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Showing content from https://openmath.org/cd/calculus1 below:

calculus1

OpenMath Content Dictionary: calculus1

Canonical URL:: http://www.openmath.org/cd/calculus1.ocd
CD Base:: http://www.openmath.org/cd
CD File:: calculus1.ocd
CD as XML Encoded OpenMath:: calculus1.omcd
Defines:: defint, diff, int, nthdiff, partialdiff, partialdiffdegree
Date:: 2009-04-01
Version:: 5
Review Date:: 2014-04-01
Status:: official

     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.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

This CD is intended to be compatible with the calculus operations in Content MathML.

Integration is just for the univariate case and is either definite or indefinite.

diff

Role:: application

Description:: This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function.

Commented Mathematical property (CMP):: diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x)

Formal Mathematical property (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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply>
   <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>y</ci></bvar>
     <apply><csymbol cd="arith1">plus</csymbol>
      <apply><ci>a</ci><ci>y</ci></apply>
      <apply><ci>b</ci><ci>y</ci></apply>
     </apply>
    </bind>
   </apply>
   <ci>x</ci>
  </apply>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply>
    <apply><csymbol cd="calculus1">diff</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>y</ci></bvar>
      <apply><ci>a</ci><ci>y</ci></apply>
     </bind>
    </apply>
    <ci>x</ci>
   </apply>
   <apply>
    <apply><csymbol cd="calculus1">diff</csymbol>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>y</ci></bvar>
      <apply><ci>b</ci><ci>y</ci></apply>
     </bind>
    </apply>
    <ci>x</ci>
   </apply>
  </apply>
 </apply>
</math>

calculus1.diff(fns1.lambda[$y -> $a($y) + $b($y)])($x) = calculus1.diff(fns1.lambda[$y -> $a($y)])($x) + calculus1.diff(fns1.lambda[$y -> $b($y)])($x)

( d d y ( a ⁡ ( y ) + b ⁡ ( y ) ) ) ⁡ ( x ) = ( d d y ( a ⁡ ( y ) ) ) ⁡ ( x ) + ( d d y ( b ⁡ ( y ) ) ) ⁡ ( x )

Commented Mathematical property (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)

Formal Mathematical property (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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply>
   <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>y</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><ci>a</ci><ci>y</ci></apply>
      <apply><ci>b</ci><ci>y</ci></apply>
     </apply>
    </bind>
   </apply>
   <ci>x</ci>
  </apply>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>y</ci></bvar>
       <apply><ci>a</ci><ci>y</ci></apply>
      </bind>
     </apply>
     <ci>x</ci>
    </apply>
    <apply><ci>b</ci><ci>x</ci></apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><ci>a</ci><ci>x</ci></apply>
    <apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>y</ci></bvar>
       <apply><ci>b</ci><ci>y</ci></apply>
      </bind>
     </apply>
     <ci>x</ci>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

calculus1.diff(fns1.lambda[$y -> $a($y) * $b($y)])($x) = calculus1.diff(fns1.lambda[$y -> $a($y)])($x) * $b($x) + $a($x) * calculus1.diff(fns1.lambda[$y -> $b($y)])($x)

( d d y ( a ⁡ ( y ) ⁢ b ⁡ ( y ) ) ) ⁡ ( x ) = ( d d y ( a ⁡ ( y ) ) ) ⁡ ( x ) ⁢ b ⁡ ( x ) + a ⁡ ( x ) ⁢ ( d d y ( b ⁡ ( y ) ) ) ⁡ ( x )

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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply>
   <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><cn type="real">1.0</cn></apply>
    </bind>
   </apply>
   <ci>y</ci>
  </apply>
  <cn type="real">1.0</cn>
 </apply>
</math>

calculus1.diff(fns1.lambda[$x -> $x + 1.0])($y) = 1.0

( d d x ( x + 1.0 ) ) ⁡ ( y ) = 1.0

Signatures:: sts

nthdiff

Role:: application

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.

Formal Mathematical property (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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="calculus1">nthdiff</csymbol><csymbol cd="alg1">zero</csymbol><ci>f</ci></apply>
  <ci>f</ci>
 </apply>
</math>

calculus1.nthdiff(alg1.zero, $f) = $f

Formal Mathematical property (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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="calculus1">nthdiff</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol><ci>n</ci><csymbol cd="alg1">one</csymbol></apply>
   <ci>f</ci>
  </apply>
  <apply><csymbol cd="calculus1">diff</csymbol>
   <apply><csymbol cd="calculus1">nthdiff</csymbol><ci>n</ci><ci>f</ci></apply>
  </apply>
 </apply>
</math>

calculus1.nthdiff($n + alg1.one, $f) = calculus1.diff(calculus1.nthdiff($n, $f))

D n + 1 ( f ) = D ( D n ( f ) )

Signatures:: sts

partialdiff

Role:: application

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.

