[Om3] Binding Integral symbols (was [Fwd: MathML CDs])

Michael Kohlhase m.kohlhase at jacobs-university.de
Thu Jul 5 06:14:17 CEST 2007


Dear James,

could you explain what you have done to the CDs, I have had a short look 
at integration, which I consider the most pressing problem in the 
alignment. The situation I see is the following: MathML has a binder 
symbol for integration and OpenMath an appliccative one. (I see that you 
have added an indefinite integral to the CD below to account for the two 
usages of <int> in MathML).

I think that the binding symbol for integrals is important to support, I 
mean a usage of the following form:

<OMBIND>
   <OMS name="int" cd="newint"/>
   <OMBVAR><OMV name="x"/></OMBVAR>
   <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
</OMBIND>

This usage of the integral allows to view it as an operation on an 
expression with a (bound) variable and does not force one to use the 
\lambda (which in my recollection mathematicians do not like, and I do 
not like, as I want to be able to use integration in logics that do not 
supply a \lambda). So I think we need a symbol for binding integral.

Please also see the my first e-mail  to the list 
[http://openmath.org/pipermail/om3/2007-June/000002.html], if we adopt 
having more than one role per symbol, then we could use the same symbol 
for the binding usage above and the apliccative below, and get by with 
one symbol.

<OMA>
   <OMS name="int" cd="newint"/>
   <OMBIND>
     <OMS cd="fn1" name="lambda"/>
     <OMBVAR><OMV name="x"/></OMBVAR>
     <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
  </OMBIND>
</OMA>

I think that  we need to support both usages in the aligned OM/MathML 
collection.

I am using the CD newint in my examples here to characterize my 
proposal. Anything I am saying also applies to other binding symbols 
like max, lim, ... and even the MathML symbols union, ... which  can be 
used in a binding form to represent terms like  \bigcup_{i=1}^n S_i. So 
we should really discuss this.

I will go into a discussion of the definite integrals in a separate e-mail.

