Examples from MathML4

Content Markup

Introduction

The Intent of Content Markup

The Structure and Scope of Content MathML Expressions

Strict Content MathML

Content Dictionaries

Content MathML Concepts

Content MathML Elements Encoding Expression Structure

Numbers `<cn>`

Rendering `<cn>`,`<sep/>`-Represented Numbers

Strict uses of `<cn>`

                        
<cn type="hexdouble">7F800000</cn>

Non-Strict uses of `<cn>`

                        <cn base="16">7FE0</cn>

                        <cn base="1000">10F</cn>

                        <cn type="rational">22<sep/>7</cn>

                        
<cn type="complex-cartesian"> 12.3 <sep/> 5 </cn>

                        
<cn type="complex-polar"> 2 <sep/> 3.1415 </cn>

Rewrite: cn sep

                  
<cn type="rational" base="b">n<sep/>d</cn>

                  
<apply><csymbol cd="nums1">rational</csymbol>
  <cn type="integer" base="b">n</cn>
  <cn type="integer" base="b">d</cn>
</apply>

Rewrite: cn based_integer

                  
<cn type="integer" base="16">FF60</cn>

                  
<apply><csymbol cd="nums1">based_integer</csymbol>
  <cn type="integer">16</cn>
  <cs>FF60</cs>
</apply>

Rewrite: cn constant

                     <cn type="constant">c</cn>

                     <csymbol cd="nums1">c2</csymbol>

Rewrite: cn presentation mathml

                     
<cn type="rational"><mi>P</mi><sep/><mi>Q</mi></cn>

                     
<apply><csymbol cd="nums1">rational</csymbol>
 <semantics>
  <ci>p</ci>
  <annotation-xml encoding="MathML-Presentation">
   <mi>P</mi>
  </annotation-xml>
 </semantics>
 <semantics>
  <ci>q</ci>
  <annotation-xml encoding="MathML-Presentation">
   <mi>Q</mi>
  </annotation-xml>
 </semantics>
</apply>

Content Identifiers `<ci>`

               <ci>x</ci>

Strict uses of `<ci>`

                  <ci type="integer">n</ci>

                  
<semantics>
  <ci>n</ci>
  <annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content">
    <csymbol cd="mathmltypes">integer_type</csymbol>
  </annotation-xml>
</semantics>

Non-Strict uses of `<ci>`

Rewrite: ci type annotation

                     <ci type="T">n</ci>

                     
<semantics>
  <ci>n</ci>
  <annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content">
    <ci>T</ci>
  </annotation-xml>
</semantics>

Rewrite: ci presentation mathml

                     <ci><mi>P</mi></ci>

                     
<semantics>
  <ci>p</ci>
  <annotation-xml encoding="MathML-Presentation">
    <mi>P</mi>
  </annotation-xml>
</semantics>

                  
<ci>
  <msup><mi>C</mi><mn>2</mn></msup>
</ci>

                  
<semantics>
  <ci>C2</ci>
  <annotation-xml encoding="MathML-Presentation">
    <msup><mi>C</mi><mn>2</mn></msup>
  </annotation-xml>
</semantics>

Rendering Content Identifiers

                  <ci type="vector">V</ci>

Content Symbols `<csymbol>`

Strict uses of `<csymbol>`

Non-Strict uses of `<csymbol>`

Rewrite: csymbol type annotation

                     <csymbol type="T">symbolname</csymbol>

                     
<semantics>
  <csymbol>symbolname</csymbol>
  <annotation-xml cd="mathmltypes" name="type" encoding="MathML-Content">
    <ci>T</ci>
  </annotation-xml>
</semantics>

Rendering Symbols

String Literals `<cs>`

               
<set>
  <cs>A</cs><cs>B</cs><cs>  </cs>
</set>

Function Application `<apply>`

Strict Content MathML

                  <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply>

                  
<apply><csymbol cd="arith1">plus</csymbol>
  <ci>x</ci>
  <ci>y</ci>
  <ci>z</ci>
</apply>

                  
<apply><csymbol cd="arith1">plus</csymbol>
  <apply><csymbol cd="arith1">times</csymbol>
    <ci>a</ci>
    <ci>x</ci>
  </apply>
  <ci>b</ci>
</apply>

                  
<apply><csymbol cd="arith1">times</csymbol>
  <apply><csymbol cd="arith1">plus</csymbol>
    <ci>F</ci>
    <ci>G</ci>
  </apply>
  <ci>x</ci>
</apply>

                  <apply><csymbol cd="arith1">plus</csymbol><ci>F</ci><ci>G</ci></apply>

                  
<apply>
  <apply><csymbol cd="arith1">plus</csymbol>
    <ci>F</ci>
    <ci>G</ci>
  </apply>
  <ci>x</ci>
</apply>

Rendering Applications

                  
<apply><ci>f</ci>
  <ci>a1</ci>
  <ci>a2</ci>
  <ci>...</ci>
  <ci>an</ci>
</apply>

                  
<apply><ci>op</ci>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>d</ci></domainofapplication>
  <ci>expression-in-x</ci>
</apply>

Bindings and Bound Variables `<bind>` and `<bvar>`

Bindings

Bound Variables

                  
<bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci id="var-x">x</ci></bvar>
  <apply><csymbol cd="relation1">lt</csymbol>
    <ci xref="var-x">x</ci>
    <cn>1</cn>
  </apply>
</bind>

               
<bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply>
    <apply><csymbol cd="arith1">plus</csymbol><ci>y</ci><ci>x</ci></apply>
  </apply>
</bind>

Renaming Bound Variables

Rendering Binding Constructions

                  
<bind><ci>b</ci>
  <bvar><ci>x1</ci></bvar>
  <bvar><ci>...</ci></bvar>
  <bvar><ci>xn</ci></bvar>
  <ci>s</ci>
</bind>

Structure Sharing `<share>`

The `share` element

                              
<apply><ci>f</ci>
  <apply><ci>f</ci>
    <apply><ci>f</ci>
      <ci>a</ci>
      <ci>a</ci>
    </apply>
    <apply><ci>f</ci>
      <ci>a</ci>
      <ci>a</ci>
    </apply>
  </apply>
  <apply><ci>f</ci>
    <apply><ci>f</ci>
      <ci>a</ci>
      <ci>a</ci>
    </apply>
    <apply><ci>f</ci>
      <ci>a</ci>
      <ci>a</ci>
    </apply>
  </apply>
</apply>

                              
<apply><ci>f</ci>
  <apply id="t1"><ci>f</ci>
    <apply id="t11"><ci>f</ci>
      <ci>a</ci>
      <ci>a</ci>
    </apply>
    <share src="#t11"/>



  </apply>
  <share src="#t1"/>









</apply>

An Acyclicity Constraint

Structure Sharing and Binding

                  
<bind id="outer"><csymbol cd="fns1">lambda</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><ci>f</ci>
    <bind id="inner"><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <share id="copy" src="#orig"/>
    </bind>
    <apply id="orig"><ci>g</ci><ci>x</ci></apply>
  </apply>
</bind>

Rendering Expressions with Structure Sharing

Attribution via `semantics`

Error Markup `<cerror>`

               
<cerror>
  <csymbol cd="aritherror">DivisionByZero</csymbol>
  <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
</cerror>

               
<apply><csymbol cd="relation1">eq</csymbol>
  <cerror>
    <csymbol cd="aritherror">DivisionByZero</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
  </cerror>
  <cn>0</cn>
</apply>

