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>

<math>
                        
<mrow><mi>hexdouble</mi><mo>⁡</mo><mrow><mo>(</mo><mrow></mrow><mo>)</mo></mrow></mrow>
                     
</math>

hexdouble ()

Non-Strict uses of `<cn>`

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

<math>
                        <msub><mrow></mrow><mn>16</mn></msub>
                     
</math>

_{16}

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

<math>
                        <msub><mrow></mrow><mn>1000</mn></msub>
                     
</math>

_{1000}

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

<math>
                        <mfrac><mrow><cn>22</cn><mn>22</mn></mrow><mrow><cn>7</cn><mn>7</mn></mrow></mfrac>
                     
</math>

\frac{2222}{77}

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

<math>
                        
<mrow><mrow><cn> 12.3 </cn><mn> 12.3 </mn></mrow><mo>+</mo><mrow><cn> 5 </cn><mn> 5 </mn></mrow><mo>⁢</mo><mi>i</mi></mrow>

                     
</math>

12.3 12.3 + 55 i

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

<math>
                        
<mrow><mrow><cn> 2 </cn><mn> 2 </mn></mrow><mo>⁢</mo><msup><mi>e</mi><mrow><mrow><cn> 3.1415 </cn><mn> 3.1415 </mn></mrow><mo>⁢</mo><mi>i</mi></mrow></msup></mrow>
                     
</math>

22 e^{3.1415 3.1415 i}

Rewrite: cn sep

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

<math>
                  
<msub><mrow><cn>n</cn><mn>n</mn></mrow><mn>b</mn></msub>
               
</math>

{n n}_{b}

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

<math>
                  
<mfrac><msub><mrow></mrow><mn>b</mn></msub><msub><mrow></mrow><mn>b</mn></msub></mfrac>
               
</math>

\frac{_{b}}{_{b}}

Rewrite: cn based_integer

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

<math>
                  
<msub><mrow></mrow><mn>16</mn></msub>
               
</math>

_{16}

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

<math>
                  
<mrow><mi>based_integer</mi><mo>⁡</mo><mrow><mo>(</mo><mrow></mrow><mo>,</mo><ms>FF60</ms><mo>)</mo></mrow></mrow>
               
</math>

based_integer (, FF60)

Rewrite: cn constant

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

<math>
                     <mrow><mi>constant</mi><mo>⁡</mo><mrow><mo>(</mo><mrow></mrow><mo>)</mo></mrow></mrow>
                  
</math>

constant ()

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

<math>
                     <mi>c2</mi>
                  
</math>

c2

Rewrite: cn presentation mathml

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

<math>
                     
<mfrac><mrow><mi>P</mi><mi>P</mi></mrow><mrow><mi>Q</mi><mi>Q</mi></mrow></mfrac>
                  
</math>

\frac{P P}{Q Q}

                     
<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>

<math>
                     
<mfrac><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mi>Q</mi></mrow></mrow></mfrac>
                  
</math>

\frac{P}{Q}

Content Identifiers `<ci>`

               <ci>x</ci>

<math>
               <mi>x</mi>
            
</math>

x

Strict uses of `<ci>`

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

<math>
                  <mi>n</mi>
               
</math>

n

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

<math>
                  
<mrow><mi>n</mi></mrow>
               
</math>

n

Non-Strict uses of `<ci>`

Rewrite: ci type annotation

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

<math>
                     <mi>n</mi>
                  
</math>

n

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

<math>
                     
<mrow><mi>n</mi></mrow>
                  
</math>

n

Rewrite: ci presentation mathml

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

<math>
                     <mrow><mi>P</mi></mrow>
                  
</math>

P

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

<math>
                     
<mrow><mrow><mi>P</mi></mrow></mrow>
                  
</math>

P

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

<math>
                  
<mrow><msup><mi>C</mi><mn>2</mn></msup></mrow>
               
</math>

C^{2}

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

<math>
                  
<mrow><mrow><msup><mi>C</mi><mn>2</mn></msup></mrow></mrow>
               
</math>

C^{2}

Rendering Content Identifiers

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

<math>
                  <mi>V</mi>
               
</math>

V

Content Symbols `<csymbol>`

Strict uses of `<csymbol>`

Non-Strict uses of `<csymbol>`

Rewrite: csymbol type annotation

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

<math>
                     <mi>symbolname</mi>
                  
</math>

symbolname

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

<math>
                     
<mrow><mi>symbolname</mi></mrow>
                  
</math>

symbolname

Rendering Symbols

String Literals `<cs>`

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

<math>
               
<mo>{</mo><mrow><ms>A</ms><mo>,</mo><ms>B</ms><mo>,</mo><ms>  </ms></mrow><mo>}</mo>
            
</math>

{A, B,}

Function Application `<apply>`

Strict Content MathML

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

<math>
                  <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow>
               
</math>

x + y

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

<math>
                  
<mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow>
               
</math>

x + y + z

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

<math>
                  
<mrow><mrow><mi>a</mi><mo>⁢</mo><mi>x</mi></mrow><mo>+</mo><mi>b</mi></mrow>
               
</math>

a x + b

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

<math>
                  
<mrow><mrow><mo>(</mo><mi>F</mi><mo>+</mo><mi>G</mi><mo>)</mo></mrow><mo>⁢</mo><mi>x</mi></mrow>
               
</math>

(F + G) x

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

<math>
                  <mrow><mi>F</mi><mo>+</mo><mi>G</mi></mrow>
               
</math>

F + G

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

<math>
                  
<mrow><mrow><mi>F</mi><mo>+</mo><mi>G</mi></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

F + G (x)

Rendering Applications

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

<math>
                  
<mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>,</mo><mi>...</mi><mo>,</mo><mi>an</mi><mo>)</mo></mrow></mrow>
               
</math>

f (a1, a2, ..., an)

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

<math>
                  
<mrow><mi>op</mi><mo>⁡</mo><mrow><mo>(</mo><mi>expression-in-x</mi><mo>)</mo></mrow></mrow>
               
</math>

op (expression-in-x)

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>

<math>
                  
<mrow><mo>∀</mo><mi>x</mi><mo>.</mo><mrow><mi>x</mi><mo>&gt;</mo><mn>1</mn></mrow></mrow>
               
</math>

\forall x . x > 1

               
<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>

<math>
               
<mrow><mo>∀</mo><mi>x</mi><mo>.</mo><mrow><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>=</mo><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mrow></mrow>
            
</math>

\forall x . x + y = y + x

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>

<math>
                  
<mrow><mi>b</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow>
               
</math>

b (s)

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>

<math>
                              
<mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
                           
</math>

f (f (f (a, a), f (a, a)), f (f (a, a), f (a, a)))

                              
<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>

<math>
                              
<mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>,</mo><mi href="null">sharenull</mi><mo>)</mo></mrow></mrow><mo>,</mo><mi href="null">sharenull</mi><mo>)</mo></mrow></mrow>
                           
</math>

f (f (f (a, a), sharenull), sharenull)

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>

<math>
                  
<mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi href="null">sharenull</mi></mrow><mo>,</mo><mrow><mi>g</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow>
               
</math>

λ x . f (λ x . sharenull, g (x))

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>

<math>
               
<merror><mi>DivisionByZero</mi><mrow><mi>x</mi><mo>/</mo><mn>0</mn></mrow></merror>
            
</math>

DivisionByZero x / 0

               
<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>

<math>
               
<mrow><merror><mi>DivisionByZero</mi><mrow><mi>x</mi><mo>/</mo><mn>0</mn></mrow></merror><mo>=</mo><mn>0</mn></mrow>
            