Michael


Michael Kohlhase wrote:
> Dear James,
>
> we should use the OM3 mailing list, so I am forwarding this here.
>
> Michael
>
>
> ------------------------------------------------------------------------
>
> Subject:
> MathML CDs
> From:
> Professor James Davenport <jhd at cs.bath.ac.uk>
> Date:
> Tue, 3 Jul 2007 17:43:45 +0100 (BST)
> To:
> m.kohlhase at iu-bremen.de, olga.caprotti at helsinki.fi
>
> To:
> m.kohlhase at iu-bremen.de, olga.caprotti at helsinki.fi
> CC:
> James Davenport <jhd at cs.bath.ac.uk>
>
>
> Enclosed p,lease find very much a first cut an integration for MathML. 
> Ignore differentiation: I haven't really looked at that yet.
> Olga thought we might have a skype confernce call: at your times, I should 
> be in 10-12 and 3.30-6 tomorrow.
> James
> ------------------------------------------------------------------------
>
> <CDSignatures xmlns="http://www.openmath.org/OpenMathCDS" type="sts" cd="calculus1" >
> <CDSStatus> official </CDSStatus>
>
> <CDSComment>
> Date:  1999-04-13
> Author: David Carlisle
> </CDSComment>
>
>
> <Signature name="diff" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> 
>  <OMA>
>   <OMS name="mapsto" cd="sts" />
>   <OMA>
>    <OMS name="mapsto" cd="sts" />
>    <OMS name="Object" cd="sts" />
>    <OMS name="Object" cd="sts" />
>   </OMA>
>   <OMA>
>    <OMS name="mapsto" cd="sts" />
>    <OMS name="Object" cd="sts" />
>    <OMS name="Object" cd="sts" />
>   </OMA>
>  </OMA>
> </OMOBJ>
> </Signature>
>
> <Signature name="nthdiff" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> 
>  <OMA>
>   <OMS name="mapsto" cd="sts" />
>   <OMS name="Z" cd="setname1" />
>   <OMA>
>    <OMS name="mapsto" cd="sts" />
>    <OMS name="Object" cd="sts" />
>    <OMS name="Object" cd="sts" />
>   </OMA>
>   <OMA>
>    <OMS name="mapsto" cd="sts" />
>    <OMS name="Object" cd="sts" />
>    <OMS name="Object" cd="sts" />
>   </OMA>
>  </OMA>
> </OMOBJ>
> </Signature>
>
>
>
>
>
> <Signature name="partialdiff" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath">                         
>  <OMA>                         
>   <OMS name="mapsto" cd="sts" />
>   <OMV name="VariableList" />  
>   <OMA>                         
>    <OMS name="mapsto" cd="sts" />
>    <OMA>                         
>     <OMS name="nary" cd="sts"/>
>     <OMS name="Object" cd="sts"/>
>    </OMA>
>    <OMS name="Object" cd="sts"/>
>   </OMA>
>   <OMA>                         
>    <OMS name="mapsto" cd="sts" />
>    <OMA>                         
>     <OMS name="nary" cd="sts"/>
>     <OMS name="Object" cd="sts"/>
>    </OMA>
>    <OMS name="Object" cd="sts"/>
>   </OMA>
>  </OMA>
> </OMOBJ>
> </Signature>
>
>
> <Signature name="lowlimit" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> 
>  <OMA>
>   <OMS name="mapsto" cd="sts" />
>   <OMV name="intqualifier"/>
>   <OMS name="Object" cd="sts" />
>  </OMA>
> </OMOBJ>
> </Signature>
>
> <Signature name="uplimit" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> 
>  <OMA>
>   <OMS name="mapsto" cd="sts" />
>   <OMV name="intqualifier"/>
>   <OMS name="Object" cd="sts" />
>  </OMA>
> </OMOBJ>
> </Signature>
>
>
> <Signature name="int" >
> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> 
>  <OMA>
>   <OMS name="mapsto" cd="sts" />
>   <OMS name="Object" cd="sts" />
>   <OMS name="Object" cd="sts" />
>   <OMA>
>      <OMS name="nassoc" cd="sts"/>
>      <OMV name="intqualifier"/>
>   </OMA>
>  </OMA>
> </OMOBJ>
> </Signature>
>
>
> </CDSignatures>
>   
> ------------------------------------------------------------------------
>
> <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 
>      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 http://www.openmath.org.
> </CDComment>
>
> <CDName> calcmml1 </CDName>
> <CDBase>http://www.openmath.org/cd</CDBase>
> <CDURL> http://www.openmath.org/cd/calcmml1.ocd </CDURL>
> <CDReviewDate>2008-03-30</CDReviewDate>
