<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> meta </CDName> <CDReviewDate>2006-03-30</CDReviewDate> <CDDate>2004-03-30</CDDate> <CDVersion>3</CDVersion> <CDRevision>0</CDRevision> <CDStatus> official </CDStatus> <CDURL> http://www.openmath.org/cd/meta.ocd </CDURL> <CDBase>http://www.openmath.org/cd</CDBase> <Description> This is a content dictionary to represent content dictionaries, so that they may be passed between OpenMath compliant application in a similar way to mathematical objects. The information written here is taken from chapter 4 of the current draft of the "OpenMath Standard". </Description> <CDDefinition> <Name> CD </Name> <Role>application</Role> <Description> The top level element for the Content Dictionary. It just acts as a container for the elements described below. </Description> </CDDefinition> <CDDefinition> <Name> CDDefinition </Name> <Role>application</Role> <Description> This symbol is used to represent the element which contains the definition of each symbol in a content dictionary. That is: it must contain a 'Name' element and a 'Description' element, and it may contain an arbitrary number of 'Example', 'FMP' or 'CMP' elements. </Description> </CDDefinition> <CDDefinition> <Name> CDName </Name> <Role>application</Role> <Description> An element which contains the string corresponding to the name of the CD. The string must match the syntax for CD names given in the OpenMath Standard. Here and elsewhere white space occurring at the beginning or end of the string will be ignored. </Description> </CDDefinition> <CDDefinition> <Name> CDURL </Name> <Role>application</Role> <Description> An optional element. If it is used it contains a string representing the URL where the canonical reference copy of this CD is stored. </Description> </CDDefinition> <CDDefinition> <Name> CDBase </Name> <Role>application</Role> <Description> An optional element. If it is used it contains a string representing the URI to be used as the base for generated canonical URI references for symbols in the CD. </Description> </CDDefinition> <CDDefinition> <Name> Example </Name> <Role>application</Role> <Description> An element which contains an arbitrary number of children, each of which is either a string or an OpenMath Object. These children give examples in natural language, or in OpenMath, of the enclosing symbol definition. </Description> </CDDefinition> <CDDefinition> <Name> CDDate </Name> <Role>application</Role> <Description> An element which contains a date as a string in the ISO-8601 YYYY-MM-DD format. This gives the date at which the Content Dictionary was last edited. </Description> </CDDefinition> <CDDefinition> <Name> CDVersion </Name> <Role>application</Role> <Description> An element which contains a version number for the CD. This should be a non negative integer. Any change to the CD that affects existing OpenMath applications that support this CD should result in an increase in the version number. </Description> </CDDefinition> <CDDefinition> <Name> CDRevision </Name> <Role>application</Role> <Description> An element which contains a revision number (or minor version number) This should be a non-negative integer starting from zero for each new version. Additional examples would be typical changes to a CD requiring a new revision number. </Description> </CDDefinition> <CDDefinition> <Name> CDReviewDate </Name> <Role>application</Role> <Description> An element which contains a date as a string in the ISO-8601 YYYY-MM-DD format. This gives the date at which the Content Dictionary is next scheduled for review. It should be expected to be stable until at least this date. </Description> </CDDefinition> <CDDefinition> <Name> CDStatus </Name> <Role>application</Role> <Description> An element giving information on the status of the CD. The content of the element must be one of the following strings. official (approved by the OpenMath Society), experimental (currently being tested), private (used by a private group of OpenMath users), or obsolete (an obsolete CD kept only for archival purposes). </Description> </CDDefinition> <CDDefinition> <Name> CDComment </Name> <Role>application</Role> <Description> This