               
<cerror>
  <csymbol cd="aritherror">DivisionByZero</csymbol>
  <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
</cerror>

               
<cerror>
  <csymbol cd="error">Illegal bound variable</csymbol>
  <cs> &lt;bvar&gt;&lt;plus/&gt;&lt;/bvar&gt; </cs>
</cerror>

Encoded Bytes `<cbytes>`

Content MathML for Specific Structures

Container Markup

Container Markup for Constructor Symbols

                     <set><ci>a</ci><ci>b</ci><ci>c</ci></set>

                     <apply><csymbol cd="set1">set</csymbol><ci>a</ci><ci>b</ci><ci>c</ci></apply>

                     
<set>
  <bvar><ci>x</ci></bvar> 
  <domainofapplication><integers/></domainofapplication>
  <apply><times/><cn>2</cn><ci>x</ci></apply>
</set>

                     
<apply><csymbol cd="set1">map</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <apply><csymbol cd="arith1">times</csymbol>
      <cn>2</cn>
      <ci>x</ci>
    </apply>
  </bind>
  <csymbol cd="setname1">Z</csymbol>
</apply>

Container Markup for Binding Constructors

                  <lambda><bvar><ci>x</ci></bvar><ci>x</ci></lambda>

                  
<bind><csymbol cd="fns1">lambda</csymbol>
 <bvar><ci>x</ci></bvar><ci>x</ci>
</bind>

Bindings with `<apply>`

                  
<apply><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><geq/><ci>x</ci><ci>x</ci></apply>
</apply>

                  
<bind><csymbol cd="logic1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">geq</csymbol><ci>x</ci><ci>x</ci></apply>
</bind>

                  
<apply><sum/>
  <bvar><ci>i</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>100</cn></uplimit>
  <apply><power/><ci>x</ci><ci>i</ci></apply>
</apply>

                  
<apply><csymbol cd="arith1">sum</csymbol>
  <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn>0</cn>
    <cn>100</cn>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="arith1">power</csymbol>
      <ci>x</ci>
      <ci>i</ci>
    </apply>
  </bind>
</apply>

Qualifiers

Uses of `<domainofapplication>`, `<interval>`, `<condition>`, `<lowlimit>` and `<uplimit>`

                  
<apply><int/>
  <domainofapplication>
    <ci type="set">C</ci>
  </domainofapplication>
  <ci type="function">f</ci>
</apply>

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <interval><cn>0</cn><cn>1</cn></interval>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>1</cn></uplimit>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><and/>
      <apply><leq/><cn>0</cn><ci>x</ci></apply>
      <apply><leq/><ci>x</ci><cn>1</cn></apply>
    </apply>
  </condition>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <domainofapplication>
    <set>
      <bvar><ci>t</ci></bvar>
      <bvar><ci>u</ci></bvar>
      <condition>
        <apply><and/>
          <apply><leq/><cn>0</cn><ci>t</ci></apply>
          <apply><leq/><ci>t</ci><cn>1</cn></apply>
          <apply><leq/><cn>0</cn><ci>u</ci></apply>
          <apply><leq/><ci>u</ci><cn>1</cn></apply>
        </apply>
      </condition>
      <list><ci>t</ci><ci>u</ci></list>
    </set>
  </domainofapplication>
  <apply><times/>
    <apply><power/><ci>x</ci><cn>2</cn></apply>
    <apply><power/><ci>y</ci><cn>3</cn></apply>
  </apply>
</apply>

Rewrite: interval qualifier

                  
<apply><ci>H</ci>
  <bvar><ci>x</ci></bvar>
  <lowlimit><ci>a</ci></lowlimit>
  <uplimit><ci>b</ci></uplimit>
  <ci>C</ci>
</apply>

                  
<apply><ci>H</ci>
  <bvar><ci>x</ci></bvar>
  <domainofapplication>
    <apply><csymbol cd="interval1">interval</csymbol>
      <ci>a</ci>
      <ci>b</ci>
    </apply>
  </domainofapplication>
  <ci>C</ci>
</apply>

Rewrite: condition

Rewrite: restriction

                  
<apply><ci>F</ci>
  <domainofapplication>
    <ci>C</ci>
  </domainofapplication>
  <ci>a1</ci>
  <ci>an</ci>
</apply>

                  
<apply>
  <apply><csymbol cd="fns1">restriction</csymbol>
    <ci>F</ci>
    <ci>C</ci>
  </apply>
  <ci>a1</ci>
  <ci>an</ci>
</apply>

Rewrite: apply bvar domainofapplication

Uses of `<degree>`

               
<apply><diff/>
  <bvar>
    <ci>x</ci>
    <degree><cn>2</cn></degree>
  </bvar>
  <apply><power/><ci>x</ci><cn>4</cn></apply>
</apply>

Uses of `<momentabout>` and `<logbase>`

Operator Classes

Rewrite: element

               <plus/>

                  <csymbol cd="arith1">plus</csymbol>

N-ary Operators (classes nary-arith, nary-functional, nary-logical, nary-linalg, nary-set, nary-constructor)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary domainofapplication

                        
<apply><union/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x</ci>
</apply>

                        
<apply><csymbol cd="fns2">apply_to_list</csymbol>
  <csymbol cd="set1">union</csymbol>
  <apply><csymbol cd="list1">map</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <ci>expression-in-x</ci>
    </bind>
    <ci>D</ci>
  </apply>
</apply>

N-ary Constructors for set and list (class nary-setlist-constructor)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary setlist domainofapplication

                        
<set>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x</ci>
</set>

                        
<apply><csymbol cd="set1">map</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <ci>expression-in-x</ci>
  </bind>
  <ci>D</ci>
</apply>

N-ary Relations (classes nary-reln, nary-set-reln)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary relations

                        
<apply><lt/>
  <ci>a</ci><ci>b</ci><ci>c</ci><ci>d</ci>
</apply>

                        
<apply><csymbol cd="fns2">predicate_on_list</csymbol>
 <csymbol cd="reln1">lt</csymbol>
 <apply><csymbol cd="list1">list</csymbol>
  <ci>a</ci><ci>b</ci><ci>c</ci><ci>d</ci>
 </apply>
</apply>

Rewrite: n-ary relations bvar

                        
<apply><lt/>
 <bvar><ci>x</ci></bvar>
 <domainofapplication><ci>R</ci></domainofapplication>
 <ci>expression-in-x</ci>
</apply>

                        
<apply><csymbol cd="fns2">predicate_on_list</csymbol>
 <csymbol cd="reln1">lt</csymbol>
 <apply><csymbol cd="list1">map</csymbol>
   <ci>R</ci>
   <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <ci>expression-in-x</ci>
   </bind>
  </apply>
</apply>

N-ary/Unary Operators (classes nary-minmax, nary-stats)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary unary set

                     
<apply><max/><ci>a1</ci><ci>a2</ci><ci>an</ci></apply>

                     
<apply><csymbol cd="minmax1">max</csymbol>
  <apply><csymbol cd="set1">set</csymbol>
    <ci>a1</ci><ci>a2</ci><ci>an</ci>
  </apply>
</apply>

Rewrite: n-ary unary domainofapplication

                        
<apply><max/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x</ci>
</apply>

                        
<apply><csymbol cd="minmax1">max</csymbol>
  <apply><csymbol cd="set1">map</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <ci>expression-in-x</ci>
    </bind>
    <ci>D</ci>
  </apply>
</apply>

