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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<link rel="stylesheet" href="https://openmath.org/public/css/hyde.css"/>
<link rel="stylesheet" href="https://openmath.org/public/css/main.css"/>
<link rel="shortcut icon" sizes="144x144" type="image/png" href="/public/favicon.png"/>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_SVG"> </script>
<script type="text/javascript"> function divfold(_Id){
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thisLevela.style.backgroundColor = "#CCCCCC";}
else{
thisLevel.style.display = "block";
thisLevela.style.backgroundColor = "#AAFFAA";}
}
</script>
<title>asymp1</title>
<link rel="stylesheet" href="https://openmath.org/cd/omcd.css" type="text/css"/>
</head>
<body class="theme-base-odk">
<div class="sidebar">
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<div class="content container">
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<h1>OpenMath Content Dictionary: asymp1</h1>
<dl>
<dt>
<span class="dt">Canonical URL:</span>
</dt>
<dd>
<a href="http://www.openmath.org/cd/asymp1.ocd">http://www.openmath.org/cd/asymp1.ocd</a>
</dd>
<dt>
<span class="dt">CD File:</span>
</dt>
<dd>
<a href="asymp1.ocd">asymp1.ocd
</a>
</dd>
<dt>
<span class="dt">CD as XML Encoded OpenMath:</span>
</dt>
<dd>
<a href="asymp1.omcd">asymp1.omcd
</a>
</dd>
<dt>
<span class="dt">Defines:</span>
</dt>
<dd>
<a href="#O">O</a>, <a href="#Omega">Omega</a>, <a href="#asymptotic">asymptotic</a>, <a href="#o">o</a>, <a href="#omega">omega</a>, <a href="#theta">theta</a>
</dd>
<dt>
<span class="dt">Date:</span>
</dt>
<dd>1999-10-19</dd>
<dt>
<span class="dt">Version:</span>
</dt>
<dd>2
(Revision 1)
</dd>
<dt>
<span class="dt">Review Date:</span>
</dt>
<dd>2017-12-31</dd>
<dt>
<span class="dt">Status:</span>
</dt>
<dd>experimental</dd>
</dl>
<hr/>
<pre>
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work. Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society. In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such. This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
c) The derived work is distributed under terms that allow the
compilation of derived works, but keep paragraphs a) and b)
intact. The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
If you have questions about this license please contact the OpenMath
society at <a href="http://www.openmath.org">http://www.openmath.org</a>.
</pre>
<pre>
Author: Bill Naylor
</pre>
<p>
This CD provides a representation of various asymptotic set constructors (O, \Omega, etc.) The constructors represent sets of functions : R -> R.
</p>
<hr/>
<h2>
<a name="O">O</a>
</h2>
<dl>
<dt>
<span class="dt">Description:</span>
</dt>
<dd>
<p>
The O symbol represents a unary function which constructs a set of certain
functions of type reals to reals. The condition f(n)=O(g(n))
is intended to express an upper bound condition on f. </p>
<p/>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Commented Mathematical property (CMP):</span>
</dt>
<dd>
O(g) =
{ f:reals -> reals | exists c in positive reals and M in the naturals
such that
forall n geq M. |f(n)| leq c*g(n)}
</dd>
</dl>
<dl>
<dt>
<span class="dt">Formal Mathematical property (FMP):</span>
</dt>
<dd>
<div>
<button id="id.om.13xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.13xml')">OpenMath XML (source)</button>
<pre id="id.om.13xml" style="display:none">
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="asymp1" name="O"/>
<OMV name="g"/>
</OMA>
<OMA>
<OMS cd="set1" name="suchthat"/>
<OMA>
<OMS cd="setname3" name="function_set"/>
<OMS cd="setname1" name="R"/>