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="x"/>
    <OMV name="y"/>
    <OMV name="z"/>
  </OMA>
  <OMV name="y"/>
</OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply>
   <apply><csymbol cd="calculus1">partialdiff</csymbol>
    <apply><csymbol cd="list1">list</csymbol>
     <cn type="integer">1</cn>
     <cn type="integer">3</cn>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <bvar><ci>y</ci></bvar>
     <bvar><ci>z</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol><ci>x</ci><ci>y</ci><ci>z</ci></apply>
    </bind>
   </apply>
   <ci>x</ci>
   <ci>y</ci>
   <ci>z</ci>
  </apply>
  <ci>y</ci>
 </apply>
</math>

calculus1.partialdiff([1 , 3], fns1.lambda[$x, $y, $z -> $x * $y * $z])($x, $y, $z) = $y

( ∂ 2 ∂ x ∂ z ( x ⁢ y ⁢ z ) ) ⁡ ( x , y , z ) = y

Signatures:: sts

partialdiffdegree

Role:: application

Description:: This symbol is used to express partial differentiation of a function of more than one variable. It has three arguments, the first is a list of integers which give the degrees by which the function is differentiated by the corresponding variable. The second is the total degree (which should therefore be the sum of the values in the first list, but may be given symbolically). The third is the function.

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="partialdiffdegree"/>
      <OMA>
        <OMS cd="list1" name="list"/>
        <OMI>1</OMI>
        <OMI>0</OMI>
        <OMI>1</OMI>
      </OMA>
      <OMI>2</OMI>
      <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="x"/>
    <OMV name="y"/>
    <OMV name="z"/>
  </OMA>
  <OMV name="y"/>
</OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply>
   <apply><csymbol cd="calculus1">partialdiffdegree</csymbol>
    <apply><csymbol cd="list1">list</csymbol>
     <cn type="integer">1</cn>
     <cn type="integer">0</cn>
     <cn type="integer">1</cn>
    </apply>
    <cn type="integer">2</cn>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <bvar><ci>y</ci></bvar>
     <bvar><ci>z</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol><ci>x</ci><ci>y</ci><ci>z</ci></apply>
    </bind>
   </apply>
   <ci>x</ci>
   <ci>y</ci>
   <ci>z</ci>
  </apply>
  <ci>y</ci>
 </apply>
</math>

calculus1.partialdiffdegree([1 , 0 , 1], 2, fns1.lambda[$x, $y, $z -> $x * $y * $z])($x, $y, $z) = $y

∂ 2 ∂ x ∂ z ( x ⁢ y ⁢ z ) = y

Signatures:: sts

int

Role:: application

Description:: This symbol is used to represent indefinite integration of unary functions. The argument is the unary function.

Commented Mathematical property (CMP):: application of integrate followed by differentiate is the identity function, that is: diff(lambda y:integral(lambda z:f(z))(y)) = f

Formal Mathematical property (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="calculus1" name="diff"/>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="y"/>
            </OMBVAR>
            <OMA>
              <OMA>
                <OMS cd="calculus1" name="int"/>
                <OMBIND>
                  <OMS cd="fns1" name="lambda"/>
                  <OMBVAR>
                    <OMV name="z"/>
                  </OMBVAR>
                  <OMA>
                    <OMV name="f"/>
                    <OMV name="z"/>
                  </OMA>
                </OMBIND>
              </OMA>
              <OMV name="y"/>
            </OMA>
          </OMBIND>
        </OMA>
        <OMV name="f"/>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="calculus1">diff</csymbol>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>y</ci></bvar>
    <apply>
     <apply><csymbol cd="calculus1">int</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><ci>f</ci><ci>z</ci></apply>
      </bind>
     </apply>
     <ci>y</ci>
    </apply>
   </bind>
  </apply>
  <ci>f</ci>
 </apply>
</math>

calculus1.diff(fns1.lambda[$y -> calculus1.int(fns1.lambda[$z -> $f($z)])($y)]) = $f

d d y ( ( ∫ f ⁡ ( z ) d z ) ⁡ ( y ) ) = f

Example:

An example which represents the equation: integral(x +-> sin(x)) w.r.t. x = x +-> -cos(x)

<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="calculus1" name="int"/>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMS cd="transc1" name="sin"/>
          <OMV name="x"/>
        </OMA>
      </OMBIND>
    </OMA>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
        <OMA>
          <OMS cd="transc1" name="cos"/>
          <OMV name="x"/>
        </OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="calculus1">int</csymbol>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
   </bind>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="arith1">unary_minus</csymbol>
    <apply><csymbol cd="transc1">cos</csymbol><ci>x</ci></apply>
   </apply>
  </bind>
 </apply>
</math>

calculus1.int(fns1.lambda[$x -> sin($x)]) = fns1.lambda[$x -> -(cos($x))]

∫ sin ⁡ ( x ) d x = λ x . - cos ⁡ ( x )

Signatures:: sts

defint

Role:: application

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.