Rewrite: n-ary unary single

                     
<apply><max/><ci>a</ci></apply>

                     
<apply><csymbol cd="minmax1">max</csymbol> <ci>a</ci> </apply>

Binary Operators (classes binary-arith, binary-logical, binary-reln, binary-linalg, binary-set)

Schema Patterns

Unary Operators (classes unary-arith, unary-linalg, unary-functional, unary-set, unary-elementary, unary-veccalc)

Schema Patterns

Constants (classes constant-arith, constant-set)

Schema Patterns

Quantifiers (class quantifier)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: quantifier

                        
<apply><exists/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x</ci>
</apply>

                        
<bind><csymbol cd="quant1">exists</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="set1">in</csymbol><ci>x</ci><ci>D</ci></apply>
  <ci>expression-in-x</ci>
  </apply>
</bind>

Other Operators (classes lambda, interval, int, diff partialdiff, sum, product, limit)

Schema Patterns

Non-strict Attributes

Rewrite: attributes

                  
<ci class="foo" xmlns:other="http://example.com" other:att="bla">x</ci>

                  
<semantics>
  <ci>x</ci>
  <annotation cd="mathmlattr"
     name="class" encoding="text/plain">foo</annotation>
  <annotation-xml cd="mathmlattr" name="foreign" encoding="MathML-Content">
    <apply><csymbol cd="mathmlattr">foreign_attribute</csymbol>
      <cs>http://example.com</cs>
      <cs>other</cs>
      <cs>att</cs>
      <cs>bla</cs>
    </apply>
  </annotation-xml>
</semantics>

Content MathML for Specific Operators and Constants

Functions and Inverses

Interval `<interval>`

                  
<interval closure="open"><ci>x</ci><cn>1</cn></interval>

                  
<interval closure="closed"><cn>0</cn><cn>1</cn></interval>

                  
<interval closure="open-closed"><cn>0</cn><cn>1</cn></interval>

                  
<interval closure="closed-open"><cn>0</cn><cn>1</cn></interval>

Inverse `<inverse>`

                  
<apply><inverse/>
  <ci> f </ci>
</apply>

                  
<apply>
  <apply><inverse/><ci type="matrix">A</ci></apply>
  <ci>a</ci>
</apply>

Lambda `<lambda>`

                  
<lambda>
  <bvar><ci> x </ci></bvar>
  <domainofapplication><integers/></domainofapplication>
  <apply><sin/><ci> x </ci></apply>
</lambda>

                  
<lambda>
  <domainofapplication><integers/></domainofapplication> 
  <sin/>
</lambda>

                  
<lambda>
  <bvar><ci>x</ci></bvar>
  <apply><sin/>
    <apply><plus/><ci>x</ci><cn>1</cn></apply>
  </apply>
</lambda>

Rewrite: lambda

                  
<lambda>
  <bvar><ci>x1</ci></bvar><bvar><ci>xn</ci></bvar>
  <ci>expression-in-x1-xn</ci>
</lambda>

                  
<bind><csymbol cd="fns1">lambda</csymbol>
  <bvar><ci>x1</ci></bvar><bvar><ci>xn</ci></bvar>
  <ci>expression-in-x1-xn</ci>
</bind>

Rewrite: lambda domainofapplication

                  
<lambda>
  <bvar><ci>x1</ci></bvar><bvar><ci>xn</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x1-xn</ci>
</lambda>

                  
<apply><csymbol cd="fns1">restriction</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x1</ci></bvar><bvar><ci>xn</ci></bvar>
    <ci>expression-in-x1-xn</ci>
  </bind>
  <ci>D</ci>
</apply>

Function composition `<compose/>`

                  
<apply><compose/><ci>f</ci><ci>g</ci><ci>h</ci></apply>

                  
<apply><eq/>
  <apply>
    <apply><compose/><ci>f</ci><ci>g</ci></apply>
    <ci>x</ci>
  </apply>
  <apply><ci>f</ci><apply><ci>g</ci><ci>x</ci></apply></apply>
</apply>

Identity function `<ident/>`

                  
<apply><eq/>
  <apply><compose/>
    <ci type="function">f</ci>
    <apply><inverse/>
      <ci type="function">f</ci>
    </apply>
  </apply>
  <ident/>
</apply>

Domain `<domain/>`

                  
<apply><eq/>
  <apply><domain/><ci>f</ci></apply>
  <reals/>
</apply>

codomain `<codomain/>`

                  
<apply><eq/>
  <apply><codomain/><ci>f</ci></apply>
  <rationals/>
</apply>

Image `<image/>`

                  
<apply><eq/>
  <apply><image/><sin/></apply>
  <interval><cn>-1</cn><cn> 1</cn></interval>
</apply>

Piecewise declaration `<piecewise>`, `<piece>`, `<otherwise>`

                  
<piecewise>
  <piece>
    <apply><minus/><ci>x</ci></apply>
    <apply><lt/><ci>x</ci><cn>0</cn></apply>
  </piece>
  <piece>
    <cn>0</cn>
    <apply><eq/><ci>x</ci><cn>0</cn></apply>
  </piece>
  <piece>
    <ci>x</ci>
    <apply><gt/><ci>x</ci><cn>0</cn></apply>
  </piece>
</piecewise>

                  
<piecewise>
  <piece>
    <cn>0</cn>
    <apply><lt/><ci>x</ci><cn>0</cn></apply>
  </piece>
  <piece>
    <cn>1</cn>
    <apply><gt/><ci>x</ci><cn>1</cn></apply>
  </piece>
  <otherwise>
    <ci>x</ci>
  </otherwise>
</piecewise>

                  
<apply><csymbol cd="piece1">piecewise</csymbol>
  <apply><csymbol cd="piece1">piece</csymbol>
    <cn>0</cn>
    <apply><csymbol cd="relation1">lt</csymbol><ci>x</ci><cn>0</cn></apply>  
  </apply>   
  <apply><csymbol cd="piece1">piece</csymbol>
    <cn>1</cn>
    <apply><csymbol cd="relation1">gt</csymbol><ci>x</ci><cn>1</cn></apply>  
  </apply>   
  <apply><csymbol cd="piece1">otherwise</csymbol>
    <ci>x</ci>
  </apply>   
</apply>

Arithmetic, Algebra and Logic

Quotient `<quotient/>`

                  
<apply><quotient/><ci>a</ci><ci>b</ci></apply>

Factorial `<factorial/>`

                  
<apply><factorial/><ci>n</ci></apply>

Division `<divide/>`

                  
<apply><divide/>
  <ci>a</ci>
  <ci>b</ci>
</apply>

Maximum `<max/>`

                  
<apply><max/><cn>2</cn><cn>3</cn><cn>5</cn></apply>

                  
<apply><max/>
  <bvar><ci>y</ci></bvar>
  <condition>
    <apply><in/>
      <ci>y</ci>
      <interval><cn>0</cn><cn>1</cn></interval>
    </apply>
  </condition>
  <apply><power/><ci>y</ci><cn>3</cn></apply>
</apply>

Minimum `<min/>`

                  
<apply><min/><ci>a</ci><ci>b</ci></apply>

                  
<apply><min/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><notin/><ci>x</ci><ci type="set">B</ci></apply>
  </condition>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>