</math>

DivisionByZero x / 0 = 0

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

<math>
               
<merror><mi>DivisionByZero</mi><mrow><mi>x</mi><mo>/</mo><mn>0</mn></mrow></merror>
            
</math>

DivisionByZero x / 0

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

<math>
               
<merror><mi>Illegal bound variable</mi><ms> &lt;bvar&gt;&lt;plus/&gt;&lt;/bvar&gt; </ms></merror>
            
</math>

Illegal bound variable <bvar><plus/></bvar>

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>

<math>
                     <mo>{</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mo>}</mo>
                  
</math>

{a, b, c}

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

<math>
                     <mo>{</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mo>}</mo>
                  
</math>

{a, b, c}

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

<math>
                     
<mo>{</mo><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>x</mi></mrow></mrow><mo>|</mo><mrow><bvar><mi>x</mi></bvar></mrow><mo>∈</mo><mrow><mrow><mi>ℤ</mi></mrow></mrow><mo>}</mo>
                  
</math>

{2 x | x \in ℤ}

                     
<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>

<math>
                     
<mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mn>2</mn><mo>⁢</mo><mi>x</mi></mrow></mrow><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></mrow>
                  
</math>

map (λ x . 2 x, Z)

Container Markup for Binding Constructors

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

<math>
                  <mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>x</mi></mrow>
               
</math>

λ x . x

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

<math>
                  
<mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>x</mi></mrow>
               
</math>

λ x . x

Bindings with `<apply>`

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

<math>
                  
<mrow><mo>∀</mo><mi>x</mi><mo>.</mo><mrow><mi>x</mi><mo>≥</mo><mi>x</mi></mrow></mrow>
               
</math>

\forall x . x \geq x

                  
<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>

<math>
                  
<mrow><mo>∀</mo><mi>x</mi><mo>.</mo><mrow><mi>x</mi><mo>≥</mo><mi>x</mi></mrow></mrow>
               
</math>

\forall x . x \geq x

                  
<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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow><ci>i</ci><mo>=</mo><mn>0</mn></mrow><mjrow><mn>100</mn></mjrow></munderover><msup><mi>x</mi><mi>i</mi></msup></mrow>
               
</math>

\sum_{i = 0}^{100} x^{i}

                  
<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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow></mrow><mjrow></mjrow></munderover><mrow><mi>integer_interval</mi><mo>⁡</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>100</mn><mo>)</mo></mrow></mrow><mrow><mo>λ</mo><mi>i</mi><mo>.</mo><msup><mi>x</mi><mi>i</mi></msup></mrow></mrow>
               
</math>

\sum_{}^{} integer_interval (0, 100) λ i . x^{i}

Qualifiers

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

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

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mi>C</mi></mrow><mrow></mrow></msubsup><mi>f</mi></mrow>
               
</math>

\int_{C}^{} f

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mi>x</mi><mn>2</mn></msup><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
               
</math>

\int_{0}^{1} x^{2} d x

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mi>x</mi><mn>2</mn></msup><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
               
</math>

\int_{0}^{1} x^{2} d x

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mrow></mrow></msubsup><msup><mi>x</mi><mn>2</mn></msup><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
               
</math>

\int_{(0 \leq x) \land (x \leq 1)}^{} x^{2} d x

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mo>{</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow><mo>|</mo><mrow><mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>t</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>t</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>u</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>u</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow></mrow><mo>}</mo></mrow><mrow></mrow></msubsup><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>y</mi><mn>3</mn></msup></mrow><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mrow>
               
</math>

\int_{{(t, u) | (0 \leq t) \land (t \leq 1) \land (0 \leq u) \land (u \leq 1)}}^{} x^{2} y^{3} d x d y

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>

<math>
                  
<mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></mrow>
               
</math>

H (C)

                  
<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>

<math>
                  
<mrow><mi>H</mi><mo>⁡</mo><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></mrow>
               
</math>

H (C)

Rewrite: condition

Rewrite: restriction

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

<math>
                  
<mrow><mi>F</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a1</mi><mo>,</mo><mi>an</mi><mo>)</mo></mrow></mrow>
               
</math>

F (a1, an)

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

<math>
                  
<mrow><mrow><mi>restriction</mi><mo>⁡</mo><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>a1</mi><mo>,</mo><mi>an</mi><mo>)</mo></mrow></mrow>
               
</math>

restriction (F, C) (a1, an)

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>

<math>
               
<mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mi>d</mi><msup><ci>x</ci><mn>2</mn></msup></mrow></mfrac>
            
</math>

\frac{d^{2} x^{4}}{d x^{2}}

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

Operator Classes

Rewrite: element

               <plus/>

<math>
               <mi>plus</mi>
            
</math>

plus

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

<math>
                  <mi>plus</mi>
               
</math>

plus

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>

<math>
                        
<mrow><mi>expression-in-x</mi></mrow>
                     
</math>

expression-in-x

                        
<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>

<math>
                        
<mrow><mi>apply_to_list</mi><mo>⁡</mo><mrow><mo>(</mo><mi>union</mi><mo>,</mo><mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>,</mo><mi>D</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
                     
</math>

apply_to_list (union, map (λ x . expression-in-x, D))

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>

<math>
                        
<mo>{</mo><mrow><mi>expression-in-x</mi></mrow><mo>|</mo><mrow><bvar><mi>x</mi></bvar></mrow><mo>∈</mo><mrow><mrow><mi>D</mi></mrow></mrow><mo>}</mo>
                     
</math>

{expression-in-x | x \in D}

                        
<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>

<math>
                        
<mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>,</mo><mi>D</mi><mo>)</mo></mrow></mrow>
                     
</math>

map (λ x . expression-in-x, D)

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>

<math>
                        
<mrow><mi>a</mi><mo>&gt;</mo><mi>b</mi><mo>&gt;</mo><mi>c</mi><mo>&gt;</mo><mi>d</mi></mrow>
                     
</math>

a > b > c > d

                        
<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>

<math>
                        
<mrow><mi>predicate_on_list</mi><mo>⁡</mo><mrow><mo>(</mo><mi>lt</mi><mo>,</mo><mo>(</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow><mo>)</mo><mo>)</mo></mrow></mrow>

                     
</math>

predicate_on_list (lt, (a, b, c, d))

Rewrite: n-ary relations bvar

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

<math>
                        
<mrow><mi>expression-in-x</mi></mrow>
                     
</math>

expression-in-x

                        
<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>

<math>
                        
<mrow><mi>predicate_on_list</mi><mo>⁡</mo><mrow><mo>(</mo><mi>lt</mi><mo>,</mo><mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mi>R</mi><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
                     
</math>

predicate_on_list (lt, map (R, λ x . expression-in-x))

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>

<math>
                     
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>,</mo><mi>an</mi><mo>)</mo></mrow></mrow>
                  
</math>

\max (a1, a2, an)

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

<math>
                     
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mo>{</mo><mrow><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>,</mo><mi>an</mi></mrow><mo>}</mo><mo>)</mo></mrow></mrow>
                  
</math>

\max ({a1, a2, an})

Rewrite: n-ary unary domainofapplication

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

<math>
                        
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mi>expression-in-x</mi><mo>)</mo></mrow></mrow>
                     
</math>

\max (expression-in-x)

                        
<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>

<math>
                        
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>,</mo><mi>D</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
                     
</math>

\max (map (λ x . expression-in-x, D))

Rewrite: n-ary unary single

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

<math>
                     
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
                  
</math>

\max (a)

