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# OpenMath Content Dictionary: integer1

Canonical URL:
http://www.openmath.org/cd/integer1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
integer1.ocd
CD as XML Encoded OpenMath:
integer1.omcd
Defines:
factorial, quotient, remainder, factorof
Date:
2004-03-30
Version:
3
Review Date:
2006-03-30
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 a collection of basic integer functions.

This CD is intended to be `compatible' with the corresponding elements in Content MathML.

## factorof

Role:
application
Description:

This is the binary OpenMath operator that is used to indicate the mathematical relationship a "is a factor of" b, where a is the first argument and b is the second. This relationship is true if and only if b mod a = 0.

Commented Mathematical property (CMP):
b is a factor of a iff remainder of a divided by b = 0
Formal Mathematical property (FMP):
$\mathrm{factorof}\left(b,a\right)⇒\mathrm{remainder}\left(a,b\right)=0$
Signatures:
sts

 [Next: factorial] [Last: remainder] [Top]

## factorial

Role:
application
Description:

The symbol to represent a unary factorial function on non-negative integers.

Commented Mathematical property (CMP):
factorial n = product [1..n]
Formal Mathematical property (FMP):
$n!=\prod _{i=1}^{n}i$
Signatures:
sts

 [Next: quotient] [Previous: factorof] [Top]

## quotient

Role:
application
Description:

The symbol to represent the integer (binary) division operator. That is, for integers a and b, quotient(a,b) denotes q such that a=b*q+r, with |r| less than |b| and a*r positive.

Commented Mathematical property (CMP):
for all a,b with a,b Integers | a = b * quotient(a,b) + remainder(a,b) and abs(remainder(a,b)) is less than abs(b) and a*remainder(a,b) >= 0
Formal Mathematical property (FMP):
$\forall a,b.a\in \mathbb{Z}\wedge b\in \mathbb{Z}⇒a=b\mathrm{quotient}\left(a,b\right)+\mathrm{remainder}\left(a,b\right)\wedge |\mathrm{remainder}\left(a,b\right)|<|b|\wedge a\mathrm{remainder}\left(a,b\right)\ge 0$
Signatures:
sts

 [Next: remainder] [Previous: factorial] [Top]

## remainder

Role:
application
Description:

The symbol to represent the integer remainder after (binary) division. For integers a and b, remainder(a,b) denotes r such that a=b*q+r, with |r| less than |b| and a*r positive.

Commented Mathematical property (CMP):
for all a,b with a,b Integers | a = b * quotient(a,b) + remainder(a,b) and abs(remainder(a,b)) is less than abs(b) and a*remainder(a,b) >= 0
Formal Mathematical property (FMP):
$\forall a,b.a\in \mathbb{Z}\wedge b\in \mathbb{Z}⇒a=b\mathrm{quotient}\left(a,b\right)+\mathrm{remainder}\left(a,b\right)\wedge |\mathrm{remainder}\left(a,b\right)|<|b|\wedge a\mathrm{remainder}\left(a,b\right)\ge 0$
Signatures:
sts

 [First: factorof] [Previous: quotient] [Top]

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