Subtraction `<minus/>`

                  
<apply><minus/><cn>3</cn></apply>

                  
<apply><minus/><ci>x</ci><ci>y</ci></apply>

Addition `<plus/>`

                  
<apply><plus/><ci>x</ci><ci>y</ci><ci>z</ci></apply>

Exponentiation `<power/>`

                  
<apply><power/><ci>x</ci><cn>3</cn></apply>

Remainder `<rem/>`

                  
<apply><rem/><ci> a </ci><ci> b </ci></apply>

Multiplication `<times/>`

                  
<apply><times/><ci>a</ci><ci>b</ci></apply>

Root `<root/>`

                  
<apply><root/>
  <degree><ci type="integer">n</ci></degree>
  <ci>a</ci>
</apply>

                  <apply><root/><ci>x</ci></apply>

                  
<apply><csymbol cd="arith1">root</csymbol>
  <ci>x</ci>
  <cn type="integer">2</cn>
</apply>

                  
<apply><root/>
  <degree><ci type="integer">n</ci></degree>
  <ci>a</ci>
</apply>

                  
<apply><csymbol cd="arith1">root</csymbol>
  <ci>a</ci>
  <cn type="integer">n</cn>
</apply>

Greatest common divisor `<gcd/>`

                  
<apply><gcd/><ci>a</ci><ci>b</ci><ci>c</ci></apply>

And `<and/>`

                  
<apply><and/><ci>a</ci><ci>b</ci></apply>

                  
<apply><and/>
  <bvar><ci>i</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><ci>n</ci></uplimit>
  <apply><gt/><apply><selector/><ci>a</ci><ci>i</ci></apply><cn>0</cn></apply>
</apply>

                  
<apply><csymbol cd="fns2">apply_to_list</csymbol>
  <csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="list1">map</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>i</ci></bvar>
      <apply><csymbol cd="relation1">gt</csymbol>
        <apply><csymbol cd="linalg1">vector_selector</csymbol>
          <ci>i</ci>
          <ci>a</ci>
        </apply>
        <cn>0</cn>
      </apply>
    </bind>
    <apply><csymbol cd="interval1">integer_interval</csymbol>
      <cn type="integer">0</cn>
      <ci>n</ci>
    </apply>
  </apply>
</apply>

Or `<or/>`

                  
<apply><or/><ci>a</ci><ci>b</ci></apply>

Exclusive Or `<xor/>`

                  
<apply><xor/><ci>a</ci><ci>b</ci></apply>

Not `<not/>`

                  
<apply><not/><ci>a</ci></apply>

Implies `<implies/>`

                  
<apply><implies/><ci>A</ci><ci>B</ci></apply>

Universal quantifier `<forall/>`

                  
<bind><forall/>
  <bvar><ci>x</ci></bvar>
  <apply><eq/>
    <apply><minus/><ci>x</ci><ci>x</ci></apply>
    <cn>0</cn>
  </apply>
</bind>

                     
<bind><forall/>
  <bvar><ci>p</ci></bvar>
  <bvar><ci>q</ci></bvar>
  <condition>
    <apply><and/>
      <apply><in/><ci>p</ci><rationals/></apply>
      <apply><in/><ci>q</ci><rationals/></apply>
      <apply><lt/><ci>p</ci><ci>q</ci></apply>
    </apply>
  </condition>
  <apply><lt/>
    <ci>p</ci>
    <apply><power/><ci>q</ci><cn>2</cn></apply>
  </apply>
</bind>

                     
<bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>p</ci></bvar>
  <bvar><ci>q</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
    <apply><csymbol cd="logic1">and</csymbol>
      <apply><csymbol cd="set1">in</csymbol>
        <ci>p</ci>
        <csymbol cd="setname1">Q</csymbol>
        </apply>
      <apply><csymbol cd="set1">in</csymbol>
        <ci>q</ci>
        <csymbol cd="setname1">Q</csymbol>
      </apply>
      <apply><csymbol cd="relation1">lt</csymbol><ci>p</ci><ci>q</ci></apply>
    </apply>
    <apply><csymbol cd="relation1">lt</csymbol>
      <ci>p</ci>
      <apply><csymbol cd="arith1">power</csymbol>
        <ci>q</ci>
        <cn>2</cn>
      </apply>
    </apply>
  </apply>
</bind>

Existential quantifier `<exists/>`

                  
<bind><exists/>
  <bvar><ci>x</ci></bvar>
  <apply><eq/>
    <apply><ci>f</ci><ci>x</ci></apply>
    <cn>0</cn>
  </apply>
</bind>

                  
<apply><exists/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication>
    <integers/>
  </domainofapplication>
   <apply><eq/>
    <apply><ci>f</ci><ci>x</ci></apply>
    <cn>0</cn>
  </apply>
</apply>

                  
<bind><csymbol cd="quant1">exists</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="set1">in</csymbol>
      <ci>x</ci>
      <csymbol cd="setname1">Z</csymbol>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
      <apply><ci>f</ci><ci>x</ci></apply>
      <cn>0</cn>
    </apply>
  </apply>
</bind>

Absolute Value `<abs/>`

                  
<apply><abs/><ci>x</ci></apply>

Complex conjugate `<conjugate/>`

                  
<apply><conjugate/>
  <apply><plus/>
    <ci>x</ci>
    <apply><times/><cn>&#x2148;</cn><ci>y</ci></apply>
  </apply>
</apply>

Argument `<arg/>`

                  
<apply><arg/>
  <apply><plus/>
    <ci> x </ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

Real part `<real/>`

                  
<apply><real/>
  <apply><plus/>
    <ci>x</ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

Imaginary part `<imaginary/>`

                  
<apply><imaginary/>
  <apply><plus/>
    <ci>x</ci>
    <apply><times/><imaginaryi/><ci>y</ci></apply>
  </apply>
</apply>

Lowest common multiple `<lcm/>`

                  
<apply><lcm/><ci>a</ci><ci>b</ci><ci>c</ci></apply>

Floor `<floor/>`

                  
<apply><floor/><ci>a</ci></apply>

Ceiling `<ceiling/>`

                  
<apply><ceiling/><ci>a</ci></apply>

Relations

Equals `<eq/>`

                  
<apply><eq/>
  <cn type="rational">2<sep/>4</cn>
  <cn type="rational">1<sep/>2</cn>
</apply>

Not Equals `<neq/>`

                  
<apply><neq/><cn>3</cn><cn>4</cn></apply>

Greater than `<gt/>`

                  
<apply><gt/><cn>3</cn><cn>2</cn></apply>

Less Than `<lt/>`

                  
<apply><lt/><cn>2</cn><cn>3</cn><cn>4</cn></apply>

Greater Than or Equal `<geq/>`

                  
<apply><geq/><cn>4</cn><cn>3</cn><cn>3</cn></apply>

                  
<apply><csymbol cd="fns2">predicate_on_list</csymbol>
 <csymbol cd="reln1">geq</csymbol>
 <apply><csymbol cd="list1">list</csymbol>
  <cn>4</cn><cn>3</cn><cn>3</cn>
 </apply>
</apply>

Less Than or Equal `<leq/>`

                  
<apply><leq/><cn>3</cn><cn>3</cn><cn>4</cn></apply>

Equivalent `<equivalent/>`

                  
<apply><equivalent/>
  <ci>a</ci>
  <apply><not/><apply><not/><ci>a</ci></apply></apply>
</apply>

Approximately `<approx/>`

                  
<apply><approx/>
  <pi/>
  <cn type="rational">22<sep/>7</cn>
</apply>

Factor Of `<factorof/>`

                  
<apply><factorof/><ci>a</ci><ci>b</ci></apply>

Calculus and Vector Calculus

Integral `<int/>`

                  
<apply><eq/>
  <apply><int/><sin/></apply>
  <cos/>
</apply>

                  
<apply><int/>
  <interval><ci>a</ci><ci>b</ci></interval>
  <cos/>
</apply>

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>1</cn></uplimit>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>