symbol is used to represent the element of a content dictionary which explains some aspect of that content dictionary. It should have one string argument which makes that explanation. </Description> </CDDefinition> <CDDefinition> <Name> CDUses </Name> <Role>application</Role> <Description> An element which contains zero or more CDNames which correspond to the CDs that this CD depends on, i.e. uses in examples and FMPs. If the CD is dependent on any other CDs they may be present here. </Description> </CDDefinition> <CDDefinition> <Name> Description </Name> <Role>application</Role> <Description> An element which contains a string corresponding to the description of either the CD or the symbol (depending on which is the enclosing element). </Description> </CDDefinition> <CDDefinition> <Name> Name </Name> <Role>application</Role> <Description> An element containing the string corresponding to the name of the symbol being defined. This must match the syntax for symbol names given in the OpenMath Standard. Here and elsewhere white space occurring at the begining or end of the string will be ignored. </Description> </CDDefinition> <CDDefinition> <Name> Role </Name> <Role>application</Role> <Description> An element containing the string corresponding to the role of the symbol being defined. </Description> </CDDefinition> <CDDefinition> <Name> CMP </Name> <Role>application</Role> <Description> An optional element (which may be repeated many times) which contains a string corresponding to a property of the symbol being defined. </Description> </CDDefinition> <CDDefinition> <Name> FMP </Name> <Role>application</Role> <Description> An optional element which contains an OpenMath Object. This corresponds to a property of the symbol being defined. </Description> </CDDefinition> </CD>
<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> arith1 </CDName> <CDBase>http://www.openmath.org/cd</CDBase> <CDURL> http://www.openmath.org/cd/arith1.ocd </CDURL> <CDReviewDate>2006-03-30</CDReviewDate> <CDStatus> official </CDStatus> <CDDate>2004-03-30</CDDate> <CDVersion>3</CDVersion> <CDRevision>0</CDRevision> <Description> This CD defines symbols for common arithmetic functions. </Description> <CDDefinition> <Name> lcm </Name> <Role>application</Role> <Description> The symbol to represent the n-ary function to return the least common multiple of its arguments. </Description> <CMP> lcm(a,b) = a*b/gcd(a,b) </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> <OMS cd="arith1" name="lcm"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="divide"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="gcd"/> <OMV name="a"/> <OMV name="b"/> </OMA> </OMA> </OMA> </OMOBJ> </FMP> <CMP> for all integers a,b | There does not exist a c>0 such that c/a is an Integer and c/b is an Integer and lcm(a,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"/> </OMBVAR> <OMA> <OMS cd="logic1" name="implies"/> <OMA> <OMS cd="logic1" name="and"/> <OMA> <OMS cd="set1" name="in"/> <OMV name="a"/> <OMS cd="setname1" name="Z"/> </OMA> <OMA> <OMS cd="set1" name="in"/> <OMV name="b"/> <OMS cd="setname1" name="Z"/> </OMA> </OMA> <OMA> <OMS cd="logic1" name="not"/> <OMBIND> <OMS cd="quant1" name="exists"/> <OMBVAR> <OMV name="c"/> </OMBVAR> <OMA> <OMS cd="logic1" name="and"/> <OMA> <OMS cd="relation1" name="gt"/> <OMV name="c"/> <OMI>0</OMI> </OMA> <OMA> <OMS cd="integer1" name="factorof"/> <OMV name="a"/> <OMV name="c"/> </OMA> <OMA> <OMS cd="integer1" name="factorof"/> <OMV name="b"/> <OMV name="c"/> </OMA> <OMA> <OMS cd="relation1" name="lt"/> <OMV name="c"/> <OMA> <OMS cd="arith1" name="lcm"/> <OMV name="a"/> <OMV name="b"/> </OMA> </OMA> </OMA> </OMBIND> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> gcd </Name> <Role>application</Role> <Description> The symbol to represent the n-ary function to return the gcd (greatest common divisor) of its arguments. </Description> <CMP> for all integers a,b | There does not exist a c such that a/c is an Integer and b/c is an Integer and c > gcd(a,b). Note that this implies that gcd(a,b) > 0 </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="logic1" name="implies"/> <OMA> <OMS