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

<math>
                     
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
                  
</math>

\max (a)

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>

<math>
                        
<mrow><mo>∃</mo><mi>x</mi><mo>∈</mo><mi>D</mi><mo>.</mo><mi>expression-in-x</mi></mrow>
                     
</math>

\exists x \in D . expression-in-x

                        
<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>

<math>
                        
<mrow><mo>∃</mo><mi>x</mi><mo>.</mo><mrow><mrow><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><mo>∧</mo><mi>expression-in-x</mi></mrow></mrow>

                     
</math>

\exists x . x \in D \land expression-in-x

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>

<math>
                  
<mi>x</mi>
               
</math>

x

                  
<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>

<math>
                  
<mrow><mi>x</mi></mrow>
               
</math>

x

Content MathML for Specific Operators and Constants

Functions and Inverses

Interval `<interval>`

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

<math>
                  
<mo>(</mo><mrow><mi>x</mi><mo>,</mo><mn>1</mn></mrow><mo>)</mo>
               
</math>

(x, 1)

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

<math>
                  
<mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo>
               
</math>

[0, 1]

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

<math>
                  
<mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo>
               
</math>

(0, 1]

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

<math>
                  
<mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo>
               
</math>

[0, 1)

Inverse `<inverse>`

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

<math>
                  
<msup><mi> f </mi><mfenced><mn>-1</mn></mfenced></msup>
               
</math>

f^{(-1)}

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

<math>
                  
<mrow><msup><mi>A</mi><mfenced><mn>-1</mn></mfenced></msup><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

A^{(-1)} (a)

Lambda `<lambda>`

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

<math>
                  
<mrow><mo>λ</mo><mi> x </mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi> x </mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

λ x \in ℤ . \sin (x)

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

<math>
                  
<mrow><mi>sin</mi><msub><mo>|</mo><mrow><mi>ℤ</mi></mrow></msub></mrow>
               
</math>

\sin |_{ℤ}

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

<math>
                  
<mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow>
               
</math>

λ x . \sin (x + 1)

Rewrite: lambda

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

<math>
                  
<mrow><mo>λ</mo><mi>x1</mi><mi>xn</mi><mo>.</mo><mi>expression-in-x1-xn</mi></mrow>
               
</math>

λ x1 xn . expression-in-x1-xn

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

<math>
                  
<mrow><mo>λ</mo><mi>x1</mi><mi>xn</mi><mo>.</mo><mi>expression-in-x1-xn</mi></mrow>
               
</math>

λ x1 xn . expression-in-x1-xn

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>

<math>
                  
<mrow><mo>λ</mo><mi>x1</mi><mi>xn</mi><mo>∈</mo><mi>D</mi><mo>.</mo><mi>expression-in-x1-xn</mi></mrow>
               
</math>

λ x1 xn \in D . expression-in-x1-xn

                  
<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>

<math>
                  
<mrow><mi>restriction</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>x1</mi><mi>xn</mi><mo>.</mo><mi>expression-in-x1-xn</mi></mrow><mo>,</mo><mi>D</mi><mo>)</mo></mrow></mrow>
               
</math>

restriction (λ x1 xn . expression-in-x1-xn, D)

Function composition `<compose/>`

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

<math>
                  
<mrow><mi>f</mi><mo>∘</mo><mi>g</mi><mo>∘</mo><mi>h</mi></mrow>
               
</math>

f \circ g \circ h

                  
<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>

<math>
                  
<mrow><mrow><mrow><mi>f</mi><mo>∘</mo><mi>g</mi></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>g</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow>
               
</math>

f \circ g (x) = f (g (x))

Identity function `<ident/>`

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

<math>
                  
<mrow><mrow><mi>f</mi><mo>∘</mo><msup><mi>f</mi><mfenced><mn>-1</mn></mfenced></msup></mrow><mo>=</mo><mi>id</mi></mrow>
               
</math>

f \circ f^{(-1)} = id

Domain `<domain/>`

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

<math>
                  
<mrow><mrow><mi>domain</mi><mo>⁡</mo><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow><mo>=</mo><mi>ℝ</mi></mrow>
               
</math>

domain (f) = ℝ

codomain `<codomain/>`

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

<math>
                  
<mrow><mrow><mi>codomain</mi><mo>⁡</mo><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow><mo>=</mo><mi>ℚ</mi></mrow>
               
</math>

codomain (f) = ℚ

Image `<image/>`

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

<math>
                  
<mrow><mrow><mi>image</mi><mo>⁡</mo><mrow><mo>(</mo><mi>sin</mi><mo>)</mo></mrow></mrow><mo>=</mo><mo>[</mo><mrow><mn>-1</mn><mo>,</mo><mn> 1</mn></mrow><mo>]</mo></mrow>
               
</math>

image (\sin) = [-1, 1]

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>

<math>
                  
<mrow><mo>{</mo><mtable><mtr><mtd><mrow><mo>-</mo><mi>x</mi></mrow></mtd><mtd><mtext>&nbsp;if&nbsp;</mtext></mtd><mtd><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>&nbsp;if&nbsp;</mtext></mtd><mtd><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mi>x</mi></mtd><mtd><mtext>&nbsp;if&nbsp;</mtext></mtd><mtd><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow>
               
</math>

{\begin{matrix} - x & if & x > 0 \\ 0 & if & x = 0 \\ x & if & x < 0 \end{matrix}

                  
<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>

<math>
                  
<mrow><mo>{</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mtext>&nbsp;if&nbsp;</mtext></mtd><mtd><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mtext>&nbsp;if&nbsp;</mtext></mtd><mtd><mrow><mi>x</mi><mo>&lt;</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>x</mi></mtd><mtd columnspan="2"><mtext>&nbsp;otherwise</mtext></mtd></mtr></mtable></mrow>
               
</math>

{\begin{matrix} 0 & if & x > 0 \\ 1 & if & x < 1 \\ x & otherwise \end{matrix}

                  
<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>

<math>
                  
<mrow><mi>piecewise</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>piece</mi><mo>⁡</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mrow><mi>x</mi><mo>&gt;</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mi>piece</mi><mo>⁡</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mrow><mi>x</mi><mo>&lt;</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mi>otherwise</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

piecewise (piece (0, x > 0), piece (1, x < 1), otherwise (x))

Arithmetic, Algebra and Logic

Quotient `<quotient/>`

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

<math>
                  
<mrow><mo>⌊</mo><mi>a</mi><mo>/</mo><mi>b</mi><mo>⌋</mo></mrow>
               
</math>

⌊ a / b ⌋

Factorial `<factorial/>`

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

<math>
                  
<mrow><mi>n</mi><mo>!</mo></mrow>
               
</math>

n!

