alg1.ocd

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

<CDComment>

     This document is distributed in the hope that it will be useful, 
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          this document, and maintains a prominent reference in the 
          work to the original source.
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          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
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          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
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</CDComment>

<CDName>alg1</CDName>
<CDBase>http://www.openmath.org/cd</CDBase>
<CDURL>http://www.openmath.org/cd/alg1.ocd</CDURL>
<CDReviewDate>2006-03-30</CDReviewDate>
<CDStatus>official</CDStatus>
<CDDate>2004-03-30</CDDate>
<CDVersion>3</CDVersion>
<CDRevision>1</CDRevision>
<CDComment>
  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
</CDComment>

<Description> 
A CD of basic algebraic concepts

At present this CD only holds definitions of zero and one. They are
deliberately defined here without specifying any particular structure
(e.g. a group) to which they correspond.
</Description>


<CDDefinition>
<Name>zero</Name>
<Role>constant</Role>
<Description>
This symbol represents the additive identity element.
</Description>
<CMP> for all a | a + 0 = 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"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMV name="a"/>
        <OMS cd="alg1" name="zero"/>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>
<CMP> for all a | 0 * 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="times"/>
        <OMS cd="alg1" name="zero"/>
        <OMV name="a"/>
      </OMA>
      <OMS cd="alg1" name="zero"/>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>
<CMP>The zero of the integers is 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="x"/>
    </OMBVAR>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="logic1" name="and"/>
        <OMA>
          <OMS cd="set1" name="in"/>
          <OMV name="x"/>
          <OMS cd="setname1" name="Z"/>
        </OMA>
        <OMA>
          <OMS cd="relation1" name="eq"/>
          <OMV name="x"/>
          <OMS cd="alg1" name="zero"/>
        </OMA>
      </OMA>
      <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="x"/>
      <OMI>0</OMI>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>
</CDDefinition>

<CDDefinition>
<Name>one</Name>
<Role>constant</Role>
<Description>
This symbol represents the multiplicative identity element.
</Description>

<CMP> for all a | 1 * a = 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"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMS cd="alg1" name="one"/>
        <OMV name="a"/>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>

<CMP> for all a | a * 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"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMV name="a"/>
        <OMS cd="alg1" name="one"/>
      </OMA>
      <OMV name="a"/>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>
<CMP>The one of the integers is 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="x"/>
    </OMBVAR>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="logic1" name="and"/>
        <OMA>
          <OMS cd="set1" name="in"/>
          <OMV name="x"/>
          <OMS cd="setname1" name="Z"/>
        </OMA>
        <OMA>
          <OMS cd="relation1" name="eq"/>
          <OMV name="x"/>
          <OMS cd="alg1" name="one"/>
        </OMA>
      </OMA>
      <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="x"/>
      <OMI>1</OMI>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>
</FMP>
</CDDefinition>

</CD>