<OMS cd="setname1" name="R"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="f"/>
</OMBVAR>
<OMBIND>
<OMS cd="quant1" name="exists"/>
<OMBVAR>
<OMV name="c"/>
<OMV name="M"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="set1" name="in"/>
<OMV name="c"/>
<OMS cd="setname1" name="R"/>
</OMA>
<OMA>
<OMS cd="relation1" name="gt"/>
<OMV name="c"/>
<OMS cd="alg1" name="zero"/>
</OMA>
<OMA>
<OMS cd="set1" name="in"/>
<OMV name="M"/>
<OMS cd="setname1" name="N"/>
</OMA>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="n"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="set1" name="in"/>
<OMV name="n"/>
<OMS cd="setname1" name="N"/>
</OMA>
<OMA>
<OMS cd="relation1" name="geq"/>
<OMV name="n"/>
<OMV name="M"/>
</OMA>
</OMA>
<OMA>
<OMS cd="relation1" name="leq"/>
<OMA>
<OMS cd="arith1" name="abs"/>
<OMA>
<OMV name="f"/>
<OMV name="n"/>
</OMA>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="c"/>
<OMA>
<OMV name="g"/>
<OMV name="n"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMA>
</OMBIND>
</OMBIND>
</OMA>
</OMA>
</OMOBJ></pre>
</div>
<div>
<button id="id.om.13cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.13cmml')">Strict Content MathML</button>
<pre id="id.om.13cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="asymp1">O</csymbol><ci>g</ci></apply>
<apply><csymbol cd="set1">suchthat</csymbol>
<apply><csymbol cd="setname3">function_set</csymbol>
<csymbol cd="setname1">R</csymbol>
<csymbol cd="setname1">R</csymbol>
</apply>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>f</ci></bvar>
<bind><csymbol cd="quant1">exists</csymbol>
<bvar><ci>c</ci></bvar>
<bvar><ci>M</ci></bvar>
<apply><csymbol cd="logic1">and</csymbol>
<apply><csymbol cd="set1">in</csymbol><ci>c</ci><csymbol cd="setname1">R</csymbol></apply>
<apply><csymbol cd="relation1">gt</csymbol><ci>c</ci><csymbol cd="alg1">zero</csymbol></apply>
<apply><csymbol cd="set1">in</csymbol><ci>M</ci><csymbol cd="setname1">N</csymbol></apply>
<bind><csymbol cd="quant1">forall</csymbol>
<bvar><ci>n</ci></bvar>
<apply><csymbol cd="logic1">implies</csymbol>
<apply><csymbol cd="logic1">and</csymbol>
<apply><csymbol cd="set1">in</csymbol><ci>n</ci><csymbol cd="setname1">N</csymbol></apply>
<apply><csymbol cd="relation1">geq</csymbol><ci>n</ci><ci>M</ci></apply>
</apply>
<apply><csymbol cd="relation1">leq</csymbol>
<apply><csymbol cd="arith1">abs</csymbol><apply><ci>f</ci><ci>n</ci></apply></apply>
<apply><csymbol cd="arith1">times</csymbol>
<ci>c</ci>
<apply><ci>g</ci><ci>n</ci></apply>
</apply>
</apply>
</apply>
</bind>
</apply>
</bind>
</bind>
</apply>
</apply>
</math></pre>
</div>
<button id="id.om.13prefa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.13pref')">Prefix</button>
<div id="id.om.13pref" style="display:none; margin-top: 0.5em">
<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/asymp1.html#O">O</a>
( <i>g</i>)
, <a href="../../../cd/set1.html#suchthat">suchthat</a>
(<a href="../../../cd/setname3.html#function_set">function_set</a>
(<a href="../../../cd/setname1.html#R">R</a>, <a href="../../../cd/setname1.html#R">R</a>)
, <a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>f</i>
] .
(<a href="../../../cd/quant1.html#exists">exists</a>
[
<i>c</i>
<i>M</i>
] .
(<a href="../../../cd/logic1.html#and">and</a>
(<a href="../../../cd/set1.html#in">in</a>
( <i>c</i>, <a href="../../../cd/setname1.html#R">R</a>)
, <a href="../../../cd/relation1.html#gt">gt</a>
( <i>c</i>, <a href="../../../cd/alg1.html#zero">zero</a>)
, <a href="../../../cd/set1.html#in">in</a>
( <i>M</i>, <a href="../../../cd/setname1.html#N">N</a>)
, <a href="../../../cd/quant1.html#forall">forall</a>
[
<i>n</i>
] .
