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Showing content from https://openmath.org/cd/fns2 below:

fns2

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, predicate_on_list, right_compose
Date:
2009-04-19
Version:
4
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 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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="fns2">kernel</csymbol><ci>f</ci></apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><ci>f</ci><ci>x</ci></apply>
   <csymbol cd="alg1">zero</csymbol>
  </apply>
 </apply>
</math>

set1.in($x, fns2.kernel($f)) ==> $f($x) = alg1.zero

x ∈ kernel ⁡ ( f ) ⇒ f ⁡ ( x ) = 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. 
<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="map"/>
          <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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>n</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="fns2">apply_to_list</csymbol>
    <csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="list1">map</csymbol>
     <cn type="integer">1</cn>
     <ci>n</ci>
     <csymbol cd="fns1">identity</csymbol>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">divide</csymbol>
    <apply><csymbol cd="arith1">times</csymbol>
     <ci>n</ci>
     <apply><csymbol cd="arith1">plus</csymbol><ci>n</ci><cn type="integer">1</cn></apply>
    </apply>
    <cn type="integer">2</cn>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$n -> fns2.apply_to_list(arith1.plus, list1.map(1, $n, fns1.identity)) = ($n * ($n + 1)) / 2]

∀   n . apply_to_list ⁡ ( + , map ⁡ ( 1 , n , Id ) ) = n ⁢ ( n + 1 ) 2

Example:
One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 } 
<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="map"/>
      <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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="fns2">apply_to_list</csymbol>
  <csymbol cd="set1">set</csymbol>
  <apply><csymbol cd="list1">map</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">10</cn>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn type="integer">2</cn></apply>
   </bind>
  </apply>
 </apply>
</math>

fns2.apply_to_list(set1.set, list1.map(1, 10, fns1.lambda[$x -> $x ^ 2]))

Signatures:
sts
predicate_on_list
Role:
application
Description:

This symbol is used to denote the chains of application or a binary predicate typified by a < b < c. In particular it is used to support the usage in MathML, where transative relations are classed as nary, but the underlying OpenMath symbols are binary.

The symbol takes two arguments; the first of which is the binary predicate, the second a list. It is true if every application of the predicate on a pair of elements of the list, taken in order, returns true, otherwise it is false.

Example:
a < b < c. 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
  <OMS cd="fns2" name="predicate_on_list"/>
  <OMS cd="relation1" name="lt"/>
  <OMA>
    <OMS cd="list1" name="list"/>
    <OMV name="a"/>
    <OMV name="b"/>
    <OMV name="c"/>
  </OMA>
 </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="fns2">predicate_on_list</csymbol>
  <csymbol cd="relation1">lt</csymbol>
  <apply><csymbol cd="list1">list</csymbol><ci>a</ci><ci>b</ci><ci>c</ci></apply>
 </apply>
</math>

fns2.predicate_on_list(relation1.lt, [$a , $b , $c])

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="fns2" name="predicate_on_list"/>
      <OMV name="p"/>
      <OMS cd="list2" name="nil"/>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns2">predicate_on_list</csymbol><ci>p</ci><csymbol cd="list2">nil</csymbol></apply>
  <csymbol cd="logic1">true</csymbol>
 </apply>
</math>

fns2.predicate_on_list($p, list2.nil) = logic1.true

predicate_on_list ⁡ ( p , nil ) = T

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="fns2" name="predicate_on_list"/>
      <OMV name="p"/>
      <OMA>
        <OMS cd="list2" name="append"/>
        <OMV name="a"/>
        <OMS cd="list2" name="nil"/>
      </OMA>
    </OMA>
    <OMS cd="logic1" name="true"/>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns2">predicate_on_list</csymbol>
   <ci>p</ci>
   <apply><csymbol cd="list2">append</csymbol><ci>a</ci><csymbol cd="list2">nil</csymbol></apply>
  </apply>
  <csymbol cd="logic1">true</csymbol>
 </apply>
</math>

fns2.predicate_on_list($p, list2.append($a, list2.nil)) = logic1.true

predicate_on_list ⁡ ( p , append ⁡ ( a , nil ) ) = T

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="fns2" name="predicate_on_list"/>
      <OMV name="p"/>
      <OMA>
        <OMS cd="list2" name="append"/>
        <OMV name="a"/>
        <OMA>
          <OMS cd="list2" name="append"/>
          <OMV name="b"/>
          <OMV name="l"/>
        </OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMV name="p"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
      <OMA>
        <OMS cd="fns2" name="predicate_on_list"/>
        <OMV name="p"/>
        <OMA>
          <OMS cd="list2" name="append"/>
          <OMV name="b"/>
          <OMV name="l"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns2">predicate_on_list</csymbol>
   <ci>p</ci>
   <apply><csymbol cd="list2">append</csymbol>
    <ci>a</ci>
    <apply><csymbol cd="list2">append</csymbol><ci>b</ci><ci>l</ci></apply>
   </apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><ci>p</ci><ci>a</ci><ci>b</ci></apply>
   <apply><csymbol cd="fns2">predicate_on_list</csymbol>
    <ci>p</ci>
    <apply><csymbol cd="list2">append</csymbol><ci>b</ci><ci>l</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

fns2.predicate_on_list($p, list2.append($a, list2.append($b, $l))) = ($p($a, $b) and fns2.predicate_on_list($p, list2.append($b, $l)))

predicate_on_list ⁡ ( p , append ⁡ ( a , append ⁡ ( b , l ) ) ) = p ⁡ ( a , b ) ∧ predicate_on_list ⁡ ( p , append ⁡ ( b , l ) )

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): 
  <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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>f</ci></bvar>
  <bvar><ci>g</ci></bvar>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <apply><csymbol cd="fns2">right_compose</csymbol><ci>f</ci><ci>g</ci></apply>
    <ci>x</ci>
   </apply>
   <apply><ci>g</ci><apply><ci>f</ci><ci>x</ci></apply></apply>
  </apply>
 </bind>
</math>

quant1.forall[$f, $g, $x -> fns2.right_compose($f, $g)($x) = $g($f($x))]

∀   f , g , x . ( right_compose ⁡ ( f , g ) ) ⁡ ( x ) = g ⁡ ( f ⁡ ( x ) )

Signatures:
sts

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