Division `<divide/>`

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

<math>
                  
<mrow><mi>a</mi><mo>/</mo><mi>b</mi></mrow>
               
</math>

a / b

Maximum `<max/>`

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

<math>
                  
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></mrow>
               
</math>

\max (2, 3, 5)

                  
<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>

<math>
                  
<mrow><mi>max</mi><mo>⁡</mo><mrow><mo>(</mo><msup><mi>y</mi><mn>3</mn></msup><mo>)</mo></mrow></mrow>
               
</math>

\max (y^{3})

Minimum `<min/>`

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

<math>
                  
<mrow><mi>min</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow>
               
</math>

\min (a, b)

                  
<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>

<math>
                  
<mrow><mi>min</mi><mo>⁡</mo><mrow><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow>
               
</math>

\min (x^{2})

Subtraction `<minus/>`

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

<math>
                  
<mrow><mo>-</mo><mn>3</mn></mrow>
               
</math>

- 3

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

<math>
                  
<mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow>
               
</math>

x - y

Addition `<plus/>`

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

<math>
                  
<mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow>
               
</math>

x + y + z

Exponentiation `<power/>`

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

<math>
                  
<msup><mi>x</mi><mn>3</mn></msup>
               
</math>

x^{3}

Remainder `<rem/>`

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

<math>
                  
<mrow><mi> a </mi><mo>mod</mo><mi> b </mi></mrow>
               
</math>

a mod b

Multiplication `<times/>`

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

<math>
                  
<mrow><mi>a</mi><mo>⁢</mo><mi>b</mi></mrow>
               
</math>

a b

Root `<root/>`

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

<math>
                  
<mroot><mi>a</mi><mi>n</mi></mroot>
               
</math>

\sqrt[n]{a}

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

<math>
                  <msqrt><mi>x</mi></msqrt>
               
</math>

\sqrt{x}

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

<math>
                  
<mroot><mi>x</mi><mrow></mrow></mroot>
               
</math>

\sqrt[]{x}

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

<math>
                  
<mroot><mi>a</mi><mi>n</mi></mroot>
               
</math>

\sqrt[n]{a}

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

<math>
                  
<mroot><mi>a</mi><mrow></mrow></mroot>
               
</math>

\sqrt[]{a}

Greatest common divisor `<gcd/>`

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

<math>
                  
<mrow><mi>gcd</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow>
               
</math>

\gcd (a, b, c)

And `<and/>`

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

<math>
                  
<mrow><mi>a</mi><mo>∧</mo><mi>b</mi></mrow>
               
</math>

a \land b

                  
<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>

<math>
                  
<mrow><mrow><mo>(</mo><msub><mi>a</mi><mrow><mi>i</mi></mrow></msub><mo>&lt;</mo><mn>0</mn><mo>)</mo></mrow></mrow>
               
</math>

(a_{i} < 0)

                  
<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>

<math>
                  
<mrow><mi>apply_to_list</mi><mo>⁡</mo><mrow><mo>(</mo><mi>and</mi><mo>,</mo><mrow><mi>map</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mo>λ</mo><mi>i</mi><mo>.</mo><mrow><mrow><mi>vector_selector</mi><mo>⁡</mo><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>&lt;</mo><mn>0</mn></mrow></mrow><mo>,</mo><mrow><mi>integer_interval</mi><mo>⁡</mo><mrow><mo>(</mo><mrow></mrow><mo>,</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

apply_to_list (and, map (λ i . vector_selector (i, a) < 0, integer_interval (, n)))

Or `<or/>`

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

<math>
                  
<mrow><mi>a</mi><mo>∨</mo><mi>b</mi></mrow>
               
</math>

a \lor b

Exclusive Or `<xor/>`

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

<math>
                  
<mrow><mi>a</mi><mo>xor</mo><mi>b</mi></mrow>
               
</math>

a xor b

Not `<not/>`

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

<math>
                  
<mrow><mi>not</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

not (a)

Implies `<implies/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⇒</mo><mi>B</mi></mrow>
               
</math>

A \Rightarrow B

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>

<math>
                  
<mrow><mo>∀</mo><mi>x</mi><mo>.</mo><mrow><mrow><mi>x</mi><mo>-</mo><mi>x</mi></mrow><mo>=</mo><mn>0</mn></mrow></mrow>
               
</math>

\forall x . x - x = 0

                     
<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>

<math>
                     
<mrow><mo>∀</mo><mrow><mrow><mi>p</mi><mo>∈</mo><mi>ℚ</mi></mrow><mo>∧</mo><mrow><mi>q</mi><mo>∈</mo><mi>ℚ</mi></mrow><mo>∧</mo><mrow><mo>(</mo><mi>p</mi><mo>&gt;</mo><mi>q</mi><mo>)</mo></mrow></mrow><mo>.</mo><mrow><mi>p</mi><mo>&gt;</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mrow>
                  
</math>

\forall p \in ℚ \land q \in ℚ \land (p > q) . p > q^{2}

                     
<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>

<math>
                     
<mrow><mo>∀</mo><mi>p</mi><mi>q</mi><mo>.</mo><mrow><mrow><mrow><mi>p</mi><mo>∈</mo><mi>Q</mi></mrow><mo>∧</mo><mrow><mi>q</mi><mo>∈</mo><mi>Q</mi></mrow><mo>∧</mo><mrow><mo>(</mo><mi>p</mi><mo>&gt;</mo><mi>q</mi><mo>)</mo></mrow></mrow><mo>⇒</mo><mrow><mi>p</mi><mo>&gt;</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mrow></mrow>
                  
</math>

\forall p q . p \in Q \land q \in Q \land (p > q) \Rightarrow p > q^{2}

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>

<math>
                  
<mrow><mo>∃</mo><mi>x</mi><mo>.</mo><mrow><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></mrow>
               
</math>

\exists x . f (x) = 0

                  
<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>

<math>
                  
<mrow><mo>∃</mo><mi>x</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mrow><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></mrow>
               
</math>

\exists x \in ℤ . f (x) = 0

                  
<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>

<math>
                  
<mrow><mo>∃</mo><mi>x</mi><mo>.</mo><mrow><mrow><mi>x</mi><mo>∈</mo><mi>Z</mi></mrow><mo>∧</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn><mo>)</mo></mrow></mrow></mrow>
               
</math>

\exists x . x \in Z \land (f (x) = 0)

Absolute Value `<abs/>`

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

<math>
                  
<mrow><mi>abs</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

abs (x)

Complex conjugate `<conjugate/>`

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

<math>
                  
<mrow><mi>conjugate</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mn>ⅈ</mn><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

conjugate (x + ⅈ y)

Argument `<arg/>`

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

<math>
                  
<mrow><mi>arg</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi> x </mi><mo>+</mo><mrow><mi>i</mi><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

\arg (x + i y)

Real part `<real/>`

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

<math>
                  
<mrow><mi>real</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi>i</mi><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

real (x + i y)

Imaginary part `<imaginary/>`

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

<math>
                  
<mrow><mi>imaginary</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mrow><mi>i</mi><mo>⁢</mo><mi>y</mi></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

imaginary (x + i y)

Lowest common multiple `<lcm/>`

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

<math>
                  
<mrow><mi>lcm</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow>
               
</math>

lcm (a, b, c)

Floor `<floor/>`

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

<math>
                  
<mrow><mi>floor</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

floor (a)

Ceiling `<ceiling/>`

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

<math>
                  
<mrow><mi>ceiling</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

ceiling (a)

Relations

Equals `<eq/>`

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

<math>
                  
<mrow><mfrac><mrow><cn>2</cn><mn>2</mn></mrow><mrow><cn>4</cn><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><cn>1</cn><mn>1</mn></mrow><mrow><cn>2</cn><mn>2</mn></mrow></mfrac></mrow>
               
</math>

\frac{22}{44} = \frac{11}{22}

Not Equals `<neq/>`

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

<math>
                  
<mrow><mn>3</mn><mo>≠</mo><mn>4</mn></mrow>
               
</math>

3 \neq 4

Greater than `<gt/>`

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

<math>
                  
<mrow><mn>3</mn><mo>&lt;</mo><mn>2</mn></mrow>
               
</math>

3 < 2

Less Than `<lt/>`

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

<math>
                  
<mrow><mn>2</mn><mo>&gt;</mo><mn>3</mn><mo>&gt;</mo><mn>4</mn></mrow>
               
</math>

2 > 3 > 4

Greater Than or Equal `<geq/>`

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

<math>
                  
<mrow><mn>4</mn><mo>≥</mo><mn>3</mn><mo>≥</mo><mn>3</mn></mrow>
               
</math>

4 \geq 3 \geq 3

                  
<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>

<math>
                  
<mrow><mi>predicate_on_list</mi><mo>⁡</mo><mrow><mo>(</mo><mi>geq</mi><mo>,</mo><mo>(</mo><mrow><mn>4</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo><mo>)</mo></mrow></mrow>
               