cd="logic1" name="and"/> <OMA> <OMS cd="set1" name="in"/> <OMV name="a"/> <OMS cd="setname1" name="Z"/> </OMA> <OMA> <OMS cd="set1" name="in"/> <OMV name="b"/> <OMS cd="setname1" name="Z"/> </OMA> </OMA> <OMA> <OMS cd="logic1" name="not"/> <OMBIND> <OMS cd="quant1" name="exists"/> <OMBVAR> <OMV name="c"/> </OMBVAR> <OMA> <OMS cd="logic1" name="and"/> <OMA> <OMS cd="set1" name="in"/> <OMA> <OMS cd="arith1" name="divide"/> <OMV name="a"/> <OMV name="c"/> </OMA> <OMS cd="setname1" name="Z"/> </OMA> <OMA> <OMS cd="set1" name="in"/> <OMA> <OMS cd="arith1" name="divide"/> <OMV name="b"/> <OMV name="c"/> </OMA> <OMS cd="setname1" name="Z"/> </OMA> <OMA> <OMS cd="relation1" name="gt"/> <OMV name="c"/> <OMA> <OMS cd="arith1" name="gcd"/> <OMV name="a"/> <OMV name="b"/> </OMA> </OMA> </OMA> </OMBIND> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> <Example> gcd(6,9) = 3 <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="gcd"/> <OMI> 6 </OMI> <OMI> 9 </OMI> </OMA> <OMI> 3 </OMI> </OMA> </OMOBJ> </Example> </CDDefinition> <CDDefinition> <Name> plus </Name> <Role>application</Role> <Description> The symbol representing an n-ary commutative function plus. </Description> <CMP> for all a,b | a + b = b + 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="arith1" name="plus"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="plus"/> <OMV name="b"/> <OMV name="a"/> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> unary_minus </Name> <Role>application</Role> <Description> This symbol denotes unary minus, i.e. the additive inverse. </Description> <CMP> for all a | a + (-a) = 0 </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"/> </OMBVAR> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="plus"/> <OMV name="a"/> <OMA> <OMS cd="arith1" name="unary_minus"/> <OMV name="a"/> </OMA> </OMA> <OMS cd="alg1" name="zero"/> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> minus </Name> <Role>application</Role> <Description> The symbol representing a binary minus function. This is equivalent to adding the additive inverse. </Description> <CMP> for all a,b | a - b = a + (-b) </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="arith1" name="minus"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="plus"/> <OMV name="a"/> <OMA> <OMS cd="arith1" name="unary_minus"/> <OMV name="b"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> times </Name> <Role>application</Role> <Description> The symbol representing an n-ary multiplication function. </Description> <Example> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="times"/> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 1 </OMI> <OMI> 2 </OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 3 </OMI> <OMI> 4 </OMI> </OMA> </OMA> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 5 </OMI> <OMI> 6 </OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 7 </OMI> <OMI> 8 </OMI> </OMA> </OMA> </OMA> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 19 </OMI> <OMI> 22 </OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 43 </OMI> <OMI> 50 </OMI> </OMA> </OMA> </OMA> </OMOBJ> </Example> <CMP> for all a,b | a * 0 = 0 and a * b = a * (b - 1) + 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="logic1" name="and"/> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMS cd="alg1" name="zero"/> </OMA> <OMS cd="alg1" name="zero"/> </OMA> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="plus"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMA> <OMS cd="arith1" name="minus"/> <OMV name="b"/> <OMS cd="alg1" name="one"/> </OMA> </OMA> <OMV name="a"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ></FMP> <CMP> for all a,b,c | a*(b+c) = a*b + a*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="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMA> <OMS cd="arith1" name="plus"/> <OMV name="b"/> <OMV name="c"/> </OMA> </OMA> <OMA> <OMS cd="arith1" name="plus"/> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="times"/> <OMV name="a"/> <OMV name="c"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ></FMP> </CDDefinition> <CDDefinition> <Name> divide </Name> <Role>application</Role> <Description> This symbol represents a (binary) division