(<a href="../../../cd/logic1.html#implies">implies</a>
(<a href="../../../cd/logic1.html#and">and</a>
(<a href="../../../cd/set1.html#in">in</a>
( <i>n</i>, <a href="../../../cd/setname1.html#N">N</a>)
, <a href="../../../cd/relation1.html#geq">geq</a>
( <i>n</i>, <i>M</i>)
)
, <a href="../../../cd/relation1.html#leq">leq</a>
(<a href="../../../cd/arith1.html#abs">abs</a>
( <i>f</i>
( <i>n</i>)
)
, <a href="../../../cd/arith1.html#times">times</a>
( <i>c</i>, <i>g</i>
( <i>n</i>)
)
)
)
)
)
)
)
)
)
</div>
<div>
<button id="id.om.13popa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.13pop')">Popcorn</button>
<div id="id.om.13pop" style="display:none; margin-top: 0.5em">
asymp1.O($g) = set1.suchthat(setname3.function_set(setname1.R, setname1.R), fns1.lambda[$f -> quant1.exists[$c, $M -> set1.in($c, setname1.R) and $c > alg1.zero and set1.in($M, setname1.N) and quant1.forall[$n -> set1.in($n, setname1.N) and $n >= $M ==> arith1.abs($f($n)) <= $c * $g($n)]]])
</div>
</div>
<div>
<button id="id.om.13mmla" style="width:20em; background-color:#AAFFAA"
onclick="divfold('id.om.13mml')">Rendered Presentation MathML</button>
<div id="id.om.13mml" style="display:block; margin-top: 0.5em">
<div style="overflow:auto;font-size:90%">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow xmlns:om="http://www.openmath.org/OpenMath">
<mrow>
<mi>O</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mi>f</mi>
<mo>∈</mo>
<mrow>
<mi>function_set</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi mathvariant="double-struck">R</mi>
<mo>,</mo>
<mi mathvariant="double-struck">R</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>|</mo>
<mrow>
<mo>∃</mo>
<mo> </mo>
<mrow>
<mi>c</mi>
<mo>,</mo>
<mi>M</mi>
</mrow>
<mo>.</mo>
<mrow>
<mrow>
<mi>c</mi>
<mo>∈</mo>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>∧</mo>
<mrow>
<mi>c</mi>
<mo>></mo>
<mn>0</mn>
</mrow>
<mo>∧</mo>
<mrow>
<mi>M</mi>
<mo>∈</mo>
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>∧</mo>
<mrow>
<mo>∀</mo>
<mo> </mo>
<mrow>
<mi>n</mi>
</mrow>
<mo>.</mo>
<mrow>
<mrow>
<mrow>
<mi>n</mi>
<mo>∈</mo>
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>∧</mo>
<mrow>
<mi>n</mi>
<mo>≥</mo>
<mi>M</mi>
</mrow>
</mrow>
<mo>⇒</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>f</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>≤</mo>
<mrow>
<mi>c</mi>
<mo>⁢</mo>
<mrow>
<mi>g</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
</math>
</div>
</div>
</div>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Signatures:</span>
</dt>
<dd>
<a href="../../sts/asymp1.html#O">
sts
</a>
</dd>
</dl>
<p/>
<hr/>
<table width="100%">
<tr>
<td align="right">
<font size="-1">
[Next: <a href="#o">o</a>]
[Last: <a href="#Omega">Omega</a>]
[<a href="#top">Top</a>]</font>
</td>
</tr>
</table>
<hr/>
<h2>
<a name="o">o</a>
</h2>
<dl>
<dt>
<span class="dt">Description:</span>
</dt>
<dd>
<p>
The o symbol represents a unary function which constructs a set
of certain functions of type reals to positive reals. The condition
f(n) = o(g(n)) is intended to express a lower bouund condition
on f. Formally we say that f(n) = o(g(n)) if and only if the
limit as n tends to infinity of f(n)/g(n) exists and is equal to 0.