> <CDStatus> experimental </CDStatus>
> <CDDate>2007-07-01</CDDate>
> <CDVersion>1</CDVersion>
> <CDRevision>0</CDRevision>
> <Description>
>     This CD is intended to be compatible with the calculus operations
>     in Content MathML, unlike calculus1 more in 'letter' than in spirit.
>
>     Integration is just for the univariate case and is either definite
>     or indefinite, but depending on qualifiers, unlike the 'proper'
>     calculus1.
> </Description>
>
> <CDDefinition>
> <Name> diff </Name>
> <Role>application</Role>
> <Description>
> This symbol is used to express ordinary differentiation of a unary
> function. The single argument is the unary function.
> </Description>
>
> <CMP> 
> diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x)
> </CMP>
>
> <FMP>
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS cd="relation1" name="eq"/>
>     <OMA>
>       <OMA>
>         <OMS cd="calculus1" name="diff"/>
>         <OMBIND>
>           <OMS cd="fns1" name="lambda"/>
>           <OMBVAR>
>             <OMV name="y"/>
>           </OMBVAR>
>           <OMA>
>             <OMS cd="arith1" name="plus"/>
>             <OMA>
>               <OMV name="a"/>
>               <OMV name="y"/>
>             </OMA>
>             <OMA>
>               <OMV name="b"/>
>               <OMV name="y"/>
>             </OMA>
>           </OMA>
>         </OMBIND>
>       </OMA>
>       <OMV name="x"/>
>     </OMA>
>     <OMA>
>       <OMS cd="arith1" name="plus"/>
>       <OMA>
>         <OMA>
>           <OMS cd="calculus1" name="diff"/>
>           <OMBIND>
>             <OMS cd="fns1" name="lambda"/>
>             <OMBVAR>
>               <OMV name="y"/>
>             </OMBVAR>
>             <OMA>
>               <OMV name="a"/>
>               <OMV name="y"/>
>             </OMA>
>           </OMBIND>
>         </OMA>
>         <OMV name="x"/>
>       </OMA>
>       <OMA>
>         <OMA>
>           <OMS cd="calculus1" name="diff"/>
>           <OMBIND>
>             <OMS cd="fns1" name="lambda"/>
>             <OMBVAR>
>               <OMV name="y"/>
>             </OMBVAR>
>             <OMA>
>               <OMV name="b"/>
>               <OMV name="y"/>
>             </OMA>
>           </OMBIND>
>         </OMA>
>         <OMV name="x"/>
>       </OMA>
>     </OMA>
>   </OMA>
> </OMOBJ>
> </FMP>
>
> <CMP> 
> diff(lambda y:a(y) * b(y))(x) = diff(lambda y:a(y))(x) * b(x) + 
>                                 a(x) * diff(lambda y:b(y))(x)
> </CMP>
>
> <FMP>
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS cd="relation1" name="eq"/>
>     <OMA>
>       <OMA>
>         <OMS cd="calculus1" name="diff"/>
>         <OMBIND>
>           <OMS cd="fns1" name="lambda"/>
>           <OMBVAR>
>             <OMV name="y"/>
>           </OMBVAR>
>           <OMA>
>             <OMS cd="arith1" name="times"/>
>             <OMA>
>               <OMV name="a"/>
>               <OMV name="y"/>
>             </OMA>
>             <OMA>
>               <OMV name="b"/>
>               <OMV name="y"/>
>             </OMA>
>           </OMA>
>         </OMBIND>
>       </OMA>
>       <OMV name="x"/>
>     </OMA>
>     <OMA>
>       <OMS cd="arith1" name="plus"/>
>       <OMA>
>         <OMS cd="arith1" name="times"/>
>         <OMA>
>           <OMA>
>             <OMS cd="calculus1" name="diff"/>
>             <OMBIND>
>               <OMS cd="fns1" name="lambda"/>
>               <OMBVAR>
>                 <OMV name="y"/>
>               </OMBVAR>
>               <OMA>
>                 <OMV name="a"/>
>                 <OMV name="y"/>
>               </OMA>
>             </OMBIND>
>           </OMA>
>           <OMV name="x"/>
>         </OMA>
>         <OMA>
>           <OMV name="b"/>
>           <OMV name="x"/>
>         </OMA>
>       </OMA>
>       <OMA>
>         <OMS cd="arith1" name="times"/>
>         <OMA>