Rewrite: int

                     
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <ci>expression-in-x</ci>
</apply>

                     
<apply>
  <apply><csymbol cd="calculus1">int</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <ci>expression-in-x</ci>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

                     
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <apply><cos/><ci>x</ci></apply>
</apply>

                     
<apply>
  <apply><csymbol cd="calculus1">int</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><cos/><ci>x</ci></apply>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

                     
<apply><int/>
  <domainofapplication><ci>C</ci></domainofapplication>
  <ci>f</ci>
</apply>

                     
<apply><csymbol cd="calculus1">defint</csymbol><ci>C</ci><ci>f</ci></apply>

Rewrite: defint

                     
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <domainofapplication><ci>D</ci></domainofapplication>
  <ci>expression-in-x</ci>
</apply>

                     
<apply><csymbol cd="calculus1">defint</csymbol>
  <ci>D</ci>  
  <bind><csymbol cd="fns1">lambda</csymbol>
  <bvar><ci>x</ci></bvar>
  <ci>expression-in-x</ci>
  </bind>
</apply>

Rewrite: defint limits

                  
<apply><int/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><ci>a</ci></lowlimit>
  <uplimit><ci>b</ci></uplimit>
  <ci>expression-in-x</ci>
</apply>

                  
<apply><csymbol cd="calculus1">defint</csymbol>
  <apply><csymbol cd="interval1">oriented_interval</csymbol>
    <ci>a</ci> <ci>b</ci>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
  <bvar><ci>x</ci></bvar>
  <ci>expression-in-x</ci>
  </bind>
</apply>

                  
<bind><int/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <condition>
    <apply><and/>
      <apply><leq/><cn>0</cn><ci>x</ci></apply>
      <apply><leq/><ci>x</ci><cn>1</cn></apply>
      <apply><leq/><cn>0</cn><ci>y</ci></apply>
      <apply><leq/><ci>y</ci><cn>1</cn></apply>
    </apply>
  </condition>
  <apply><times/>
    <apply><power/><ci>x</ci><cn>2</cn></apply>
    <apply><power/><ci>y</ci><cn>3</cn></apply>
  </apply>
</bind>

                  
<apply><csymbol cd="calculus1">defint</csymbol>
 <apply><csymbol cd="set1">suchthat</csymbol>
  <apply><csymbol cd="set1">cartesianproduct</csymbol>
   <csymbol cd="setname1">R</csymbol>
   <csymbol cd="setname1">R</csymbol>
  </apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="arith1">leq</csymbol><cn>0</cn><ci>x</ci></apply>
   <apply><csymbol cd="arith1">leq</csymbol><ci>x</ci><cn>1</cn></apply>
   <apply><csymbol cd="arith1">leq</csymbol><cn>0</cn><ci>y</ci></apply>
   <apply><csymbol cd="arith1">leq</csymbol><ci>y</ci><cn>1</cn></apply>
  </apply>
  <bind><csymbol cd="fns11">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <bvar><ci>y</ci></bvar>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><cn>2</cn></apply>
    <apply><csymbol cd="arith1">power</csymbol><ci>y</ci><cn>3</cn></apply>
   </apply>
  </bind>
 </apply>
</apply>

Differentiation `<diff/>`

                  <apply><diff/><ci>f</ci></apply>

                  
<apply><eq/>
  <apply><diff/>
    <bvar><ci>x</ci></bvar>
    <apply><sin/><ci>x</ci></apply>
  </apply>
  <apply><cos/><ci>x</ci></apply>
</apply>

                  
<apply><diff/>
  <bvar><ci>x</ci><degree><cn>2</cn></degree></bvar>
  <apply><power/><ci>x</ci><cn>4</cn></apply>
</apply>

Rewrite: diff

                     
<apply><diff/>
  <bvar><ci>x</ci></bvar>
  <ci>expression-in-x</ci>
</apply>

                     
<apply>
  <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <ci>E</ci>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

                     
<apply><diff/>
  <bvar><ci>x</ci></bvar>
  <apply><sin/><ci>x</ci></apply>
</apply>

                     
<apply>
  <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

Rewrite: nthdiff

                  
<apply><diff/>
  <bvar><ci>x</ci><degree><ci>n</ci></degree></bvar>
  <ci>expression-in-x</ci>
</apply>

                  
<apply>
  <apply><csymbol cd="calculus1">nthdiff</csymbol>
    <ci>n</ci>
    <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <ci>expression-in-x</ci>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

                  
<apply><diff/>
  <bvar><degree><cn>2</cn></degree><ci>x</ci></bvar>
  <apply><sin/><ci>x</ci></apply>
</apply>

                  
<apply>
  <apply><csymbol cd="calculus1">nthdiff</csymbol>
    <cn>2</cn>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><csymbol cd="transc1">sin</csymbol><ci>x</ci></apply>
    </bind>
  </apply>
  <ci>x</ci>
</apply>

Partial Differentiation `<partialdiff/>`

                  
<apply><partialdiff/>
  <list><cn>1</cn><cn>1</cn><cn>3</cn></list>
  <ci type="function">f</ci>
</apply>

                  
<apply><partialdiff/>
  <list><cn>1</cn><cn>1</cn><cn>3</cn></list>
  <lambda>
   <bvar><ci>x</ci></bvar>
   <bvar><ci>y</ci></bvar>
   <bvar><ci>z</ci></bvar>
   <apply><ci>f</ci><ci>x</ci><ci>y</ci><ci>z</ci></apply>
  </lambda>
</apply>

                  
<apply><partialdiff/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <apply><ci type="function">f</ci><ci>x</ci><ci>y</ci></apply>
</apply>

                  
<apply><partialdiff/>
  <bvar><ci>x</ci><degree><ci>m</ci></degree></bvar>
  <bvar><ci>y</ci><degree><ci>n</ci></degree></bvar>
  <degree><ci>k</ci></degree>
  <apply><ci type="function">f</ci>
    <ci>x</ci>
    <ci>y</ci>
  </apply>
</apply>

Rewrite: partialdiffdegree

                     
<apply><partialdiff/>
  <bvar><ci>x1</ci><degree><ci>n1</ci></degree></bvar>
  <bvar><ci>xk</ci><degree><ci>nk</ci></degree></bvar>
  <degree><ci>total-n1-nk</ci></degree>
  <ci>expression-in-x1-xk</ci>
</apply>

                  
<apply>
  <apply><csymbol cd="calculus1">partialdiffdegree</csymbol>
    <apply><csymbol cd="list1">list</csymbol>
      <ci>n1</ci> <ci>nk</ci>
    </apply>
    <ci>total-n1-nk</ci>
    <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x1</ci></bvar>
    <bvar><ci>xk</ci></bvar>
    <ci>expression-in-x1-xk</ci>
   </bind>
  </apply>
  <ci>x1</ci>
  <ci>xk</ci>
</apply>

                     
<apply><csymbol cd="arith1">plus</csymbol>
  <ci>n1</ci> <ci>nk</ci>
</apply>

                     
<apply><partialdiff/>
  <bvar><ci>x</ci><degree><ci>n</ci></degree></bvar>
  <bvar><ci>y</ci><degree><ci>m</ci></degree></bvar>
  <apply><sin/>
    <apply><times/><ci>x</ci><ci>y</ci></apply>
  </apply>
</apply>