</p>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Commented Mathematical property (CMP):</span>
</dt>
<dd>
o(g) =
{f : reals -> reals | the limit as x tends to infinity of f(x)/g(x) is 0}
</dd>
</dl>
<dl>
<dt>
<span class="dt">Formal Mathematical property (FMP):</span>
</dt>
<dd>
<div>
<button id="id.om.82xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.82xml')">OpenMath XML (source)</button>
<pre id="id.om.82xml" style="display:none">
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="asymp1" name="o"/>
<OMV name="g"/>
</OMA>
<OMA>
<OMS cd="set1" name="suchthat"/>
<OMA>
<OMS cd="setname3" name="function_set"/>
<OMS cd="setname1" name="R"/>
<OMS cd="setname1" name="R"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="f"/>
</OMBVAR>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="limit1" name="limit"/>
<OMS cd="nums1" name="infinity"/>
<OMS cd="limit1" name="below"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="divide"/>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
<OMA>
<OMV name="g"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMS cd="alg1" name="zero"/>
</OMA>
</OMBIND>
</OMA>
</OMA>
</OMOBJ></pre>
</div>
<div>
<button id="id.om.82cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.82cmml')">Strict Content MathML</button>
<pre id="id.om.82cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="asymp1">o</csymbol><ci>g</ci></apply>
<apply><csymbol cd="set1">suchthat</csymbol>
<apply><csymbol cd="setname3">function_set</csymbol>
<csymbol cd="setname1">R</csymbol>
<csymbol cd="setname1">R</csymbol>
</apply>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>f</ci></bvar>
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="limit1">limit</csymbol>
<csymbol cd="nums1">infinity</csymbol>
<csymbol cd="limit1">below</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="arith1">divide</csymbol>
<apply><ci>f</ci><ci>x</ci></apply>
<apply><ci>g</ci><ci>x</ci></apply>
</apply>
</bind>
</apply>
<csymbol cd="alg1">zero</csymbol>
</apply>
</bind>
</apply>
</apply>
</math></pre>
</div>
<button id="id.om.82prefa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.82pref')">Prefix</button>
<div id="id.om.82pref" style="display:none; margin-top: 0.5em">
<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/asymp1.html#o">o</a>
( <i>g</i>)
, <a href="../../../cd/set1.html#suchthat">suchthat</a>
(<a href="../../../cd/setname3.html#function_set">function_set</a>
(<a href="../../../cd/setname1.html#R">R</a>, <a href="../../../cd/setname1.html#R">R</a>)
, <a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>f</i>
] .
(<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/limit1.html#limit">limit</a>
(<a href="../../../cd/nums1.html#infinity">infinity</a>, <a href="../../../cd/limit1.html#below">below</a>, <a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>x</i>
] .
(<a href="../../../cd/arith1.html#divide">divide</a>
( <i>f</i>
( <i>x</i>)
, <i>g</i>
( <i>x</i>)
)
)
)
, <a href="../../../cd/alg1.html#zero">zero</a>)
)
)
)
</div>
<div>
<button id="id.om.82popa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.82pop')">Popcorn</button>
<div id="id.om.82pop" style="display:none; margin-top: 0.5em">
asymp1.o($g) = set1.suchthat(setname3.function_set(setname1.R, setname1.R), fns1.lambda[$f -> limit1.limit(nums1.infinity, limit1.below, fns1.lambda[$x -> $f($x) / $g($x)]) = alg1.zero])
</div>
</div>
<div>
<button id="id.om.82mmla" style="width:20em; background-color:#AAFFAA"
onclick="divfold('id.om.82mml')">Rendered Presentation MathML</button>
<div id="id.om.82mml" style="display:block; margin-top: 0.5em">
<div style="overflow:auto;font-size:90%">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow xmlns:om="http://www.openmath.org/OpenMath">
<mrow>
<mi>o</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mi>f</mi>
<mo>∈</mo>
<mrow>
<mi>function_set</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi mathvariant="double-struck">R</mi>
<mo>,</mo>
<mi mathvariant="double-struck">R</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>|</mo>
<mrow>
<mrow>
<munder>
<mi>limit</mi>
<mrow>
<mrow>
<mi>x</mi>
</mrow>
<mo>→</mo>
<msub>
<mi>∞</mi>
<mo>-</mo>
</msub>
</mrow>
</munder>
<mspace width="0.1em"/>
<mfrac>
<mrow>
<mi>f</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>g</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>}</mo>
</mrow>
</mrow>
</math>
</div>
</div>
</div>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Signatures:</span>
</dt>
<dd>
<a href="../../sts/asymp1.html#o">
sts
</a>
</dd>
</dl>
<p/>
<hr/>
<table width="100%">
<tr>
<td align="right">
<font size="-1">
[Next: <a href="#theta">theta</a>]
[Previous: <a href="#O">O</a>]
[<a href="#top">Top</a>]</font>
</td>
</tr>
</table>
<hr/>
<h2>
<a name="theta">theta</a>
</h2>
<dl>
<dt>
<span class="dt">Description:</span>
</dt>
<dd>
<p>
The theta symbol represents a unary function which constructs a set
of certain functions of type reals to positive reals. The theta
symbol represents a set of functions which all have the same 'rate of
growth'. Formally we say that f(x) = theta(g(x)) if and only if
there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for
all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x).