>           <OMV name="a"/>
>           <OMV name="x"/>
>         </OMA>
>         <OMA>
>           <OMA>
>             <OMS cd="calculus1" name="diff"/>
>             <OMBIND>
>               <OMS cd="fns1" name="lambda"/>
>               <OMBVAR>
>                 <OMV name="y"/>
>               </OMBVAR>
>               <OMA>
>                 <OMV name="b"/>
>                 <OMV name="y"/>
>               </OMA>
>             </OMBIND>
>           </OMA>
>           <OMV name="x"/>
>         </OMA>
>       </OMA>
>     </OMA>
>   </OMA>
> </OMOBJ>
> </FMP>
>
> <Example>
> This represents the equation:
>   derivative(x + 1.0) = 1.0
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS cd="relation1" name="eq"/>
>     <OMA>
>       <OMA>
>         <OMS cd="calculus1" name="diff"/>
>         <OMBIND>
>           <OMS cd="fns1" name="lambda"/>
>           <OMBVAR>
>             <OMV name="x"/>
>           </OMBVAR>
>           <OMA>
>             <OMS cd="arith1" name="plus"/>
>             <OMV name="x"/>
>             <OMF dec="1.0"/>
>           </OMA>
>         </OMBIND>
>       </OMA>
>       <OMV name="y"/>
>     </OMA>
>     <OMF dec="1.0"/>
>   </OMA>
> </OMOBJ>
> </Example>
> </CDDefinition>
>
> <CDDefinition>
> <Name> nthdiff </Name>
> <Role>application</Role>
> <Description>
> This symbol is used to express the nth-iterated ordinary differentiation of 
> a unary function. The first argument is n, and the second the unary function.
> </Description>
>
> <FMP>
>   <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS name="eq" cd="relation1"/>
>     <OMA>
>       <OMS name="nthdiff" cd="calculus1"/>
>       <OMS name="zero" cd="alg1"/>
>       <OMV name="f"/>
>     </OMA>
>     <OMV name="f"/>
>   </OMA>
>   </OMOBJ>
> </FMP>
>
> <FMP>
>   <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS name="eq" cd="relation1"/>
>     <OMA>
>       <OMS name="nthdiff" cd="calculus1"/>
>       <OMA>
>         <OMS name="plus" cd="arith1"/>
>         <OMV name="n"/>
>         <OMS name="one" cd="alg1"/>
>       </OMA>
>       <OMV name="f"/>
>     </OMA>
>     <OMA>
>       <OMS name="diff" cd="calculus1"/>
>       <OMA>
>         <OMS name="nthdiff" cd="calculus1"/>
>         <OMV name="n"/>
>         <OMV name="f"/>
>       </OMA>
>     </OMA>
>   </OMA>
>   </OMOBJ>
> </FMP>
>
> </CDDefinition>
>
> <CDDefinition>
> <Name> partialdiff </Name>
> <Role>application</Role>
> <Description>
> This symbol is used to express partial differentiation of a function
> of more than one variable. It has two arguments, the first is a list of 
> integers which index the variables of the function, the second is the function.
> </Description>
> <Example>
> An example to represent the equation:
>  \partial^2{xyz}/ \partial{x}\partial{z} = y
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
> <OMA>
>   <OMS cd="relation1" name="eq"/>
>   <OMA>
>     <OMA>
>       <OMS cd="calculus1" name="partialdiff"/>
>       <OMA>
>         <OMS cd="list1" name="list"/>
>         <OMI> 1 </OMI>
>         <OMI> 3 </OMI>
>       </OMA>
>       <OMBIND>
>         <OMS cd="fns1" name="lambda"/>
>         <OMBVAR>
>           <OMV name="x"/>
>           <OMV name="y"/>
>           <OMV name="z"/>
>         </OMBVAR>
>         <OMA>
>           <OMS cd="arith1" name="times"/>
>           <OMV name="x"/>
>           <OMV name="y"/>
>           <OMV name="z"/>
>         </OMA>
>       </OMBIND>
>     </OMA>
>     <OMV name="y"/>
>   </OMA>
>   <OMV name="y"/>
> </OMA>
> </OMOBJ>
> </Example>
> </CDDefinition>
>
> <CDDefinition>
> <Name> lowlimit </Name>
> <Role>qualifier</Role>
> <Description> 
> This symbol is used to represent the lower limit of indefinite integration
> of unary functions.
> </Description>
> </CDDefinition>
>
> <CDDefinition>
> <Name> uplimit </Name>
> <Role>qualifier</Role>
> <Description> 
> This symbol is used to represent the upper limit of indefinite integration