</math>

predicate_on_list (geq, (4, 3, 3))

Less Than or Equal `<leq/>`

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

<math>
                  
<mrow><mn>3</mn><mo>≤</mo><mn>3</mn><mo>≤</mo><mn>4</mn></mrow>
               
</math>

3 \leq 3 \leq 4

Equivalent `<equivalent/>`

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

<math>
                  
<mrow><mi>a</mi><mo>≡</mo><mrow><mi>not</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>not</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow>
               
</math>

a \equiv not (not (a))

Approximately `<approx/>`

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

<math>
                  
<mrow><mi>π</mi><mo>≃</mo><mfrac><mrow><cn>22</cn><mn>22</mn></mrow><mrow><cn>7</cn><mn>7</mn></mrow></mfrac></mrow>
               
</math>

π ≃ \frac{2222}{77}

Factor Of `<factorof/>`

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

<math>
                  
<mrow><mi>a</mi><mo>⇒</mo><mi>b</mi></mrow>
               
</math>

a \Rightarrow b

Calculus and Vector Calculus

Integral `<int/>`

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

<math>
                  
<mrow><mrow><msubsup><mo>∫</mo><mrow></mrow><mrow></mrow></msubsup><mi>sin</mi></mrow><mo>=</mo><mi>cos</mi></mrow>
               
</math>

\int_{}^{} \sin = \cos

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

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mi>cos</mi></mrow>
               
</math>

\int_{a}^{b} \cos

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mi>x</mi><mn>2</mn></msup><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
               
</math>

\int_{0}^{1} x^{2} d x

Rewrite: int

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

<math>
                     
<mrow><msubsup><mo>∫</mo><mrow></mrow><mrow></mrow></msubsup><mi>expression-in-x</mi><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
                  
</math>

\int_{}^{} expression-in-x d x

                     
<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>

<math>
                     
<mrow><mrow><msubsup><mo>∫</mo><mrow></mrow><mrow></mrow></msubsup><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
                  
</math>

\int_{}^{} λ x . expression-in-x (x)

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

<math>
                     
<mrow><msubsup><mo>∫</mo><mrow></mrow><mrow></mrow></msubsup><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
                  
</math>

\int_{}^{} \cos (x) d x

                     
<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>

<math>
                     
<mrow><mrow><msubsup><mo>∫</mo><mrow></mrow><mrow></mrow></msubsup><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>

                  
</math>

\int_{}^{} λ x . \cos (x) (x)

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

<math>
                     
<mrow><msubsup><mo>∫</mo><mrow><mi>C</mi></mrow><mrow></mrow></msubsup><mi>f</mi></mrow>
                  
</math>

\int_{C}^{} f

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

<math>
                     
<mrow><mi>defint</mi><mo>⁡</mo><mrow><mo>(</mo><mi>C</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></mrow>
                  
</math>

defint (C, f)

Rewrite: defint

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

<math>
                     
<mrow><msubsup><mo>∫</mo><mrow><mi>D</mi></mrow><mrow></mrow></msubsup><mi>expression-in-x</mi><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
                  
</math>

\int_{D}^{} expression-in-x d x

                     
<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>

<math>
                     
<mrow><mi>defint</mi><mo>⁡</mo><mrow><mo>(</mo><mi>D</mi><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>)</mo></mrow></mrow>
                  
</math>

defint (D, λ x . expression-in-x)

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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></msubsup><mi>expression-in-x</mi><mrow><mi>d</mi><mi>x</mi></mrow></mrow>
               
</math>

\int_{a}^{b} expression-in-x d x

                  
<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>

<math>
                  
<mrow><mi>defint</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>oriented_interval</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>)</mo></mrow></mrow>
               
</math>

defint (oriented_interval (a, b), λ x . expression-in-x)

                  
<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>

<math>
                  
<mrow><msubsup><mo>∫</mo><mrow><mrow><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>y</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>y</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></mrow></mrow><mrow></mrow></msubsup><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>y</mi><mn>3</mn></msup></mrow><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mrow>
               
</math>

\int_{(0 \leq x) \land (x \leq 1) \land (0 \leq y) \land (y \leq 1)}^{} x^{2} y^{3} d x d y

                  
<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>

<math>
                  
<mrow><mi>defint</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>suchthat</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>R</mi><mo>×</mo><mi>R</mi></mrow><mo>,</mo><mrow><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>y</mi><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mi>y</mi><mo>≤</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mi>y</mi><mo>.</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>y</mi><mn>3</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

defint (suchthat (R \times R, (0 \leq x) \land (x \leq 1) \land (0 \leq y) \land (y \leq 1), λ x y . x^{2} y^{3}))

Differentiation `<diff/>`

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

<math>
                  <msup><mrow><mi>f</mi></mrow><mo>′</mo></msup>
               
</math>

f^{'}

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

<math>
                  
<mrow><mfrac><mrow><mi>d</mi><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

\frac{d \sin (x)}{d x} = \cos (x)

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

<math>
                  
<mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mi>d</mi><msup><ci>x</ci><mn>2</mn></msup></mrow></mfrac>
               
</math>

\frac{d^{2} x^{4}}{d x^{2}}

Rewrite: diff

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

<math>
                     
<mfrac><mrow><mi>d</mi><mi>expression-in-x</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac>
                  
</math>

\frac{d expression-in-x}{d x}

                     
<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>

<math>
                     
<mrow><msup><mrow><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>E</mi></mrow></mrow><mo>′</mo></msup><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
                  
</math>

{λ x . E}^{'} (x)

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

<math>
                     
<mfrac><mrow><mi>d</mi><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac>
                  
</math>

\frac{d \sin (x)}{d x}

                     
<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>

<math>
                     
<mrow><msup><mrow><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>′</mo></msup><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
                  
</math>

{λ x . \sin (x)}^{'} (x)

Rewrite: nthdiff

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

<math>
                  
<mfrac><mrow><msup><mi>d</mi><mi>n</mi></msup><mi>expression-in-x</mi></mrow><mrow><mi>d</mi><msup><ci>x</ci><mi>n</mi></msup></mrow></mfrac>
               
</math>

\frac{d^{n} expression-in-x}{d x^{n}}

                  
<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>

<math>
                  
<mrow><mrow><mi>nthdiff</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>)</mo></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

nthdiff (n, λ x . expression-in-x) (x)

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

<math>
                  
<mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>d</mi><msup><ci>x</ci><mn>2</mn></msup></mrow></mfrac>
               
</math>

\frac{d^{2} \sin (x)}{d x^{2}}

                  
<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>

<math>
                  
<mrow><mrow><mi>nthdiff</mi><mo>⁡</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

nthdiff (2, λ x . \sin (x)) (x)

Partial Differentiation `<partialdiff/>`

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

<math>
                  
<mrow><msub><mi>D</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mi>f</mi></mrow>
               
</math>

D_{1, 1, 3} f

                  
<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>

<math>
                  
<mfrac><mrow><msup><mo>∂</mo><mn>3</mn></msup><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>z</mi></mrow></mfrac>
               