function denoting the first argument right-divided by the second, i.e. divide(a,b)=a*inverse(b). It is the inverse of the multiplication function defined by the symbol times in this CD. </Description> <CMP> whenever not(a=0) then a/a = 1 </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"/> </OMBVAR> <OMA> <OMS cd="logic1" name="implies"/> <OMA> <OMS cd="relation1" name="neq"/> <OMV name="a"/> <OMS cd="alg1" name="zero"/> </OMA> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="divide"/> <OMV name="a"/> <OMV name="a"/> </OMA> <OMS cd="alg1" name="one"/> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> power </Name> <Role>application</Role> <Description> This symbol represents a power function. The first argument is raised to the power of the second argument. When the second argument is not an integer, powering is defined in terms of exponentials and logarithms for the complex and real numbers. This operator can represent general powering. </Description> <CMP> x\in C implies x^a = exp(a ln x) </CMP> <FMP> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="logic1" name="implies"/> <OMA> <OMS cd="set1" name="in"/> <OMV name="x"/> <OMS cd="setname1" name="C"/> </OMA> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS name="power" cd="arith1"/> <OMV name="x"/> <OMV name="a"/> </OMA> <OMA> <OMS name="exp" cd="transc1"/> <OMA> <OMS name="times" cd="arith1"/> <OMV name="a"/> <OMA> <OMS name="ln" cd="transc1"/> <OMV name="x"/> </OMA> </OMA> </OMA> </OMA> </OMA> </OMOBJ> </FMP> <CMP> if n is an integer then x^0 = 1, x^n = x * x^(n-1) </CMP> <FMP> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="logic1" name="implies"/> <OMA> <OMS cd="set1" name="in"/> <OMV name="n"/> <OMS cd="setname1" name="Z"/> </OMA> <OMA> <OMS cd="logic1" name="and"/> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMI>0</OMI> </OMA> <OMS cd="alg1" name="one"/> </OMA> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMV name="n"/> </OMA> <OMA> <OMS cd="arith1" name="times"/> <OMV name="x"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMA> <OMS cd="arith1" name="minus"/> <OMV name="n"/> <OMI>1</OMI> </OMA> </OMA> </OMA> </OMA> </OMA> </OMA> </OMOBJ> </FMP> <Example> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="power"/> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 1 </OMI> <OMI> 2 </OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 3 </OMI> <OMI> 4 </OMI> </OMA> </OMA> <OMI>3</OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 37 </OMI> <OMI> 54 </OMI> </OMA> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMI> 81 </OMI> <OMI> 118 </OMI> </OMA> </OMA> </OMA> </OMOBJ> </Example> <Example> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="power"/> <OMS cd="nums1" name="e"/> <OMA> <OMS cd="arith1" name="times"/> <OMS cd="nums1" name="i"/> <OMS cd="nums1" name="pi"/> </OMA> </OMA> <OMA> <OMS cd="arith1" name="unary_minus"/> <OMS cd="alg1" name="one"/> </OMA> </OMA> </OMOBJ> </Example> </CDDefinition> <CDDefinition> <Name> abs </Name> <Role>application</Role> <Description> A unary operator which represents the absolute value of its argument. The argument should be numerically valued. In the complex case this is often referred to as the modulus. </Description> <CMP> for all x,y | abs(x) + abs(y) >= abs(x+y) </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="x"/> <OMV name="y"/> </OMBVAR> <OMA> <OMS cd="relation1" name="geq"/> <OMA> <OMS cd="arith1" name="plus"/> <OMA> <OMS cd="arith1" name="abs"/> <OMV name="x"/> </OMA> <OMA> <OMS cd="arith1" name="abs"/> <OMV name="y"/> </OMA> </OMA> <OMA> <OMS cd="arith1" name="abs"/> <OMA> <OMS cd="arith1" name="plus"/> <OMV name="x"/> <OMV name="y"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> root </Name> <Role>application</Role> <Description> A binary operator which represents its first argument "lowered" to its n'th root where n is the second argument. This is the inverse of the operation represented by the power symbol defined in this CD. Care should be taken as to the precise meaning of this operator, in particular which root is represented, however it is here to represent the general notion of taking n'th roots. As inferred