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Showing content from https://openmath.org/cd/interval1 below:

interval1

OpenMath Content Dictionary: interval1

Canonical URL:: http://www.openmath.org/cd/interval1.ocd
CD Base:: http://www.openmath.org/cd
CD File:: interval1.ocd
CD as XML Encoded OpenMath:: interval1.omcd
Defines:: integer_interval, interval, interval_cc, interval_co, interval_oc, interval_oo, oriented_interval
Date:: 2009-04-01
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 symbols which describe both discrete and continuous 1-dimensional intervals (with open/closed end points). There is also an oriented_interval, for use in integration

integer_interval

Role:: application

Description:: A symbol to denote a discrete 1 dimensional interval from the first argument to the second (inclusive), where the discretisation occurs at unit intervals. The arguments are the start and the end points of the interval in that order.

Example:

The integer interval 1, 2, ..., 10.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="integer_interval"/>
    <OMI>1</OMI>
    <OMI>10</OMI>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">integer_interval</csymbol>
  <cn type="integer">1</cn>
  <cn type="integer">10</cn>
 </apply>
</math>

interval1.integer_interval(1, 10)

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="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="n"/>
      <OMA>
        <OMS name="integer_interval" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="n"/>
        <OMS cd="setname1" name="Z"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="a"/>
        <OMV name="n"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="n"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>n</ci>
   <apply><csymbol cd="interval1">integer_interval</csymbol><ci>a</ci><ci>b</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>n</ci><csymbol cd="setname1">Z</csymbol></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>a</ci><ci>n</ci></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>n</ci><ci>b</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(set1.in($n, interval1.integer_interval($a, $b)), set1.in($n, setname1.Z) and relation1.le($a, $n) and relation1.le($n, $b))

n ∈ [ a , b ] ≡ ( n ∈ Z ∧ le ⁡ ( a , n ) ∧ le ⁡ ( n , b ) )

Signatures:: sts

interval

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the interval in that order.

Example:

The interval 1.0, ..., 10.0.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="interval"/>
    <OMF dec="1.0"/>
    <OMF dec="10.0"/>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">interval</csymbol>
  <cn type="real">1.0</cn>
  <cn type="real">10.0</cn>
 </apply>
</math>

interval1.interval(1.0, 10.0)

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="subset"/>
    <OMA>
      <OMS name="interval" cd="interval1"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS name="interval_cc" cd="interval1"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">subset</csymbol>
  <apply><csymbol cd="interval1">interval</csymbol><ci>a</ci><ci>b</ci></apply>
  <apply><csymbol cd="interval1">interval_cc</csymbol><ci>a</ci><ci>b</ci></apply>
 </apply>
</math>

set1.subset(interval1.interval($a, $b), interval1.interval_cc($a, $b))

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="subset"/>
    <OMA>
      <OMS name="interval_oo" cd="interval1"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMS name="interval" cd="interval1"/>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">subset</csymbol>
  <apply><csymbol cd="interval1">interval_oo</csymbol><ci>a</ci><ci>b</ci></apply>
  <apply><csymbol cd="interval1">interval</csymbol><ci>a</ci><ci>b</ci></apply>
 </apply>
</math>

set1.subset(interval1.interval_oo($a, $b), interval1.interval($a, $b))

Signatures:: sts

oriented_interval

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the integration, in either order.

Example:

<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="defintint"/>
      <OMA>
        <OMS name="oriented_interval" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
      <OMV name="f"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMA>
        <OMS cd="calculus1" name="defintint"/>
        <OMA>
          <OMS name="oriented_interval" cd="interval1"/>
          <OMV name="b"/>
          <OMV name="a"/>
        </OMA>
        <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">defintint</csymbol>
   <apply><csymbol cd="interval1">oriented_interval</csymbol><ci>a</ci><ci>b</ci></apply>
   <ci>f</ci>
  </apply>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="calculus1">defintint</csymbol>
    <apply><csymbol cd="interval1">oriented_interval</csymbol><ci>b</ci><ci>a</ci></apply>
    <ci>f</ci>
   </apply>
  </apply>
 </apply>
</math>

calculus1.defintint(interval1.oriented_interval($a, $b), $f) = calculus1.defintint(interval1.oriented_interval($b, $a), $f)

defintint ⁡ ( oriented_interval ⁡ ( a , b ) , f ) = defintint ⁡ ( oriented_interval ⁡ ( b , a ) , f )

Signatures:: sts

interval_oo

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval with both end points excluded from the interval. The arguments are the start and the end points of the interval in that order.

