentclass{article}
\usepackage[utf8]{inputenc}

\title{The Hoare Triple}
\author{Krystal Maughan }
\date{April 2017}

\begin{document}

\maketitle

\section{The Hoare Triple}
The Hoare triple
$\left\{Q\right\}$ $S$ $\left\{R\right\}$
holds if and only if $S$
, when executed
starting in a state that satisfies
predicate $Q$,
completes in a finite
amount of time in a state
that satisfies predicate $R$.
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Homework 2.2.2.1
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By examination, decide whether the following Hoare triples
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hold (evaluate to TRUE) and which of the predicates
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are stronger / weaker:
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1. $\left\{x > 4\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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TRUE
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2. $\left\{x = 10\right\}$ $ y:= x + 1$ $\left\{y > 5\right\}$
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TRUE
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$x = 10$ is stronger than $x > 4$
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TRUE
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3. $\left\{x = 5\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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TRUE
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$ x > 4 $ is weaker than $ x = 5$
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TRUE
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4. $\left\{x \geq 5\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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TRUE
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$x > 4$ is at least as weak as $ x \geq 4$
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TRUE
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5. $\left\{x = 4\right\}$ $y :=x + 1$ $\left\{y > 5\right\}$
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FALSE
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$x > 4$ is weaker than $ x = 4$
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FALSE
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6. $\left\{x \geq 4\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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FALSE
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$x > 4$ is weaker than $ x \geq 4$
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FALSE
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7. $\left\{x > 4\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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TRUE
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$y \geq 5$ is weaker than $y > 5$
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TRUE
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\section{The Hoare Triple Homework 2.2.2.2}

Homework 2.2.2.2
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What do you notice about the relationship
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between the preconditions, $P$, that make the
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Hoare triple
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$\left\{P\right\}$ $y := x + 1$ $\left\{y > 5\right\}$
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TRUE and the predicate $x > 4$?
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Of all $P$ for which the Hoare triple holds
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(evaluates to TRUE), $x > 4$ is the weakest.
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\end{document}