> of unary functions.
> </Description>
> </CDDefinition>
>
> <CDDefinition>
> <Name> int </Name>
> <Role>application</Role>
> <Description> 
> This symbol is used to represent indefinite integration of unary functions.
> The arguments are the unary function, followed by any qualifiers.
> </Description>
>
> <CMP>
>   In single-argument mode, it is equivalent to the calculus1 version
> </CMP>
>
> <FMP type="extensional">
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMS name="eq" cd="relation1"/>
>   <OMA>
>     <OMS name="int" cd="calcmml1"/>
>     <OMV name="f"/>
>   </OMA>
>   <OMA>
>     <OMS name="int" cd="calculus1"/>
>     <OMV name="f"/>
>   <OMA>
>   </OMA>
> </OMOBJ>
> </FMP>
>
> <CMP> with upper and lower limits, it is equivalent to defint from calculus1.
>       unfortunately, it seems that we need to state this both ways round!
> </CMP>
>
> <FMP type="extensional">
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMS name="eq" cd="relation1"/>
>   <OMA>
>     <OMS name="int" cd="calcmml1"/>
>     <OMV name="f"/>
>     <OMA> 
>       <OMS cd="calcmml1" name="lowlimit"/> 
>       <OMV name="l"/> 
>     </OMA>
>     <OMA> 
>       <OMS cd="calcmml1" name="uplimit"/> 
>       <OMV name="u"/> 
>     </OMA>
>   </OMA>
>   <OMA>
>     <OMS name="defint" cd="calculus1"/>
>     <OMA> 
>       <OMS cd="interval1" name="interval"/> 
>       <OMV name="l"/> 
>       <OMV name="u"/> 
>     </OMA>
>     <OMV name="f"/>
>   <OMA>
> </OMOBJ>
> </FMP>
>
> <FMP type="extensional">
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMS name="eq" cd="relation1"/>
>   <OMA>
>     <OMS name="int" cd="calcmml1"/>
>     <OMV name="f"/>
>     <OMA> 
>       <OMS cd="calcmml1" name="uplimit"/> 
>       <OMV name="u"/> 
>     </OMA>
>     <OMA> 
>       <OMS cd="calcmml1" name="lowlimit"/> 
>       <OMV name="l"/> 
>     </OMA>
>   </OMA>
>   <OMA>
>     <OMS name="defint" cd="calculus1"/>
>     <OMA> 
>       <OMS cd="interval1" name="interval"/> 
>       <OMV name="l"/> 
>       <OMV name="u"/> 
>     </OMA>
>     <OMV name="f"/>
>   <OMA>
> </OMOBJ>
> </FMP>
> </CDDefinition>
>
>
>
>
> <CDDefinition>
> <Name> defint </Name>
> <Role>application</Role>
> <Description> 
> This symbol is used to represent definite integration of unary
> functions. It takes two arguments; the first being the range (e.g. a
> set) of integration, and the second the function.
> </Description>
>
> <CMP> for all a,b | integral from a to b = -integral from b to a </CMP>
>
> <FMP>
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMBIND>
>     <OMS cd="quant1" name="forall"/>
>     <OMBVAR>
>       <OMV name="a"/>
>       <OMV name="b"/>
>     </OMBVAR>
>     <OMA>
>       <OMS cd="relation1" name="eq"/>
>       <OMA>
>         <OMS cd="calculus1" name="defint"/>
>         <OMA>
>           <OMS cd="interval1" name="interval"/>
>           <OMV name="a"/>
>           <OMV name="b"/>
>         </OMA>
>         <OMBIND>
>           <OMS cd="fns1" name="lambda"/>
>           <OMBVAR>
>             <OMV name="x"/>
>           </OMBVAR>
>           <OMA>
>             <OMV name="f"/>
>             <OMV name="x"/>
>           </OMA>
>         </OMBIND>
>       </OMA>
>       <OMA>
>         <OMS cd="arith1" name="unary_minus"/>
>         <OMA>
>           <OMS cd="calculus1" name="defint"/>
>           <OMA>
>             <OMS cd="interval1" name="interval"/>
>             <OMV name="b"/>
>             <OMV name="a"/>
>           </OMA>
>           <OMBIND>
>             <OMS cd="fns1" name="lambda"/>
>             <OMBVAR>
>               <OMV name="x"/>
>             </OMBVAR>
>             <OMA>
>               <OMV name="f"/>
>               <OMV name="x"/>
>             </OMA>
>           </OMBIND>
>         </OMA>
>       </OMA>
>     </OMA>
>   </OMBIND>
> </OMOBJ>
> </FMP>
>
> <CMP>
> for all a &lt; b &lt; c | integral over [a,c] = integral over [a,b] + 