by the signature relevant to this symbol, the function represented by this symbol is the single valued function, the specific root returned is the one indicated by the first CMP. Note also that the converse of the second CMP is not valid in general. </Description> <CMP> x\in C implies root(x,n) = exp(ln(x)/n) </CMP> <FMP> <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="logic1" name="implies"/> <OMA> <OMS cd="set1" name="in"/> <OMV name="x"/> <OMS cd="setname1" name="C"/> </OMA> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="root"/> <OMV name="x"/> <OMV name="n"/> </OMA> <OMA> <OMS name="exp" cd="transc1"/> <OMA> <OMS name="divide" cd="arith1"/> <OMA> <OMS name="ln" cd="transc1"/> <OMV name="x"/> </OMA> <OMV name="n"/> </OMA> </OMA> </OMA> </OMA> </OMOBJ> </FMP> <CMP> for all a,n | power(root(a,n),n) = a (if the root exists!) </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="n"/> </OMBVAR> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="power"/> <OMA> <OMS cd="arith1" name="root"/> <OMV name="a"/> <OMV name="n"/> </OMA> <OMV name="n"/> </OMA> <OMV name="a"/> </OMA> </OMBIND> </OMOBJ> </FMP> </CDDefinition> <CDDefinition> <Name> sum </Name> <Role>application</Role> <Description> An operator taking two arguments, the first being the range of summation, e.g. an integral interval, the second being the function to be summed. Note that the sum may be over an infinite interval. </Description> <Example> This represents the summation of the reciprocals of all the integers between 1 and 10 inclusive. <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="arith1" name="sum"/> <OMA> <OMS cd="interval1" name="integer_interval"/> <OMI> 1 </OMI> <OMI> 10 </OMI> </OMA> <OMBIND> <OMS cd="fns1" name="lambda"/> <OMBVAR> <OMV name="x"/> </OMBVAR> <OMA> <OMS cd="arith1" name="divide"/> <OMI> 1 </OMI> <OMV name="x"/> </OMA> </OMBIND> </OMA> </OMOBJ> </Example> </CDDefinition> <CDDefinition> <Name> product </Name> <Role>application</Role> <Description> An operator taking two arguments, the first being the range of multiplication e.g. an integral interval, the second being the function to be multiplied. Note that the product may be over an infinite interval. </Description> <Example> This represents the statement that the factorial of n is equal to the product of all the integers between 1 and n inclusive. <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="integer1" name="factorial"/> <OMV name="n"/> </OMA> <OMA> <OMS cd="arith1" name="product"/> <OMA> <OMS cd="interval1" name="integer_interval"/> <OMI> 1 </OMI> <OMV name="n"/> </OMA> <OMBIND> <OMS cd="fns1" name="lambda"/> <OMBVAR> <OMV name="i"/> </OMBVAR> <OMV name="i"/> </OMBIND> </OMA> </OMA> </OMOBJ> </Example> </CDDefinition> </CD>
<CDSignatures xmlns="http://www.openmath.org/OpenMathCDS" type="sts" cd="arith1"> <CDSComment> Date: 1999-11-26 Author: David Carlisle </CDSComment> <Signature name="lcm"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMA> <OMS name="nassoc" cd="sts"/> <OMV name="SemiGroup"/> </OMA> <OMV name="SemiGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="gcd"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMA> <OMS name="nassoc" cd="sts"/> <OMV name="SemiGroup"/> </OMA> <OMV name="SemiGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="plus"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMA> <OMS name="nassoc" cd="sts"/> <OMV name="AbelianSemiGroup"/> </OMA> <OMV name="AbelianSemiGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="unary_minus"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMV name="AbelianGroup"/> <OMV name="AbelianGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="minus"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMV name="AbelianGroup"/> <OMV name="AbelianGroup"/> <OMV name="AbelianGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="times"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMA> <OMS name="nassoc" cd="sts"/> <OMV name="SemiGroup"/> </OMA> <OMV name="SemiGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="divide"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMV name="AbelianGroup"/> <OMV name="AbelianGroup"/> <OMV