                     
<apply>
  <apply><csymbol cd="calculus1">partialdiffdegree</csymbol>
    <apply><csymbol cd="list1">list</csymbol>
      <ci>n</ci><ci>m</ci>
    </apply>
    <apply><csymbol cd="arith1">plus</csymbol>
      <ci>n</ci><ci>m</ci>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <bvar><ci>y</ci></bvar>
      <apply><csymbol cd="transc1">sin</csymbol>
        <apply><csymbol cd="arith1">times</csymbol>
          <ci>x</ci><ci>y</ci>
        </apply>
      </apply>
    </bind>
    <ci>x</ci>
    <ci>y</ci>
  </apply>
</apply>

Divergence `<divergence/>`

                  
<apply><divergence/><ci>a</ci></apply>

                  
<apply><divergence/>
  <ci type="vector">E</ci>
</apply>

                  
<apply><divergence/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>z</ci></bvar>
  <vector>
    <apply><plus/><ci>x</ci><ci>y</ci></apply>
    <apply><plus/><ci>x</ci><ci>z</ci></apply>
    <apply><plus/><ci>z</ci><ci>y</ci></apply>
  </vector>
</apply>

Gradient `<grad/>`

                  
<apply><grad/><ci type="function">f</ci></apply>

                  
<apply><grad/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>z</ci></bvar>
  <apply><times/><ci>x</ci><ci>y</ci><ci>z</ci></apply>
</apply>

Curl `<curl/>`

                  
<apply><curl/><ci>a</ci></apply>

Laplacian `<laplacian/>`

                  
<apply><laplacian/><ci type="vector">E</ci></apply>

                  
<apply><laplacian/>
  <bvar><ci>x</ci></bvar>
  <bvar><ci>y</ci></bvar>
  <bvar><ci>z</ci></bvar>
  <apply><ci>f</ci><ci>x</ci><ci>y</ci></apply>
</apply>

Theory of Sets

Set `<set>`

                  
<set>
  <ci>a</ci><ci>b</ci><ci>c</ci>
</set>

                  
<set>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><lt/><ci>x</ci><cn>5</cn></apply>
  </condition>
  <ci>x</ci>
</set>

                  
<set>
  <bvar><ci type="set">S</ci></bvar>
  <condition>
    <apply><in/><ci>S</ci><ci type="list">T</ci></apply>
  </condition>
  <ci>S</ci>
</set>

                  
<set>
  <bvar><ci> x </ci></bvar>
  <condition>
    <apply><and/>
      <apply><lt/><ci>x</ci><cn>5</cn></apply>
      <apply><in/><ci>x</ci><naturalnumbers/></apply>
    </apply>
  </condition>
  <ci>x</ci>
</set>

List `<list>`

                  
<list>
  <ci>a</ci><ci>b</ci><ci>c</ci>
</list>

                  
<list order="numeric">
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><lt/><ci>x</ci><cn>5</cn></apply>
  </condition>
</list>

Union `<union/>`

                  
<apply><union/><ci>A</ci><ci>B</ci></apply>

                  
<apply><union/>
  <bvar><ci type="set">S</ci></bvar>
  <domainofapplication>
    <ci type="list">L</ci>
  </domainofapplication>
  <ci type="set"> S</ci>
</apply>

Intersect `<intersect/>`

                  
<apply><intersect/>
  <ci type="set"> A </ci>
  <ci type="set"> B </ci>
</apply>

                  
<apply><intersect/>
  <bvar><ci type="set">S</ci></bvar>
  <domainofapplication><ci type="list">L</ci></domainofapplication>
  <ci type="set"> S </ci>
</apply>

Set inclusion `<in/>`

                  
<apply><in/><ci>a</ci><ci type="set">A</ci></apply>

Set exclusion `<notin/>`

                  
<apply><notin/><ci>a</ci><ci type="set">A</ci></apply>

Subset `<subset/>`

                  
<apply><subset/>
  <ci type="set">A</ci>
  <ci type="set">B</ci>
</apply>

Proper Subset `<prsubset/>`

                  
<apply><prsubset/>
  <ci type="set">A</ci>
  <ci type="set">B</ci>
</apply>

Not Subset `<notsubset/>`

                  
<apply><notsubset/>
  <ci type="set">A</ci>
  <ci type="set">B</ci>
</apply>

Not Proper Subset `<notprsubset/>`

                  
<apply><notprsubset/>
  <ci type="set">A</ci>
  <ci type="set">B</ci>
</apply>

Set Difference `<setdiff/>`

                  
<apply><setdiff/>
  <ci type="set">A</ci>
  <ci type="set">B</ci>
</apply>

Cardinality `<card/>`

                  
<apply><eq/>
  <apply><card/><ci>A</ci></apply>
  <cn>5</cn>
</apply>

Cartesian product `<cartesianproduct/>`

                  
<apply><cartesianproduct/><ci>A</ci><ci>B</ci></apply>

Sequences and Series

Sum `<sum/>`

                  
<apply><sum/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><ci>a</ci></lowlimit>
  <uplimit><ci>b</ci></uplimit>
  <apply><ci>f</ci><ci>x</ci></apply>
</apply>

                  
<apply><sum/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><in/><ci>x</ci><ci type="set">B</ci></apply>
  </condition>
  <apply><ci type="function">f</ci><ci>x</ci></apply>
</apply>

                  
<apply><sum/>
  <domainofapplication>
    <ci type="set">B</ci>
  </domainofapplication>
  <ci type="function">f</ci>
</apply>

                  
<apply><sum/>
  <bvar><ci>i</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>100</cn></uplimit>
  <apply><power/><ci>x</ci><ci>i</ci></apply>
</apply>

                  
<apply><csymbol cd="arith1">sum</csymbol>
  <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn>0</cn>
    <cn>100</cn>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><ci>i</ci></apply>
  </bind>
</apply>

Product `<product/>`

                  
<apply><product/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><ci>a</ci></lowlimit>
  <uplimit><ci>b</ci></uplimit>
  <apply><ci type="function">f</ci>
    <ci>x</ci>
  </apply>
</apply>

                  
<apply><product/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><in/>
      <ci>x</ci>
      <ci type="set">B</ci>
    </apply>
  </condition>
  <apply><ci>f</ci><ci>x</ci></apply>
</apply>

                  
<apply><product/>
  <bvar><ci>i</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <uplimit><cn>100</cn></uplimit>
  <apply><power/><ci>x</ci><ci>i</ci></apply>
</apply>

                  
<apply><csymbol cd="arith1">product</csymbol>
  <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn>0</cn>
    <cn>100</cn>
  </apply>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="arith1">power</csymbol><ci>x</ci><ci>i</ci></apply>
  </bind>
</apply>

Limits `<limit/>`

                  
<apply><limit/>
  <bvar><ci>x</ci></bvar>
  <lowlimit><cn>0</cn></lowlimit>
  <apply><sin/><ci>x</ci></apply>
</apply>

                  
<apply><limit/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><tendsto/><ci>x</ci><cn>0</cn></apply>
  </condition>
  <apply><sin/><ci>x</ci></apply>
</apply>

                  
<apply><limit/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><tendsto type="above"/><ci>x</ci><ci>a</ci></apply>
  </condition>
  <apply><sin/><ci>x</ci></apply>
</apply>

Rewrite: limits condition

                  
<apply><limit/>
  <bvar><ci>x</ci></bvar>
  <condition>
    <apply><tendsto/><ci>x</ci><cn>0</cn></apply>
  </condition>
  <ci>expression-in-x</ci>
</apply>