Commented Mathematical property (CMP):: for all a,b | integral from a to b = -integral from b to a

Formal Mathematical property (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="ordered_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="ordered_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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="calculus1">defint</csymbol>
    <apply><csymbol cd="interval1">ordered_interval</csymbol><ci>a</ci><ci>b</ci></apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <apply><ci>f</ci><ci>x</ci></apply>
    </bind>
   </apply>
   <apply><csymbol cd="arith1">unary_minus</csymbol>
    <apply><csymbol cd="calculus1">defint</csymbol>
     <apply><csymbol cd="interval1">ordered_interval</csymbol><ci>b</ci><ci>a</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><ci>f</ci><ci>x</ci></apply>
     </bind>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$a, $b -> calculus1.defint(interval1.ordered_interval($a, $b), fns1.lambda[$x -> $f($x)]) = -(calculus1.defint(interval1.ordered_interval($b, $a), fns1.lambda[$x -> $f($x)]))]

∀ a , b . ∫ a b f ⁡ ( x ) d x = - ∫ b a f ⁡ ( x ) d x

Commented Mathematical property (CMP):: for all a < b < c | integral over [a,c] = integral over [a,b] + integral over [b,c]

Formal Mathematical property (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="ordered_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="ordered_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="ordered_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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <bvar><ci>c</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="relation1">lt</csymbol><ci>a</ci><ci>b</ci></apply>
    <apply><csymbol cd="relation1">lt</csymbol><ci>b</ci><ci>c</ci></apply>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="calculus1">defint</csymbol>
     <apply><csymbol cd="interval1">ordered_interval</csymbol><ci>a</ci><ci>c</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><ci>f</ci><ci>x</ci></apply>
     </bind>
    </apply>
    <apply><csymbol cd="arith1">plus</csymbol>
     <apply><csymbol cd="calculus1">defint</csymbol>
      <apply><csymbol cd="interval1">ordered_interval</csymbol><ci>a</ci><ci>b</ci></apply>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>x</ci></bvar>
       <apply><ci>f</ci><ci>x</ci></apply>
      </bind>
     </apply>
     <apply><csymbol cd="calculus1">defint</csymbol>
      <apply><csymbol cd="interval1">ordered_interval</csymbol><ci>b</ci><ci>c</ci></apply>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>x</ci></bvar>
       <apply><ci>f</ci><ci>x</ci></apply>
      </bind>
     </apply>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

forall

[

a b c

] . (

implies

(

and

(

) ,

(

) ) ,

(

defint

(

ordered_interval

(

) ,

lambda

[

] . (

(

) ) ) ,

plus

(

defint

(

ordered_interval

(

) ,

lambda

[

] . (

(

) ) ) ,

defint

(

ordered_interval

(

) ,

lambda

[

] . (

(

) ) ) ) ) ) )

quant1.forall[$a, $b, $c -> $a < $b and $b < $c ==> calculus1.defint(interval1.ordered_interval($a, $c), fns1.lambda[$x -> $f($x)]) = calculus1.defint(interval1.ordered_interval($a, $b), fns1.lambda[$x -> $f($x)]) + calculus1.defint(interval1.ordered_interval($b, $c), fns1.lambda[$x -> $f($x)])]

∀ a , b , c . a < b ∧ b < c ⇒ ∫ a c f ⁡ ( x ) d x = ∫ a b f ⁡ ( x ) d x + ∫ b c f ⁡ ( x ) d x

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="ordered_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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="calculus1">defint</csymbol>
  <apply><csymbol cd="interval1">ordered_interval</csymbol>
   <cn type="real">-1.0</cn>
   <cn type="real">1.0</cn>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
  </bind>
 </apply>
</math>

calculus1.defint(interval1.ordered_interval(-1.0, 1.0), fns1.lambda[$x -> sin($x)])

∫ -1.0 1.0 sin ⁡ ( x ) d x

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>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="calculus1">defint</csymbol>
  <ci>C</ci>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><ci>f</ci><ci>x</ci></apply>
  </bind>
 </apply>
</math>

calculus1.defint($C, fns1.lambda[$x -> $f($x)])

Signatures:: sts

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