</math>

\frac{\partial^{3} f (x, y, z)}{\partial x \partial x \partial z}

                  
<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>

<math>
                  
<mfrac><mrow><msup><mo>∂</mo><mrow><mn>2</mn></mrow></msup><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac>
               
</math>

\frac{\partial^{2} f (x, y)}{\partial x \partial y}

                  
<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>

<math>
                  
<mfrac><mrow><msup><mo>∂</mo><mrow><mi>k</mi></mrow></msup><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mi>m</mi></msup><mo>∂</mo><msup><mi>y</mi><mi>n</mi></msup></mrow></mfrac>
               
</math>

\frac{\partial^{k} f (x, y)}{\partial x^{m} \partial y^{n}}

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>

<math>
                     
<mfrac><mrow><msup><mo>∂</mo><mrow><mi>total-n1-nk</mi></mrow></msup><mi>expression-in-x1-xk</mi></mrow><mrow><mo>∂</mo><msup><mi>x1</mi><mi>n1</mi></msup><mo>∂</mo><msup><mi>xk</mi><mi>nk</mi></msup></mrow></mfrac>
                  
</math>

\frac{\partial^{total-n1-nk} expression-in-x1-xk}{\partial {x1}^{n1} \partial {xk}^{nk}}

                  
<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>

<math>
                  
<mrow><mrow><mi>partialdiffdegree</mi><mo>⁡</mo><mrow><mo>(</mo><mo>(</mo><mrow><mi>n1</mi><mo>,</mo><mi>nk</mi></mrow><mo>)</mo><mo>,</mo><mi>total-n1-nk</mi><mo>,</mo><mrow><mo>λ</mo><mi>x1</mi><mi>xk</mi><mo>.</mo><mi>expression-in-x1-xk</mi></mrow><mo>)</mo></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mi>x1</mi><mo>,</mo><mi>xk</mi><mo>)</mo></mrow></mrow>
               
</math>

partialdiffdegree ((n1, nk), total-n1-nk, λ x1 xk . expression-in-x1-xk) (x1, xk)

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

<math>
                     
<mrow><mi>n1</mi><mo>+</mo><mi>nk</mi></mrow>
                  
</math>

n1 + nk

                     
<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>

<math>
                     
<mfrac><mrow><msup><mo>∂</mo><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msup><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mo>∂</mo><msup><mi>x</mi><mi>n</mi></msup><mo>∂</mo><msup><mi>y</mi><mi>m</mi></msup></mrow></mfrac>
                  
</math>

\frac{\partial^{n + m} \sin (x y)}{\partial x^{n} \partial y^{m}}

                     
<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>

<math>
                     
<mrow><mrow><mi>partialdiffdegree</mi><mo>⁡</mo><mrow><mo>(</mo><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow><mo>)</mo><mo>,</mo><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mi>y</mi><mo>.</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>⁡</mo><mrow><mo>(</mo><mo>)</mo></mrow></mrow>
                  
</math>

partialdiffdegree ((n, m), n + m, λ x y . \sin (x y), x, y) ()

Divergence `<divergence/>`

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

<math>
                  
<mrow><mi>divergence</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

divergence (a)

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

<math>
                  
<mrow><mi>divergence</mi><mo>⁡</mo><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow>
               
</math>

divergence (E)

                  
<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>

<math>
                  
<mrow><mi>divergence</mi><mo>⁡</mo><mrow><mo>(</mo><vector>
    <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow>
    <mrow><mi>x</mi><mo>+</mo><mi>z</mi></mrow>
    <mrow><mi>z</mi><mo>+</mo><mi>y</mi></mrow>
  </vector><mo>)</mo></mrow></mrow>
               
</math>

divergence (x + y x + z z + y)

Gradient `<grad/>`

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

<math>
                  
<mrow><mi>grad</mi><mo>⁡</mo><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow>
               
</math>

grad (f)

                  
<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>

<math>
                  
<mrow><mi>grad</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>⁢</mo><mi>y</mi><mo>⁢</mo><mi>z</mi></mrow><mo>)</mo></mrow></mrow>
               
</math>

grad (x y z)

Curl `<curl/>`

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

<math>
                  
<mrow><mi>curl</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

curl (a)

Laplacian `<laplacian/>`

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

<math>
                  
<mrow><mi>laplacian</mi><mo>⁡</mo><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow>
               
</math>

laplacian (E)

                  
<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>

<math>
                  
<mrow><mi>laplacian</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

laplacian (f (x, y))

Theory of Sets

Set `<set>`

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

<math>
                  
<mo>{</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mo>}</mo>
               
</math>

{a, b, c}

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

<math>
                  
<mo>{</mo><mrow><mi>x</mi></mrow><mo>|</mo><mrow><mrow><mrow><mi>x</mi><mo>&gt;</mo><mn>5</mn></mrow></mrow></mrow><mo>}</mo>
               
</math>

{x | x > 5}

                  
<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>

<math>
                  
<mo>{</mo><mrow><mi>S</mi></mrow><mo>|</mo><mrow><mrow><mrow><mi>S</mi><mo>∈</mo><mi>T</mi></mrow></mrow></mrow><mo>}</mo>
               
</math>

{S | S \in T}

                  
<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>

<math>
                  
<mo>{</mo><mrow><mi>x</mi></mrow><mo>|</mo><mrow><mrow><mrow><mrow><mo>(</mo><mi>x</mi><mo>&gt;</mo><mn>5</mn><mo>)</mo></mrow><mo>∧</mo><mrow><mi>x</mi><mo>∈</mo><mi>ℕ</mi></mrow></mrow></mrow></mrow><mo>}</mo>
               
</math>

{x | (x > 5) \land x \in ℕ}

List `<list>`

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

<math>
                  
<mo>(</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mo>)</mo>
               
</math>

(a, b, c)

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

<math>
                  
<mo>(</mo><mrow></mrow><mo>|</mo><mrow><mrow><mrow><mi>x</mi><mo>&gt;</mo><mn>5</mn></mrow></mrow></mrow><mo>)</mo>
               
</math>

(| x > 5)

Union `<union/>`

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

<math>
                  
<mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow>
               
</math>

A \cup B

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

<math>
                  
<mrow><mi> S</mi></mrow>
               
</math>

S

Intersect `<intersect/>`

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

<math>
                  
<mrow><mi> A </mi><mo>∩</mo><mi> B </mi></mrow>
               
</math>

A \cap B

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

<math>
                  
<mrow><mi> S </mi></mrow>
               
</math>

S

Set inclusion `<in/>`

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

<math>
                  
<mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow>
               
</math>

a \in A

Set exclusion `<notin/>`

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

<math>
                  
<mrow><mi>a</mi><mo>∉</mo><mi>A</mi></mrow>
               
</math>

a \notin A

Subset `<subset/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow>
               
</math>

A \subseteq B

Proper Subset `<prsubset/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⊂</mo><mi>B</mi></mrow>
               
</math>

A \subset B

Not Subset `<notsubset/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⊈</mo><mi>B</mi></mrow>
               
</math>

A ⊈ B

Not Proper Subset `<notprsubset/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⊄</mo><mi>B</mi></mrow>
               
</math>

A ⊄ B

Set Difference `<setdiff/>`

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

<math>
                  
<mrow><mi>A</mi><mo>∖</mo><mi>B</mi></mrow>
               
</math>

A ∖ B

Cardinality `<card/>`

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

<math>
                  
<mrow><mrow><mi>card</mi><mo>⁡</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>5</mn></mrow>
               