                  
<apply><csymbol cd="limit1">limit</csymbol>
  <cn>0</cn>
  <csymbol cd="limit1">null</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <ci>expression-in-x</ci>
  </bind>
</apply>

Tends To `<tendsto/>`

                  
<apply><tendsto type="above"/>
  <apply><power/><ci>x</ci><cn>2</cn></apply>
   <apply><power/><ci>a</ci><cn>2</cn></apply>
</apply>

                  
<apply><tendsto/>
  <vector><ci>x</ci><ci>y</ci></vector>
   <vector>
     <apply><ci type="function">f</ci><ci>x</ci><ci>y</ci></apply>
     <apply><ci type="function">g</ci><ci>x</ci><ci>y</ci></apply>
   </vector>
</apply>

Rewrite: tendsto

                  
<tendsto/>

                  
<semantics>
 <ci>tendsto</ci>
 <annotation-xml encoding="MathML-Content">
  <tendsto/>
 </annotation-xml>
</semantics>

Elementary classical functions

Common trigonometric functions `<sin/>`, `<cos/>`, `<tan/>`, `<sec/>`, `<csc/>`, `<cot/>`

                  
<apply><sin/><ci>x</ci></apply>

                  
<apply><sin/>
  <apply><plus/>
    <apply><cos/><ci>x</ci></apply>
    <apply><power/><ci>x</ci><cn>3</cn></apply>
  </apply>
</apply>

Common inverses of trigonometric functions `<arcsin/>`, `<arccos/>`, `<arctan/>`, `<arcsec/>`, `<arccsc/>`, `<arccot/>`

                  
<apply><arcsin/><ci>x</ci></apply>

                  
<mrow>
 <mi>arcsin</mi>
 <mo>&#x2061;</mo>
 <mi>x</mi>
</mrow>

                  
<mrow>
 <msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup>
 <mo>&#x2061;</mo>
 <mi>x</mi>
</mrow>

Common hyperbolic functions `<sinh/>`, `<cosh/>`, `<tanh/>`, `<sech/>`, `<csch/>`, `<coth/>`

                  
<apply><sinh/><ci>x</ci></apply>

Common inverses of hyperbolic functions `<arcsinh/>`, `<arccosh/>`, `<arctanh/>`, `<arcsech/>`, `<arccsch/>`, `<arccoth/>`

                  
<apply><arcsinh/><ci>x</ci></apply>

                  
<mrow>
 <mi>arcsinh</mi>
 <mo>&#x2061;</mo>
 <mi>x</mi>
</mrow>

                  
<mrow>
 <msup><mi>sinh</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup>
 <mo>&#x2061;</mo>
 <mi>x</mi>
</mrow>

Exponential `<exp/>`

                  
<apply><exp/><ci>x</ci></apply>

Natural Logarithm `<ln/>`

                  
<apply><ln/><ci>a</ci></apply>

Logarithm `<log/>` , `<logbase>`

                  
<apply><log/>
  <logbase><cn>3</cn></logbase>
  <ci>x</ci>
</apply>

                  
<apply><log/><ci>x</ci></apply>

                  <apply><plus/>
  <apply>
    <log/>
    <logbase><cn>2</cn></logbase>
    <ci>x</ci>
  </apply>
  <apply>
    <log/>
    <ci>y</ci>
  </apply>
</apply>

                  <apply>
  <csymbol cd="arith1">plus</csymbol>
  <apply>
    <csymbol cd="transc1">log</csymbol>
    <cn>2</cn>
    <ci>x</ci>
  </apply>
  <apply>
    <csymbol cd="transc1">log</csymbol>
    <cn>10</cn>
    <ci>y</ci>
  </apply>
</apply>

Statistics

Mean `<mean/>`

                  
<apply><mean/>
  <cn>3</cn><cn>4</cn><cn>3</cn><cn>7</cn><cn>4</cn>
</apply>

                  
<apply><mean/><ci>X</ci></apply>

Standard Deviation `<sdev/>`

                  
<apply><sdev/>
  <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn>
</apply>

                  
<apply><sdev/>
  <ci type="discrete_random_variable">X</ci>
</apply>

Variance `<variance/>`

                  
<apply><variance/>
  <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn>
</apply>

                  
<apply><variance/>
  <ci type="discrete_random_variable"> X</ci>
</apply>

Median `<median/>`

                  
<apply><median/>
  <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn>
</apply>

Mode `<mode/>`

                  
<apply><mode/>
  <cn>3</cn><cn>4</cn><cn>2</cn><cn>2</cn>
</apply>

Moment `<moment/>`, `<momentabout>`

                  
<apply><moment/>
  <degree><cn>3</cn></degree>
  <momentabout><mean/></momentabout>
  <cn>6</cn><cn>4</cn><cn>2</cn><cn>2</cn><cn>5</cn>
</apply>

                  
<apply><moment/>
  <degree><cn>3</cn></degree>
  <momentabout><ci>p</ci></momentabout>
  <ci>X</ci>
</apply>

                  
<apply><moment/>
  <degree><cn>3</cn></degree>
  <momentabout><ci>p</ci></momentabout>
  <ci>X</ci>
</apply>

                  
<apply><csymbol cd="s_dist1">moment</csymbol>
  <cn>3</cn>
  <ci>p</ci>
  <ci>X</ci>
</apply>

Linear Algebra

Vector `<vector>`

                  
<vector>
  <apply><plus/><ci>x</ci><ci>y</ci></apply>
  <cn>3</cn>
  <cn>7</cn>
</vector>

Matrix `<matrix>`

                  
<matrix>
  <bvar><ci type="integer">i</ci></bvar>
  <bvar><ci type="integer">j</ci></bvar>
  <condition>
    <apply><and/>
      <apply><in/>
        <ci>i</ci>
        <interval><ci>1</ci><ci>5</ci></interval>
      </apply>
      <apply><in/>
        <ci>j</ci>
        <interval><ci>5</ci><ci>9</ci></interval>
      </apply>
    </apply>
  </condition>
  <apply><power/><ci>i</ci><ci>j</ci></apply>
</matrix>

Matrix row `<matrixrow>`

Determinant `<determinant/>`

                  
<apply><determinant/>
  <ci type="matrix">A</ci>
</apply>

Transpose `<transpose/>`

                  
<apply><transpose/>
  <ci type="matrix">A</ci>
</apply>

Selector `<selector/>`

                  
<apply><selector/><ci type="vector">V</ci><cn>1</cn></apply>

                  
<apply><eq/>
  <apply><selector/>
    <matrix>
      <matrixrow><cn>1</cn><cn>2</cn></matrixrow>
      <matrixrow><cn>3</cn><cn>4</cn></matrixrow>
    </matrix>
    <cn>1</cn>
  </apply>
  <matrix>
    <matrixrow><cn>1</cn><cn>2</cn></matrixrow>
  </matrix>
</apply>

Vector product `<vectorproduct/>`

                  
<apply><eq/>
  <apply><vectorproduct/>
    <ci type="vector"> A </ci>
    <ci type="vector"> B </ci>
 </apply>
  <apply><times/>
    <ci>a</ci>
    <ci>b</ci>
    <apply><sin/><ci>&#x3b8;</ci></apply>
    <ci type="vector"> N </ci>
  </apply>
</apply>