</p>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Commented Mathematical property (CMP):</span>
</dt>
<dd>
f(x) = theta(g(x)) if and only if
there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for
all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x)
</dd>
</dl>
<dl>
<dt>
<span class="dt">Formal Mathematical property (FMP):</span>
</dt>
<dd>
<div>
<button id="id.om.122xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.122xml')">OpenMath XML (source)</button>
<pre id="id.om.122xml" style="display:none">
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="asymp1" name="theta"/>
<OMV name="g"/>
</OMA>
<OMA>
<OMS cd="set1" name="suchthat"/>
<OMA>
<OMS cd="setname3" name="function_set"/>
<OMS cd="setname1" name="R"/>
<OMS cd="setname1" name="R"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="f"/>
</OMBVAR>
<OMBIND>
<OMS cd="quant1" name="exists"/>
<OMBVAR>
<OMV name="c_1"/>
<OMV name="c_2"/>
<OMV name="x_0"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="neq"/>
<OMV name="c_1"/>
<OMS cd="alg1" name="zero"/>
</OMA>
<OMA>
<OMS cd="relation1" name="neq"/>
<OMV name="c_2"/>
<OMS cd="alg1" name="zero"/>
</OMA>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="gt"/>
<OMV name="x"/>
<OMV name="x_0"/>
</OMA>
<OMA>
<OMS cd="relation1" name="lt"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="c_1"/>
<OMA>
<OMV name="g"/>
<OMV name="x"/>
</OMA>
</OMA>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMA>
<OMA>
<OMS cd="relation1" name="lt"/>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="c_2"/>
<OMA>
<OMV name="g"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMA>
</OMBIND>
</OMBIND>
</OMA>
</OMA>
</OMOBJ></pre>
</div>
<div>
<button id="id.om.122cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.122cmml')">Strict Content MathML</button>
<pre id="id.om.122cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="asymp1">theta</csymbol><ci>g</ci></apply>
<apply><csymbol cd="set1">suchthat</csymbol>
<apply><csymbol cd="setname3">function_set</csymbol>
<csymbol cd="setname1">R</csymbol>
<csymbol cd="setname1">R</csymbol>
</apply>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>f</ci></bvar>
<bind><csymbol cd="quant1">exists</csymbol>
<bvar><ci>c_1</ci></bvar>
<bvar><ci>c_2</ci></bvar>
<bvar><ci>x_0</ci></bvar>
<apply><csymbol cd="logic1">and</csymbol>
<apply><csymbol cd="relation1">neq</csymbol><ci>c_1</ci><csymbol cd="alg1">zero</csymbol></apply>
<apply><csymbol cd="relation1">neq</csymbol><ci>c_2</ci><csymbol cd="alg1">zero</csymbol></apply>
<bind><csymbol cd="quant1">forall</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="logic1">and</csymbol>
<apply><csymbol cd="relation1">gt</csymbol><ci>x</ci><ci>x_0</ci></apply>
<apply><csymbol cd="relation1">lt</csymbol>
<apply><csymbol cd="arith1">times</csymbol>
<ci>c_1</ci>
<apply><ci>g</ci><ci>x</ci></apply>
</apply>
<apply><ci>f</ci><ci>x</ci></apply>
</apply>
<apply><csymbol cd="relation1">lt</csymbol>
<apply><ci>f</ci><ci>x</ci></apply>
<apply><csymbol cd="arith1">times</csymbol>
<ci>c_2</ci>
<apply><ci>g</ci><ci>x</ci></apply>
</apply>
</apply>
</apply>
</bind>
</apply>
</bind>
</bind>
</apply>
</apply>
</math></pre>
</div>
<button id="id.om.122prefa" style="width:20em; background-color:#CCCCCC"