</math>

card (A) = 5

Cartesian product `<cartesianproduct/>`

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

<math>
                  
<mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow>
               
</math>

A \times B

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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow><ci>x</ci><mo>=</mo><mi>a</mi></mrow><mjrow><mi>b</mi></mjrow></munderover><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

\sum_{x = a}^{b} f (x)

                  
<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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow><mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow></mrow><mjrow></mjrow></munderover><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

\sum_{x \in B}^{} f (x)

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

<math>
                  
<mrow><munderover><mo>∑</mo><mrow><mi>B</mi></mrow><mjrow></mjrow></munderover><mi>f</mi></mrow>
               
</math>

\sum_{B}^{} f

                  
<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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow><ci>i</ci><mo>=</mo><mn>0</mn></mrow><mjrow><mn>100</mn></mjrow></munderover><msup><mi>x</mi><mi>i</mi></msup></mrow>
               
</math>

\sum_{i = 0}^{100} x^{i}

                  
<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>

<math>
                  
<mrow><munderover><mo>∑</mo><mrow></mrow><mjrow></mjrow></munderover><mrow><mi>integer_interval</mi><mo>⁡</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>100</mn><mo>)</mo></mrow></mrow><mrow><mo>λ</mo><mi>i</mi><mo>.</mo><msup><mi>x</mi><mi>i</mi></msup></mrow></mrow>
               
</math>

\sum_{}^{} integer_interval (0, 100) λ i . x^{i}

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>

<math>
                  
<mrow><munderover><mo>∏</mo><mrow><ci>x</ci><mo>=</mo><mi>a</mi></mrow><mjrow><mi>b</mi></mjrow></munderover><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

\prod_{x = a}^{b} f (x)

                  
<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>

<math>
                  
<mrow><munderover><mo>∏</mo><mrow><mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow></mrow><mjrow></mjrow></munderover><mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow>
               
</math>

\prod_{x \in B}^{} f (x)

                  
<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>

<math>
                  
<mrow><munderover><mo>∏</mo><mrow><ci>i</ci><mo>=</mo><mn>0</mn></mrow><mjrow><mn>100</mn></mjrow></munderover><msup><mi>x</mi><mi>i</mi></msup></mrow>
               
</math>

\prod_{i = 0}^{100} x^{i}

                  
<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>

<math>
                  
<mrow><munderover><mo>∏</mo><mrow></mrow><mjrow></mjrow></munderover><mrow><mi>integer_interval</mi><mo>⁡</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>100</mn><mo>)</mo></mrow></mrow><mrow><mo>λ</mo><mi>i</mi><mo>.</mo><msup><mi>x</mi><mi>i</mi></msup></mrow></mrow>
               
</math>

\prod_{}^{} integer_interval (0, 100) λ i . x^{i}

Limits `<limit/>`

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

<math>
                  
<mrow><mi>limit</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

limit (\sin (x))

                  
<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>

<math>
                  
<mrow><mi>limit</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

limit (\sin (x))

                  
<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>

<math>
                  
<mrow><mi>limit</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow>
               
</math>

limit (\sin (x))

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>

<math>
                  
<mrow><mi>limit</mi><mo>⁡</mo><mrow><mo>(</mo><mi>expression-in-x</mi><mo>)</mo></mrow></mrow>
               
</math>

limit (expression-in-x)

                  
<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>

<math>
                  
<mrow><mi>limit</mi><mo>⁡</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>null</mi><mo>,</mo><mrow><mo>λ</mo><mi>x</mi><mo>.</mo><mi>expression-in-x</mi></mrow><mo>)</mo></mrow></mrow>
               
</math>

limit (0, null, λ x . expression-in-x)

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>

<math>
                  
<mrow><msup><mi>x</mi><mn>2</mn></msup><mo>↘</mo><msup><mi>a</mi><mn>2</mn></msup></mrow>
               
</math>

x^{2} ↘ a^{2}

                  
<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>

<math>
                  
<mrow><vector><mi>x</mi><mi>y</mi></vector><mo>→</mo><vector>
     <mrow><mi>f</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow>
     <mrow><mi>g</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow>
   </vector></mrow>
               
</math>

x y \to f (x, y) g (x, y)

Rewrite: tendsto

                  
<tendsto/>

<math>
                  
<mi>tendsto</mi>

               
</math>

tendsto

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

<math>
                  
<mrow><mi>tendsto</mi></mrow>
               
</math>

tendsto

Elementary classical functions

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

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

<math>
                  
<mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

\sin (x)

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

<math>
                  
<mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow>
               
</math>

\sin (\cos (x) + x^{3})

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

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

<math>
                  
<mrow><mi>arcsin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

\arcsin (x)

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

<math>
                  
<mrow>
 <mi>arcsin</mi>
 <mo>⁡</mo>
 <mi>x</mi>
<mi>arcsin</mi><mo>⁡</mo><mi>x</mi></mrow></math>

\arcsin x \arcsin x

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

<math>
                  
<mrow>
 <msup><mi>sin</mi></msup>
 <mo>⁡</mo>
 <mi>x</mi>
<msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mn>1</mn></mrow></msup><mo>⁡</mo><mi>x</mi></mrow></math>

\sin x \sin^{- 1 - 1} x

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

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

<math>
                  
<mrow><mi>sinh</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

\sinh (x)

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

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

<math>
                  
<mrow><mi>arcsinh</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

arcsinh (x)

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

<math>
                  
<mrow>
 <mi>arcsinh</mi>
 <mo>⁡</mo>
 <mi>x</mi>
<mi>arcsinh</mi><mo>⁡</mo><mi>x</mi></mrow></math>

arcsinh x arcsinh x

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

<math>
                  
<mrow>
 <msup><mi>sinh</mi></msup>
 <mo>⁡</mo>
 <mi>x</mi>
<msup><mi>sinh</mi><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mn>1</mn></mrow></msup><mo>⁡</mo><mi>x</mi></mrow></math>

\sinh x \sinh^{- 1 - 1} x

Exponential `<exp/>`

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

<math>
                  
<mrow><mi>exp</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
               
</math>

\exp (x)

Natural Logarithm `<ln/>`

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

<math>
                  
<mrow><mi>ln</mi><mo>⁡</mo><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow>
               
</math>

\ln (a)

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

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

<math>
                  
<mrow><msub><mi>log</mi><mn>3</mn></msub><mi>x</mi></mrow>
               
</math>

\log_{3} x

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

<math>
                  
<mrow><mi>log</mi><mi>x</mi></mrow>
               
</math>

\log x

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

<math>
                  <mrow><mrow><msub><mi>log</mi><mn>2</mn></msub><mi>x</mi></mrow><mo>+</mo><mrow><mi>log</mi><mi>y</mi></mrow></mrow>


               
</math>

\log_{2} x + \log y

                  <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>

<math>
                  <mrow><mrow><mi>log</mi><mn>2</mn></mrow><mo>+</mo><mrow><mi>log</mi><mn>10</mn></mrow></mrow>
               
</math>

\log 2 + \log 10

Statistics

Mean `<mean/>`

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

<math>
                  
<mrow><mi>mean</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></mrow>
               
</math>

mean (3, 4, 3, 7, 4)

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

<math>
                  
<mrow><mi>mean</mi><mo>⁡</mo><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow>
               
</math>

mean (X)

Standard Deviation `<sdev/>`

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

<math>
                  
<mrow><mi>sdev</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow>
               
</math>

sdev (3, 4, 2, 2)