Scalar product `<scalarproduct/>`

                  
<apply><eq/>
  <apply><scalarproduct/>
    <ci type="vector">A</ci>
    <ci type="vector">B</ci>
  </apply>
  <apply><times/>
    <ci>a</ci>
    <ci>b</ci>
    <apply><cos/><ci>&#x3b8;</ci></apply>
  </apply>
</apply>

Outer product `<outerproduct/>`

                  
<apply><outerproduct/>
  <ci type="vector">A</ci>
  <ci type="vector">B</ci>
</apply>

Constant and Symbol Elements

integers `<integers/>`

                  
<apply><in/>
  <cn type="integer"> 42 </cn>
  <integers/>
</apply>

reals `<reals/>`

                  
<apply><in/>
  <cn type="real"> 44.997</cn>
  <reals/>
</apply>

Rational Numbers `<rationals/>`

                  
<apply><in/>
  <cn type="rational"> 22 <sep/>7</cn>
  <rationals/>
</apply>

Natural Numbers `<naturalnumbers/>`

                  
<apply><in/>
  <cn type="integer">1729</cn>
  <naturalnumbers/>
</apply>

complexes `<complexes/>`

                  
<apply><in/>
  <cn type="complex-cartesian">17<sep/>29</cn>
  <complexes/>
</apply>

primes `<primes/>`

                  
<apply><in/>
  <cn type="integer">17</cn>
  <primes/>
</apply>

Exponential e `<exponentiale/>`

                  
<apply><eq/>
  <apply><ln/><exponentiale/></apply>
  <cn>1</cn>
</apply>

Imaginary i `<imaginaryi/>`

                  
<apply><eq/>
  <apply><power/><imaginaryi/><cn>2</cn></apply>
  <cn>-1</cn>
</apply>

Not A Number `<notanumber/>`

                  
<apply><eq/>
  <apply><divide/><cn>0</cn><cn>0</cn></apply>
  <notanumber/>
</apply>

True `<true/>`

                  
<apply><eq/>
  <apply><or/>
    <true/>
     <ci type="boolean">P</ci>
  </apply>
  <true/>
</apply>

False `<false/>`

                  
<apply><eq/>
  <apply><and/>
    <false/>
    <ci type="boolean">P</ci>
  </apply>
  <false/>
</apply>

Empty Set `<emptyset/>`

                  
<apply><neq/>
  <integers/>
  <emptyset/>
</apply>

pi `<pi/>`

                  
<apply><approx/>
  <pi/>
  <cn type="rational">22<sep/>7</cn>
</apply>

Euler gamma `<eulergamma/>`

                  
<apply><approx/>
  <eulergamma/>
  <cn>0.5772156649</cn>
</apply>

infinity `<infinity/>`

                  <infinity/>

Examples from MathML4

Content Markup

Introduction

The Intent of Content Markup

The Structure and Scope of Content MathML Expressions

Strict Content MathML

Content Dictionaries

Content MathML Concepts

Content MathML Elements Encoding Expression Structure

Numbers <cn>

Rendering <cn>,<sep/>-Represented Numbers

Strict uses of <cn>

Non-Strict uses of <cn>

Rewrite: cn sep

Rewrite: cn based_integer

Rewrite: cn constant

Rewrite: cn presentation mathml

Content Identifiers <ci>

Strict uses of <ci>

Non-Strict uses of <ci>

Rewrite: ci type annotation

Rewrite: ci presentation mathml

Rendering Content Identifiers

Content Symbols <csymbol>

Strict uses of <csymbol>

Non-Strict uses of <csymbol>

Rewrite: csymbol type annotation

Rendering Symbols

String Literals <cs>

Function Application <apply>

Strict Content MathML

Rendering Applications

Bindings and Bound Variables <bind> and <bvar>

Bindings

Bound Variables

Renaming Bound Variables

Rendering Binding Constructions

Structure Sharing <share>

The share element

An Acyclicity Constraint

Structure Sharing and Binding

Rendering Expressions with Structure Sharing

Attribution via semantics

Error Markup <cerror>

Encoded Bytes <cbytes>

Content MathML for Specific Structures

Container Markup

Container Markup for Constructor Symbols

Container Markup for Binding Constructors

Bindings with <apply>

Qualifiers

Uses of <domainofapplication>, <interval>, <condition>, <lowlimit> and <uplimit>

Rewrite: interval qualifier

Rewrite: condition

Rewrite: restriction

Rewrite: apply bvar domainofapplication

Uses of <degree>

Uses of <momentabout> and <logbase>

Operator Classes

Rewrite: element

N-ary Operators (classes nary-arith, nary-functional, nary-logical, nary-linalg, nary-set, nary-constructor)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary domainofapplication

N-ary Constructors for set and list (class nary-setlist-constructor)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary setlist domainofapplication

N-ary Relations (classes nary-reln, nary-set-reln)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary relations

Rewrite: n-ary relations bvar

N-ary/Unary Operators (classes nary-minmax, nary-stats)

Schema Patterns

Rewriting to Strict Content MathML

Rewrite: n-ary unary set

Rewrite: n-ary unary domainofapplication

Rewrite: n-ary unary single

Binary Operators (classes binary-arith, binary-logical, binary-reln, binary-linalg, binary-set)

Numbers `<cn>`

Rendering `<cn>`,`<sep/>`-Represented Numbers

Strict uses of `<cn>`

Non-Strict uses of `<cn>`

Content Identifiers `<ci>`

Strict uses of `<ci>`

Non-Strict uses of `<ci>`

Content Symbols `<csymbol>`

Strict uses of `<csymbol>`

Non-Strict uses of `<csymbol>`

String Literals `<cs>`

Function Application `<apply>`

Bindings and Bound Variables `<bind>` and `<bvar>`

Structure Sharing `<share>`

The `share` element

Attribution via `semantics`

Error Markup `<cerror>`

Encoded Bytes `<cbytes>`

Bindings with `<apply>`

Uses of `<domainofapplication>`, `<interval>`, `<condition>`, `<lowlimit>` and `<uplimit>`

Uses of `<degree>`

Uses of `<momentabout>` and `<logbase>`

Interval `<interval>`

Inverse `<inverse>`

Lambda `<lambda>`

Function composition `<compose/>`

Identity function `<ident/>`

Domain `<domain/>`

codomain `<codomain/>`

Image `<image/>`

Piecewise declaration `<piecewise>`, `<piece>`, `<otherwise>`

Quotient `<quotient/>`

Factorial `<factorial/>`

Division `<divide/>`

Maximum `<max/>`

Minimum `<min/>`

Subtraction `<minus/>`

Addition `<plus/>`

Exponentiation `<power/>`

Remainder `<rem/>`

Multiplication `<times/>`

Root `<root/>`

Greatest common divisor `<gcd/>`

And `<and/>`

Or `<or/>`

Exclusive Or `<xor/>`

Not `<not/>`

Implies `<implies/>`

Universal quantifier `<forall/>`

Existential quantifier `<exists/>`

Absolute Value `<abs/>`

Complex conjugate `<conjugate/>`

Argument `<arg/>`

Real part `<real/>`

Imaginary part `<imaginary/>`

Lowest common multiple `<lcm/>`

Floor `<floor/>`

Ceiling `<ceiling/>`

Equals `<eq/>`

Not Equals `<neq/>`

Greater than `<gt/>`

Less Than `<lt/>`

Greater Than or Equal `<geq/>`

Less Than or Equal `<leq/>`

Equivalent `<equivalent/>`

Approximately `<approx/>`

Factor Of `<factorof/>`

Integral `<int/>`

Differentiation `<diff/>`