name="AbelianGroup"/> </OMA> </OMOBJ> </Signature> <Signature name="power"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> </OMA> </OMOBJ> </Signature> <Signature name="abs"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMS name="C" cd="setname1"/> <OMS name="R" cd="setname1"/> </OMA> </OMOBJ> </Signature> <Signature name="root"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> <OMS name="NumericalValue" cd="sts"/> </OMA> </OMOBJ> </Signature> <Signature name="sum"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMV name="IntegerRange"/> <OMA> <OMS name="mapsto" cd="sts"/> <OMS name="Z" cd="setname1"/> <OMV name="AbelianMonoid"/> </OMA> <OMV name="AbelianMonoid"/> </OMA> </OMOBJ> </Signature> <Signature name="product"> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMA> <OMS name="mapsto" cd="sts"/> <OMV name="IntegerRange"/> <OMA> <OMS name="mapsto" cd="sts"/> <OMS name="Z" cd="setname1"/> <OMV name="AbelianMonoid"/> </OMA> <OMV name="AbelianMonoid"/> </OMA> </OMOBJ> </Signature> </CDSignatures>
<CDGroup xmlns="http://www.openmath.org/OpenMathCDG"> <CDGroupName>mathml</CDGroupName> <CDGroupVersion> 2 </CDGroupVersion> <CDGroupRevision> 0 </CDGroupRevision> <CDGroupURL> http://www.openmath.org/cdfiles/cdgroups/mathml.cdg</CDGroupURL> <CDGroupDescription> MathML compatibility CD Group </CDGroupDescription> <CDComment>Algebra</CDComment> <CDGroupMember> <CDName>alg1</CDName> <CDURL>http://www.openmath.org/cd/alg1.ocd</CDURL> </CDGroupMember> <CDComment>Arithmetic</CDComment> <CDGroupMember> <CDName>arith1</CDName> <CDURL>http://www.openmath.org/cd/arith1.ocd</CDURL> </CDGroupMember> <CDComment>Constructor for Floating Point Numbers</CDComment> <CDGroupMember> <CDName>bigfloat1</CDName> <CDURL>http://www.openmath.org/cd/bigfloat1.ocd</CDURL> </CDGroupMember> <CDComment>Calculus</CDComment> <CDGroupMember> <CDName>calculus1</CDName> <CDURL>http://www.openmath.org/cd/calculus1.ocd</CDURL> </CDGroupMember> <CDComment>Operations on and constructors for complex numbers</CDComment> <CDGroupMember> <CDName>complex1</CDName> <CDURL>http://www.openmath.org/cd/complex1.ocd</CDURL> </CDGroupMember> <CDComment>Functions on functions</CDComment> <CDGroupMember> <CDName>fns1</CDName> <CDURL>http://www.openmath.org/cd/fns1.ocd</CDURL> </CDGroupMember> <CDComment>Integer arithmetic</CDComment> <CDGroupMember> <CDName>integer1</CDName> <CDURL>http://www.openmath.org/cd/integer1.ocd</CDURL> </CDGroupMember> <CDComment>Intervals</CDComment> <CDGroupMember> <CDName>interval1</CDName> <CDURL>http://www.openmath.org/cd/interval1.ocd</CDURL> </CDGroupMember> <CDComment>Linear Algebra - vector & matrix constructors, those symbols which are independent of orientation, but in MathML</CDComment> <CDGroupMember> <CDName>linalg1</CDName> <CDURL>http://www.openmath.org/cd/linalg1.ocd</CDURL> </CDGroupMember> <CDComment>Linear Algebra - vector & matrix constructors, those symbols which are dependent of orientation, and in MathML</CDComment> <CDGroupMember> <CDName>linalg2</CDName> <CDURL>http://www.openmath.org/cd/linalg2.ocd</CDURL> </CDGroupMember> <CDComment>Limits of unary functions</CDComment> <CDGroupMember> <CDName>limit1</CDName> <CDURL>http://www.openmath.org/cd/limit1.ocd</CDURL> </CDGroupMember> <CDComment>List constructors</CDComment> <CDGroupMember> <CDName>list1</CDName> <CDURL>http://www.openmath.org/cd/list1.ocd</CDURL> </CDGroupMember> <CDComment>Basic logical operators</CDComment> <CDGroupMember> <CDName>logic1</CDName> <CDURL>http://www.openmath.org/cd/logic1.ocd</CDURL> </CDGroupMember> <CDComment> MathML Numerical Types </CDComment> <CDGroupMember> <CDName>mathmltypes</CDName> <CDURL>http://www.openmath.org/cd/mathmltypes.ocd</CDURL> </CDGroupMember> <CDComment>Minima and maxima</CDComment> <CDGroupMember> <CDName>minmax1</CDName> <CDURL>http://www.openmath.org/cd/minmax1.ocd</CDURL> </CDGroupMember> <CDComment>Multset-theoretic operators and constructors</CDComment> <CDGroupMember> <CDName>multiset1</CDName> <CDURL>http://www.openmath.org/cd/multiset1.ocd</CDURL> </CDGroupMember> <CDComment>Symbols for creating numbers, including some defined constants (which can be seen as nullary constructors)</CDComment> <CDGroupMember> <CDName>nums1</CDName> <CDURL>http://www.openmath.org/cd/nums1.ocd</CDURL> </CDGroupMember> <CDComment>Symbols for creating piecewise definitions</CDComment> <CDGroupMember> <CDName>piece1</CDName> <CDURL>http://www.openmath.org/cd/piece1.ocd</CDURL> </CDGroupMember> <CDComment>The basic quantifiers forall and exists.