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

<math>
                  
<mrow><mi>sdev</mi><mo>⁡</mo><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow>
               
</math>

sdev (X)

Variance `<variance/>`

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

<math>
                  
<mrow><mi>variance</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow>
               
</math>

variance (3, 4, 2, 2)

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

<math>
                  
<mrow><mi>variance</mi><mo>⁡</mo><mrow><mo>(</mo><mi> X</mi><mo>)</mo></mrow></mrow>
               
</math>

variance (X)

Median `<median/>`

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

<math>
                  
<mrow><mi>median</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow>
               
</math>

median (3, 4, 2, 2)

Mode `<mode/>`

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

<math>
                  
<mrow><mi>mode</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow>
               
</math>

mode (3, 4, 2, 2)

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>

<math>
                  
<mrow><mi>moment</mi><mo>⁡</mo><mrow><mo>(</mo><momentabout><mi>mean</mi></momentabout><mo>,</mo><mn>6</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></mrow>
               
</math>

moment (mean, 6, 4, 2, 2, 5)

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

<math>
                  
<mrow><mi>moment</mi><mo>⁡</mo><mrow><mo>(</mo><momentabout><mi>p</mi></momentabout><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow>
               
</math>

moment (p, X)

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

<math>
                  
<mrow><mi>moment</mi><mo>⁡</mo><mrow><mo>(</mo><momentabout><mi>p</mi></momentabout><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow>
               
</math>

moment (p, X)

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

<math>
                  
<mrow><mi>moment</mi><mo>⁡</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>p</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow>
               
</math>

moment (3, p, X)

Linear Algebra

Vector `<vector>`

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

<math>
                  
<vector>
  <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow>
  <mn>3</mn>
  <mn>7</mn>
</vector>
               
</math>

x + y 37

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>

<math>
                  
<mrow><mo>[</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>|</mo><msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><msup><mi>i</mi><mi>j</mi></msup><mo>;</mo><mrow><mrow><mrow><mi>i</mi><mo>∈</mo><mo>[</mo><mrow><mi>1</mi><mo>,</mo><mi>5</mi></mrow><mo>]</mo></mrow><mo>∧</mo><mrow><mi>j</mi><mo>∈</mo><mo>[</mo><mrow><mi>5</mi><mo>,</mo><mi>9</mi></mrow><mo>]</mo></mrow></mrow></mrow><mo>]</mo></mrow>
               
</math>

[m_{i, j} | m_{i, j} = i^{j}; i \in [1, 5] \land j \in [5, 9]]

Matrix row `<matrixrow>`

Determinant `<determinant/>`

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

<math>
                  
<mrow><mi>determinant</mi><mo>⁡</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow>
               
</math>

determinant (A)

Transpose `<transpose/>`

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

<math>
                  
<mrow><mi>transpose</mi><mo>⁡</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow>
               
</math>

transpose (A)

Selector `<selector/>`

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

<math>
                  
<msub><mi>V</mi><mrow><mn>1</mn></mrow></msub>
               
</math>

V_{1}

                  
<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>

<math>
                  
<mrow><msub><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></mrow>
               
</math>

{(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix})}_{1} = (\begin{matrix} 1 & 2 \end{matrix})

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>

<math>
                  
<mrow><mrow><mi> A </mi><mo>×</mo><mi> B </mi></mrow><mo>=</mo><mrow><mi>a</mi><mo>⁢</mo><mi>b</mi><mo>⁢</mo><mrow><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mi> N </mi></mrow></mrow>
               
</math>

A \times B = a b \sin (θ) N

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>

<math>
                  
<mrow><mrow><mi>A</mi><mo>.</mo><mi>B</mi></mrow><mo>=</mo><mrow><mi>a</mi><mo>⁢</mo><mi>b</mi><mo>⁢</mo><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mrow></mrow></mrow>
               
</math>

A . B = a b \cos (θ)

Outer product `<outerproduct/>`

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

<math>
                  
<mrow><mi>A</mi><mo>⊗</mo><mi>B</mi></mrow>
               
</math>

A \otimes B

Constant and Symbol Elements

integers `<integers/>`

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

<math>
                  
<mrow><mrow></mrow><mo>∈</mo><mi>ℤ</mi></mrow>
               
</math>

\in ℤ

reals `<reals/>`

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

<math>
                  
<mrow><mrow><mi>real</mi><mo>⁡</mo><mrow><mo>(</mo><mrow></mrow><mo>)</mo></mrow></mrow><mo>∈</mo><mi>ℝ</mi></mrow>
               
</math>

real () \in ℝ

Rational Numbers `<rationals/>`

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

<math>
                  
<mrow><mfrac><mrow><cn> 22 </cn><mn> 22 </mn></mrow><mrow><cn>7</cn><mn>7</mn></mrow></mfrac><mo>∈</mo><mi>ℚ</mi></mrow>
               
</math>

\frac{2222}{77} \in ℚ

Natural Numbers `<naturalnumbers/>`

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

<math>
                  
<mrow><mrow></mrow><mo>∈</mo><mi>ℕ</mi></mrow>
               
</math>

\in ℕ

complexes `<complexes/>`

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

<math>
                  
<mrow><mrow><mrow><cn>17</cn><mn>17</mn></mrow><mo>+</mo><mrow><cn>29</cn><mn>29</mn></mrow><mo>⁢</mo><mi>i</mi></mrow><mo>∈</mo><mi>ℂ</mi></mrow>
               
</math>

1717 + 2929 i \in ℂ

primes `<primes/>`

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

<math>
                  
<mrow><mrow></mrow><mo>∈</mo><mi>ℙ</mi></mrow>
               
</math>

\in ℙ

Exponential e `<exponentiale/>`

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

<math>
                  
<mrow><mrow><mi>ln</mi><mo>⁡</mo><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>1</mn></mrow>
               
</math>

\ln (e) = 1

Imaginary i `<imaginaryi/>`

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

<math>
                  
<mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mn>-1</mn></mrow>
               
</math>

i^{2} = -1

Not A Number `<notanumber/>`

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

<math>
                  
<mrow><mrow><mn>0</mn><mo>/</mo><mn>0</mn></mrow><mo>=</mo><mi>NaN</mi></mrow>
               
</math>

0 / 0 = NaN

True `<true/>`

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

<math>
                  
<mrow><mrow><mi>true</mi><mo>∨</mo><mi>P</mi></mrow><mo>=</mo><mi>true</mi></mrow>
               
</math>

true \lor P = true

False `<false/>`

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

<math>
                  
<mrow><mrow><mi>false</mi><mo>∧</mo><mi>P</mi></mrow><mo>=</mo><mi>false</mi></mrow>
               
</math>

false \land P = false

Empty Set `<emptyset/>`

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

<math>
                  
<mrow><mi>ℤ</mi><mo>≠</mo><mi>∅</mi></mrow>
               
</math>

ℤ \neq \emptyset

pi `<pi/>`

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

<math>
                  
<mrow><mi>π</mi><mo>≃</mo><mfrac><mrow><cn>22</cn><mn>22</mn></mrow><mrow><cn>7</cn><mn>7</mn></mrow></mfrac></mrow>
               
</math>

π ≃ \frac{2222}{77}

Euler gamma `<eulergamma/>`

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

<math>
                  
<mrow><mi>γ</mi><mo>≃</mo><mn>0.5772156649</mn></mrow>
               
</math>

γ ≃ 0.5772156649

infinity `<infinity/>`

                  <infinity/>

<math>
                  <mi>∞</mi>
               
</math>

\infty

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/>`