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# 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.
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.
society at http://www.openmath.org.
```

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):
$x\in \mathrm{kernel}\left(f\right)⇒f\left(x\right)=0$
Signatures:
sts

 [Next: apply_to_list] [Last: right_compose] [Top]

## 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.
$\forall n.\mathrm{apply_to_list}\left(+,\mathrm{map}\left(1,n,\mathrm{Id}\right)\right)=\frac{n\left(n+1\right)}{2}$
Example:
One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 }
$\mathrm{list}\left({x}^{2}|x\in 10\right)$
Signatures:
sts

 [Next: predicate_on_list] [Previous: kernel] [Top]

## 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.
$a
Formal Mathematical property (FMP):
$\mathrm{predicate_on_list}\left(p,\mathrm{nil}\right)=T$
Formal Mathematical property (FMP):
$\mathrm{predicate_on_list}\left(p,\mathrm{append}\left(a,\mathrm{nil}\right)\right)=T$
Formal Mathematical property (FMP):
$\mathrm{predicate_on_list}\left(p,\mathrm{append}\left(a,\mathrm{append}\left(b,l\right)\right)\right)=p\left(a,b\right)\wedge \mathrm{predicate_on_list}\left(p,\mathrm{append}\left(b,l\right)\right)$
Signatures:
sts

 [Next: right_compose] [Previous: apply_to_list] [Top]

## 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):
$\forall f,g,x.\left(\mathrm{right_compose}\left(f,g\right)\right)\left(x\right)=g\left(f\left(x\right)\right)$
Signatures:
sts

 [First: kernel] [Previous: predicate_on_list] [Top]

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