Example:

The continuous open interval (1,10).

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="interval_oo"/>
    <OMI>1</OMI>
    <OMI>10</OMI>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">interval_oo</csymbol>
  <cn type="integer">1</cn>
  <cn type="integer">10</cn>
 </apply>
</math>

interval1.interval_oo(1, 10)

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="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS name="interval_oo" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
        <OMV name="a"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
        <OMV name="x"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_oo</csymbol><ci>a</ci><ci>b</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>x</ci><csymbol cd="setname1">R</csymbol></apply>
   <apply><csymbol cd="relation1">lt</csymbol><ci>a</ci><ci>x</ci></apply>
   <apply><csymbol cd="relation1">lt</csymbol><ci>x</ci><ci>b</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(set1.in($x, interval1.interval_oo($a, $b)), set1.in($x, setname1.R) and $a < $x and $x < $b)

x ∈ ( a , b ) ≡ ( x ∈ R ∧ a < x ∧ x < b )

Signatures:: sts

interval_cc

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval with both end points included in the interval. The arguments are the start and the end points of the interval in that order.

Example:

The continuous closed interval [1,10].

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="interval_cc"/>
    <OMI>1</OMI>
    <OMI>10</OMI>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">interval_cc</csymbol>
  <cn type="integer">1</cn>
  <cn type="integer">10</cn>
 </apply>
</math>

interval1.interval_cc(1, 10)

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="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS name="interval_cc" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="a"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="x"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_cc</csymbol><ci>a</ci><ci>b</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>x</ci><csymbol cd="setname1">R</csymbol></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>a</ci><ci>x</ci></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>x</ci><ci>b</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(set1.in($x, interval1.interval_cc($a, $b)), set1.in($x, setname1.R) and relation1.le($a, $x) and relation1.le($x, $b))

x ∈ [ a , b ] ≡ ( x ∈ R ∧ le ⁡ ( a , x ) ∧ le ⁡ ( x , b ) )

Signatures:: sts

interval_oc

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval with the first point excluded from the interval, but the last included. The arguments are the start and the end points of the interval in that order.

Example:

The continuous interval open at the lower bound and closed at the higher bound (1,10].

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="interval_oc"/>
    <OMI>1</OMI>
    <OMI>10</OMI>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">interval_oc</csymbol>
  <cn type="integer">1</cn>
  <cn type="integer">10</cn>
 </apply>
</math>

interval1.interval_oc(1, 10)

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="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS name="interval_oc" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
        <OMV name="a"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="x"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_oc</csymbol><ci>a</ci><ci>b</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>x</ci><csymbol cd="setname1">R</csymbol></apply>
   <apply><csymbol cd="relation1">lt</csymbol><ci>a</ci><ci>x</ci></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>x</ci><ci>b</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(set1.in($x, interval1.interval_oc($a, $b)), set1.in($x, setname1.R) and $a < $x and relation1.le($x, $b))

x ∈ ( a , b ] ≡ ( x ∈ R ∧ a < x ∧ le ⁡ ( x , b ) )

Signatures:: sts

interval_co

Role:: application

Description:: A symbol to denote a continuous 1-dimensional interval with the first point included in the interval, but the last excluded. The arguments are the start and the end points of the interval in that order.

Example:

The continuous interval closed at the lower bound and open at the higher bound [1,10).

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="interval1" name="interval_co"/>
    <OMI>1</OMI>
    <OMI>10</OMI>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="interval1">interval_co</csymbol>
  <cn type="integer">1</cn>
  <cn type="integer">10</cn>
 </apply>
</math>

interval1.interval_co(1, 10)

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="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS name="interval_co" cd="interval1"/>
        <OMV name="a"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="le"/>
        <OMV name="a"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
        <OMV name="x"/>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="interval1">interval_co</csymbol><ci>a</ci><ci>b</ci></apply>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>x</ci><csymbol cd="setname1">R</csymbol></apply>
   <apply><csymbol cd="relation1">le</csymbol><ci>a</ci><ci>x</ci></apply>
   <apply><csymbol cd="relation1">lt</csymbol><ci>x</ci><ci>b</ci></apply>
  </apply>
 </apply>
</math>

logic1.equivalent(set1.in($x, interval1.interval_co($a, $b)), set1.in($x, setname1.R) and relation1.le($a, $x) and $x < $b)

x ∈ [ a , b ) ≡ ( x ∈ R ∧ le ⁡ ( a , x ) ∧ x < b )

Signatures:: sts

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