OpenMath Content Dictionary: fns2

Canonical URL:

http://www.openmath.org/cd/fns2.ocd

CD Base:

http://www.openmath.org/cd

CD File:

fns2.ocd

CD as XML Encoded OpenMath:

fns2.omcd

Defines:

apply_to_list, kernel, right_compose

Date:

2004-03-30

Version:

3

Review Date:

2006-03-30

Status:

official

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          work to the original source.
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          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
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          Dictionary Group whose name is, for example, `math' containing
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This CD holds further functions concerning functions themselves. A particularly interesting function is

apply_to_list

which applies an nary function to all the elements in a specified list. For example, such a function can be used to form sums and products in conjunction with plus and times respectively.

kernel

Role:

application

Description:

This symbol denotes the kernel of a given function. This may be defined as the subset of the range of the given function which maps to the identity element of the image of the given function, however no semantics are assumed. The kernel of a function is also known as the null space of the function.

Commented Mathematical property (CMP):

x in the kernal of f implies that f(x) = 0

Formal Mathematical property (FMP):

xml prefix mathml

<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"/>
      <OMA>
        <OMS cd="fns2" name="kernel"/>
        <OMV name="f"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMV name="f"/>
	<OMV name="x"/>
      </OMA>
      <OMS cd="alg1" name="zero"/>
    </OMA>
  </OMA>
</OMOBJ>

implies (in ( x, kernel ( f) ) , eq ( f ( x) , zero) )

$$x\in \mathrm{kernel}\left(f\right)\Rightarrow f\left(x\right)=0$$

Signatures:

sts

apply_to_list

Role:

application

Description:

This symbol is used to denote the repeated application of an n-ary function on the elements of a given list. For example when used with plus or times this can represent sums and products.

The symbol takes two arguments; the first of which is the n-ary function, the second a list.

Example:

For all n 1 + 2 + ... + n = n(n+1)/2.

xml prefix mathml

<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="n"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="fns2" name="apply_to_list"/>
        <OMS cd="arith1" name="plus"/>
        <OMA>
          <OMS cd="list1" name="make_list"/>
          <OMI> 1 </OMI>
          <OMV name="n"/>
          <OMS cd="fns1" name="identity"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
        <OMA>
  	<OMS cd="arith1" name="times"/>
          <OMV name="n"/>
          <OMA>
            <OMS cd="arith1" name="plus"/>
            <OMV name="n"/>
            <OMI> 1 </OMI>
          </OMA>
        </OMA>
        <OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>

forall [ n ] . (eq (apply_to_list (plus, make_list ( 1 , n, identity) ) , divide (times ( n, plus ( n, 1 ) ) , 2 ) ) )

$$\forall \hspace{0.17em}n.\mathrm{apply\_to\_list}(+,\mathrm{make\_list}(1,n,\mathrm{Id}))=\frac{n(n+1)}{2}$$

Example:

One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 }

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="fns2" name="apply_to_list"/>
    <OMS cd="set1" name="set"/>
    <OMA>
      <OMS cd="list1" name="make_list"/>
      <OMI> 1 </OMI>
      <OMI> 10 </OMI>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="x"/>
        </OMBVAR>
        <OMA>
         <OMS cd="arith1" name="power"/>
          <OMV name="x"/>
          <OMI> 2 </OMI>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

apply_to_list (set, make_list ( 1 , 10 , lambda [ x ] . (power ( x, 2 ) ) ) )

$$\mathrm{make\_list}(1,10,\lambda \hspace{0.17em}x.x^2)$$

Signatures:

sts

right_compose

Role:

application

Description:

This symbol represents a function forming the right-composition of its two functional arguments.

Commented Mathematical property (CMP):

right_compose(f,g)(x) = g(f(x))

Formal Mathematical property (FMP):

xml prefix mathml

<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="f"/>
        <OMV name="g"/>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMA>
            <OMS cd="fns2" name="right_compose"/>
            <OMV name="f"/>
            <OMV name="g"/>
          </OMA>
          <OMV name="x"/>
        </OMA>
        <OMA>
          <OMV name="g"/>
          <OMA>
            <OMV name="f"/>
            <OMV name="x"/>
          </OMA>
        </OMA>
      </OMA>
    </OMBIND>
  </OMOBJ>

forall [ f g x ] . (eq (right_compose ( f, g) ( x) , g ( f ( x) ) ) )

$$\forall \hspace{0.17em}f,g,x.\left(\mathrm{right\_compose}(f,g)\right)\left(x\right)=g\left(f\left(x\right)\right)$$

Signatures:

sts