asymp1.ocd

<CD xmlns="http://www.openmath.org/OpenMathCD">
<CDComment>

     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 
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     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
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          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
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       c) The derived work is distributed under terms that allow the
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          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
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</CDComment>

<CDName>asymp1</CDName>
<CDURL>http://www.openmath.org/cd/asymp1.ocd</CDURL>
<CDDate>1999-10-19</CDDate>
<CDReviewDate>2017-12-31</CDReviewDate>
<CDVersion>2</CDVersion>
<CDRevision>1</CDRevision>
<CDComment>
  Author: Bill Naylor
</CDComment>
<CDStatus>experimental</CDStatus>

<Description>
 This CD provides a representation of various asymptotic set constructors (O, \Omega, etc.) The constructors represent sets of functions : R -> R.
</Description>

<CDDefinition>
<Name>O</Name>
<Description> 
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.   

</Description>

<CMP>
  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)}
</CMP>

<FMP>
<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>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>o</Name>
<Description>
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.
</Description>
<CMP>
o(g) = 
{f : reals -> reals | the limit as x tends to infinity of f(x)/g(x) is 0}
</CMP>

<FMP>
<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>
</FMP>

</CDDefinition>

<CDDefinition>
<Name>theta</Name>
<Description>
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) &lt; f(x) &lt; c_2*g(x).
</Description>

<CMP>
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) &lt; f(x) &lt; c_2*g(x)
</CMP>

<FMP>
<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>
</FMP>

</CDDefinition>

<CDDefinition>
<Name>asymptotic</Name>
<Description>
The asymptotic symbol represents a binary relation between two
functions of type reals to reals. The asymptotic relation between two
functions returns true if the two functions have the same rate of
growth and more precisely there ratio  approaches 1 as the variable
approaches infinity. Formally we say that f(x) is asymptotic to g(x)
if and only if the limit as x tends to infinity of f(x)/g(x) = 1.
</Description>

<CMP>
f(x) is
asymptotic g(x) if and only if the limit as x tends to infinity of
f(x)/g(x) = 1
</CMP>

<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="asymp1" name="asymptotic"/>
      <OMV name="f"/>
      <OMV name="g"/>
    </OMA>
    <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="one"/>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>

</CDDefinition>

<CDDefinition>
<Name>omega</Name>
<Description>
The omega symbol represents a unary function which constructs a set
of certain functions of type reals to positive reals. The omega
symbol represents a set of functions such that for any function in
the set omega(g(x)), f(x); it is not true that f(x) is in
o(g(x)). Formally we say that f(x) = omega(g(x)) if and only
if there is an epsilon > 0 and an infinite  sequence x_1, x_2, x_3,
... such that for all j then abs(f(x_j)) > epsilon * g(x_j). 
</Description>

<CMP>
f(x) is omega(g(x)) if and only if
it is not true that f(x) is o(g(x))
</CMP>

<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="omega"/>
      <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="logic1" name="not"/>
	  <OMA>
	    <OMS cd="set1" name="in"/>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	    <OMA>
	      <OMS cd="asymp1" name="o"/>
	      <OMV name="g"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>

<CMP>
f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an
infinite  sequence x_1, x_2, x_3, ... such that for all j then
abs(f(x_j)) > epsilon * g(x_j).
</CMP>

<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="f"/>
      <OMA>
        <OMS cd="asymp1" name="omega"/>
	<OMV name="g"/>
      </OMA>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
	  <OMV name="epsilon"/>
	  <OMV name="seq"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
	<OMA>
	  <OMS cd="relation1" name="gt"/>
	  <OMV name="epsilon"/>
	  <OMS cd="alg1" name="zero"/>
	</OMA>
	<OMBIND>
	  <OMS cd="quant1" name="forall"/>
	  <OMBVAR>
	    <OMV name="j"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="relation1" name="gt"/>
	    <OMA>
	      <OMS cd="arith1" name="abs"/>
	      <OMA>
	        <OMV name="f"/>
		<OMA>
		  <OMS cd="linalg1" name="vector_selector"/>
		  <OMV name="j"/>
		  <OMV name="seq"/>
		</OMA>
	      </OMA>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMV name="epsilon"/>
	      <OMA>
	        <OMV name="g"/>
		<OMA>
		  <OMS cd="linalg1" name="vector_selector"/>
		  <OMV name="j"/>
		  <OMV name="seq"/>
		</OMA>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMBIND>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>
</FMP>

</CDDefinition>

<CDDefinition>
<Name>Omega</Name>
<Description>
The Omega symbol represents a unary function which constructs a set
of certain functions of type reals to positive reals. The Omega
symbol represents a set of functions such that for any function in
the set Omega(g(x)), f(x); it is not true that f(x) is in
O(g(x)).
</Description>
<CMP>
f(x) is Omega(g(x)) if and only if
it is not true that f(x) is O(g(x))
</CMP>

<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="asymp1" name="Omega"/>
      <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="logic1" name="not"/>
	  <OMA>
	    <OMS cd="set1" name="in"/>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	    <OMA>
	      <OMS cd="asymp1" name="O"/>
	      <OMV name="g"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>
</FMP>
</CDDefinition>

</CD>