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<a name="Properties-of-the-poly_005fint-comparisons"></a>
<div class="header">
<p>
Next: <a href="Comparing-potentially_002dunordered-poly_005fints.html#Comparing-potentially_002dunordered-poly_005fints" accesskey="n" rel="next">Comparing potentially-unordered <code>poly_int</code>s</a>, Previous: <a href="Comparison-functions-for-poly_005fint.html#Comparison-functions-for-poly_005fint" accesskey="p" rel="prev">Comparison functions for <code>poly_int</code></a>, Up: <a href="Comparisons-involving-poly_005fint.html#Comparisons-involving-poly_005fint" accesskey="u" rel="up">Comparisons involving <code>poly_int</code></a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Option-Index.html#Option-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Properties-of-the-poly_005fint-comparisons-1"></a>
<h4 class="subsection">10.3.2 Properties of the <code>poly_int</code> comparisons</h4>
<p>All “maybe” relations except <code>maybe_ne</code> are transitive, so for example:
</p>
<div class="smallexample">
<pre class="smallexample">maybe_lt (<var>a</var>, <var>b</var>) && maybe_lt (<var>b</var>, <var>c</var>) implies maybe_lt (<var>a</var>, <var>c</var>)
</pre></div>
<p>for all <var>a</var>, <var>b</var> and <var>c</var>. <code>maybe_lt</code>, <code>maybe_gt</code>
and <code>maybe_ne</code> are irreflexive, so for example:
</p>
<div class="smallexample">
<pre class="smallexample">!maybe_lt (<var>a</var>, <var>a</var>)
</pre></div>
<p>is true for all <var>a</var>. <code>maybe_le</code>, <code>maybe_eq</code> and <code>maybe_ge</code>
are reflexive, so for example:
</p>
<div class="smallexample">
<pre class="smallexample">maybe_le (<var>a</var>, <var>a</var>)
</pre></div>
<p>is true for all <var>a</var>. <code>maybe_eq</code> and <code>maybe_ne</code> are symmetric, so:
</p>
<div class="smallexample">
<pre class="smallexample">maybe_eq (<var>a</var>, <var>b</var>) == maybe_eq (<var>b</var>, <var>a</var>)
maybe_ne (<var>a</var>, <var>b</var>) == maybe_ne (<var>b</var>, <var>a</var>)
</pre></div>
<p>for all <var>a</var> and <var>b</var>. In addition:
</p>
<div class="smallexample">
<pre class="smallexample">maybe_le (<var>a</var>, <var>b</var>) == maybe_lt (<var>a</var>, <var>b</var>) || maybe_eq (<var>a</var>, <var>b</var>)
maybe_ge (<var>a</var>, <var>b</var>) == maybe_gt (<var>a</var>, <var>b</var>) || maybe_eq (<var>a</var>, <var>b</var>)
maybe_lt (<var>a</var>, <var>b</var>) == maybe_gt (<var>b</var>, <var>a</var>)
maybe_le (<var>a</var>, <var>b</var>) == maybe_ge (<var>b</var>, <var>a</var>)
</pre></div>
<p>However:
</p>
<div class="smallexample">
<pre class="smallexample">maybe_le (<var>a</var>, <var>b</var>) && maybe_le (<var>b</var>, <var>a</var>) does not imply !maybe_ne (<var>a</var>, <var>b</var>) [== known_eq (<var>a</var>, <var>b</var>)]
maybe_ge (<var>a</var>, <var>b</var>) && maybe_ge (<var>b</var>, <var>a</var>) does not imply !maybe_ne (<var>a</var>, <var>b</var>) [== known_eq (<var>a</var>, <var>b</var>)]
</pre></div>
<p>One example is again ‘<samp><var>a</var> == 3 + 4<var>x</var></samp>’
and ‘<samp><var>b</var> == 1 + 5<var>x</var></samp>’, where ‘<samp>maybe_le (<var>a</var>, <var>b</var>)</samp>’,
‘<samp>maybe_ge (<var>a</var>, <var>b</var>)</samp>’ and ‘<samp>maybe_ne (<var>a</var>, <var>b</var>)</samp>’
all hold. <code>maybe_le</code> and <code>maybe_ge</code> are therefore not antisymetric
and do not form a partial order.
</p>
<p>From the above, it follows that:
</p>
<ul>
<li> All “known” relations except <code>known_ne</code> are transitive.
</li><li> <code>known_lt</code>, <code>known_ne</code> and <code>known_gt</code> are irreflexive.
</li><li> <code>known_le</code>, <code>known_eq</code> and <code>known_ge</code> are reflexive.
</li></ul>
<p>Also:
</p>
<div class="smallexample">
<pre class="smallexample">known_lt (<var>a</var>, <var>b</var>) == known_gt (<var>b</var>, <var>a</var>)
known_le (<var>a</var>, <var>b</var>) == known_ge (<var>b</var>, <var>a</var>)
known_lt (<var>a</var>, <var>b</var>) implies !known_lt (<var>b</var>, <var>a</var>) [asymmetry]
known_gt (<var>a</var>, <var>b</var>) implies !known_gt (<var>b</var>, <var>a</var>)
known_le (<var>a</var>, <var>b</var>) && known_le (<var>b</var>, <var>a</var>) == known_eq (<var>a</var>, <var>b</var>) [== !maybe_ne (<var>a</var>, <var>b</var>)]
known_ge (<var>a</var>, <var>b</var>) && known_ge (<var>b</var>, <var>a</var>) == known_eq (<var>a</var>, <var>b</var>) [== !maybe_ne (<var>a</var>, <var>b</var>)]
</pre></div>
<p><code>known_le</code> and <code>known_ge</code> are therefore antisymmetric and are
partial orders. However:
</p>
<div class="smallexample">
<pre class="smallexample">known_le (<var>a</var>, <var>b</var>) does not imply known_lt (<var>a</var>, <var>b</var>) || known_eq (<var>a</var>, <var>b</var>)
known_ge (<var>a</var>, <var>b</var>) does not imply known_gt (<var>a</var>, <var>b</var>) || known_eq (<var>a</var>, <var>b</var>)
</pre></div>
<p>For example, ‘<samp>known_le (4, 4 + 4<var>x</var>)</samp>’ holds because the runtime
indeterminate <var>x</var> is a nonnegative integer, but neither
<code>known_lt (4, 4 + 4<var>x</var>)</code> nor <code>known_eq (4, 4 + 4<var>x</var>)</code> hold.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Comparing-potentially_002dunordered-poly_005fints.html#Comparing-potentially_002dunordered-poly_005fints" accesskey="n" rel="next">Comparing potentially-unordered <code>poly_int</code>s</a>, Previous: <a href="Comparison-functions-for-poly_005fint.html#Comparison-functions-for-poly_005fint" accesskey="p" rel="prev">Comparison functions for <code>poly_int</code></a>, Up: <a href="Comparisons-involving-poly_005fint.html#Comparisons-involving-poly_005fint" accesskey="u" rel="up">Comparisons involving <code>poly_int</code></a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Option-Index.html#Option-Index" title="Index" rel="index">Index</a>]</p>
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