</CDComment> <CDGroupMember> <CDName>quant1</CDName> <CDURL>http://www.openmath.org/cd/quant1.ocd</CDURL> </CDGroupMember> <CDComment>Common arithmetic relations</CDComment> <CDGroupMember> <CDName>relation1</CDName> <CDURL>http://www.openmath.org/cd/relation1.ocd</CDURL> </CDGroupMember> <CDComment>Number sets</CDComment> <CDGroupMember> <CDName>setname1</CDName> <CDURL>http://www.openmath.org/cd/setname1.ocd</CDURL> </CDGroupMember> <CDComment>Rounding</CDComment> <CDGroupMember> <CDName>rounding1</CDName> <CDURL>http://www.openmath.org/cd/rounding1.ocd</CDURL> </CDGroupMember> <CDComment>Set-theoretic operators and constructors</CDComment> <CDGroupMember> <CDName>set1</CDName> <CDURL>http://www.openmath.org/cd/set1.ocd</CDURL> </CDGroupMember> <CDComment>Basic data orientated statistical operators</CDComment> <CDGroupMember> <CDName>s_data1</CDName> <CDURL>http://www.openmath.org/cd/s_data1.ocd</CDURL> </CDGroupMember> <CDComment>Basic random variable orientated statistical operators</CDComment> <CDGroupMember> <CDName>s_dist1</CDName> <CDURL>http://www.openmath.org/cd/s_dist1.ocd</CDURL> </CDGroupMember> <CDComment>Basic transcendental functions</CDComment> <CDGroupMember> <CDName>transc1</CDName> <CDURL>http://www.openmath.org/cd/transc1.ocd</CDURL> </CDGroupMember> <CDComment>Vector calculus functions</CDComment> <CDGroupMember> <CDName>veccalc1</CDName> <CDURL>http://www.openmath.org/cd/veccalc1.ocd</CDURL> </CDGroupMember> <CDComment>Alternative encoding symbols for compatibility with the MathML Semantic mapping constructs.</CDComment> <CDGroupMember> <CDName>altenc</CDName> <CDURL>http://www.openmath.org/cd/altenc.ocd</CDURL> </CDGroupMember> </CDGroup>
<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> error </CDName> <CDBase>http://www.openmath.org/cd</CDBase> <CDURL> http://www.openmath.org/cd/error.ocd </CDURL> <CDReviewDate>2006-03-30</CDReviewDate> <CDStatus> official </CDStatus> <CDDate>2004-03-30</CDDate> <CDVersion>3</CDVersion> <CDRevision>0</CDRevision> <CDDefinition> <Name> unhandled_symbol </Name> <Role>error</Role> <Description> This symbol represents the error which is raised when an application reads a symbol which is present in the mentioned content dictionary, but which it has not implemented. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unhandled_symbol and the unhandled symbol as in the example below. </Description> <Example> The application does not implement the Complex numbers: <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OME> <OMS cd="error" name="unhandled_symbol"/> <OMS cd="setname1" name="C"/> </OME> </OMOBJ> </Example> </CDDefinition> <CDDefinition> <Name> unexpected_symbol </Name> <Role>error</Role> <Description> This symbol represents the error which is raised when an application reads a symbol which is not present in the mentioned content dictionary. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unexpected_symbol and the unexpected symbol as in the example below. </Description> <Example> The application received a mistyped symbol <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OME> <OMS cd="error" name="unexpected_symbol"/> <OMS cd="arith1" name="plurse"/> </OME> </OMOBJ> </Example> </CDDefinition> <CDDefinition> <Name> unsupported_CD </Name> <Role>error</Role> <Description> This symbol represents the error which is raised when an application reads a symbol where the mentioned content dictionary is not present. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unsupported_CD and the symbol from the unsupported Content Dictionary as in the example below. </Description> <Example> The application does not know about the CD specfun1 <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"> <OME> <OMS cd="error" name="unsupported_CD"/> <OMS cd="specfun1" name="BesselJ"/> </OME> </OMOBJ> </Example> </CDDefinition> </CD>