> integral over [b,c]
> </CMP>
>
> <FMP>
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMBIND>
>     <OMS cd="quant1" name="forall"/>
>     <OMBVAR>
>       <OMV name="a"/>
>       <OMV name="b"/>
>       <OMV name="c"/>
>     </OMBVAR>
>     <OMA>
>       <OMS cd="logic1" name="implies"/>
>       <OMA>
>         <OMS cd="logic1" name="and"/>
>         <OMA>
>           <OMS cd="relation1" name="lt"/>
>           <OMV name="a"/>
>   	<OMV name="b"/>
>         </OMA>
>         <OMA>
>           <OMS cd="relation1" name="lt"/>
> 	  <OMV name="b"/>
> 	  <OMV name="c"/>
>         </OMA>
>       </OMA>
>       <OMA>
>         <OMS cd="relation1" name="eq"/>
> 	<OMA>
> 	  <OMS cd="calculus1" name="defint"/>
> 	  <OMA>
> 	    <OMS cd="interval1" name="interval"/>
> 	    <OMV name="a"/>
> 	    <OMV name="c"/>
> 	  </OMA>
> 	  <OMBIND>
> 	    <OMS cd="fns1" name="lambda"/>
> 	    <OMBVAR>
> 	      <OMV name="x"/>
> 	    </OMBVAR>
> 	    <OMA>
> 	      <OMV name="f"/>
> 	      <OMV name="x"/>
> 	    </OMA>
> 	  </OMBIND>
> 	</OMA>
> 	<OMA>
> 	  <OMS cd="arith1" name="plus"/>
> 	  <OMA>
> 	    <OMS cd="calculus1" name="defint"/>
> 	    <OMA>
> 	      <OMS cd="interval1" name="interval"/>
> 	      <OMV name="a"/>
> 	      <OMV name="b"/>
> 	    </OMA>
> 	    <OMBIND>
> 	      <OMS cd="fns1" name="lambda"/>
> 	      <OMBVAR>
> 	        <OMV name="x"/>
> 	      </OMBVAR>
> 	      <OMA>
> 	        <OMV name="f"/>
> 	        <OMV name="x"/>
> 	      </OMA>
> 	    </OMBIND>
> 	  </OMA>
> 	  <OMA>
> 	    <OMS cd="calculus1" name="defint"/>
> 	    <OMA>
> 	      <OMS cd="interval1" name="interval"/>
> 	      <OMV name="b"/>
> 	      <OMV name="c"/>
> 	    </OMA>
> 	    <OMBIND>
> 	      <OMS cd="fns1" name="lambda"/>
> 	      <OMBVAR>
> 	        <OMV name="x"/>
> 	      </OMBVAR>
> 	      <OMA>
> 	        <OMV name="f"/>
> 	        <OMV name="x"/>
> 	      </OMA>
> 	    </OMBIND>
> 	  </OMA>
> 	</OMA>
>       </OMA>
>     </OMA>
>   </OMBIND>
> </OMOBJ>
> </FMP>
> <Example>
> An example to represent the definite integration of sin(x) between the points 
> -1.0 and 1.0.
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS cd="calculus1" name="defint"/>
>     <OMA>
>       <OMS cd="interval1" name="interval"/>
>       <OMF dec="-1.0"/>
>       <OMF dec="1.0"/>
>     </OMA>
>     <OMBIND>
>       <OMS cd="fns1" name="lambda"/>
>       <OMBVAR>
>         <OMV name="x"/>
>       </OMBVAR>
>       <OMA>
>          <OMS cd="transc1" name="sin"/>
> 	<OMV name="x"/>
>       </OMA>
>     </OMBIND>
>   </OMA>
> </OMOBJ>
> </Example>
>
> <Example>
> An example to represent the definite integration of f(x), for x in the
> set C:
> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
>   <OMA>
>     <OMS name="defint" cd="calculus1"/>
>     <OMV name="C"/>
>     <OMBIND>
>       <OMS name="lambda" cd="fns1"/>
>       <OMBVAR>
>         <OMV name="x"/>
>       </OMBVAR>
>       <OMA>
>         <OMV name="f"/>
>         <OMV name="x"/>
>       </OMA>
>     </OMBIND>
>   </OMA>
> </OMOBJ>
>
> </Example>
> </CDDefinition>
>
> </CD>
>   
> ------------------------------------------------------------------------
>
> _______________________________________________
> Om3 mailing list
> Om3 at openmath.org
> http://openmath.org/mailman/listinfo/om3
>   

-- 
----------------------------------------------------------------------
 Prof. Dr. Michael Kohlhase,       Office: Research 1, Room 62 
 Professor of Computer Science     Campus Ring 12, 
 School of Engineering & Science   D-28759 Bremen, Germany
 Jacobs University Bremen*         tel/fax: +49 421 200-3140/-493140
 m.kohlhase at jacobs-university.de http://kwarc.info/kohlhase 
 skype: m.kohlhase   * International University Bremen until Feb. 2007
----------------------------------------------------------------------



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