MathAction SandBoxTensorAlgebra2

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  - SandBoxTensorAlgebra2 <-- You are here.

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(1) -> <spad>

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)abbrev domain TENSALG TensorAlgebra
TensorAlgebra(M,R,B) : Exports == Implementation where

  M:FreeModuleCategory(R, B)
  R:CommutativeRing
  B:OrderedSet

  OF     ==> OutputForm
  NNI    ==> NonNegativeInteger
  FMB    ==> FreeMonoid B
  CTOF   ==> CoercibleTo OutputForm
  FMCRB  ==> FreeModuleCategory(R,FMB)
  GRALR  ==> GradedAlgebra(R,NNI)
  XFABR  ==> XFreeAlgebra(B,R)
  XDPBR  ==> XDistributedPolynomial(B,R)
  TERM   ==> Record(k:FMB,c:R)  

  Exports == Join(FMCRB, XFABR, GRALR) with

    coerce : B -> %
    convert : FMB -> OutputForm

  Implementation ==  XDPBR add 

    Rep := XDPBR

    product(x,y) == x*y  -- GradedAlgebra, pro forma

    convert(x:FMB):OutputForm ==
      x=1$FMB => empty()$OF 
      length(x)$FMB = 1 => x::OF
      length(x)$FMB = 2 => tensor(first(x)::OF,rest(x)::OF)$OF
      tensor(first(x)::OF, convert(rest x))

    coerce(x:%):OutputForm ==
      zero? x => empty()$OF
      x=1$% => outputForm(1)$OF
      c:R:=leadingCoefficient(x)
      if c=1 then cof:=empty()$OF else cof:=c::OF 
      kof:OF:=cof * convert(leadingSupport(x))
      zero? reductum(x) => kof
      kof + reductum(x)::OF</spad>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/579490114000940455-25px001.spad
      using old system compiler.
   TENSALG abbreviates domain TensorAlgebra 
------------------------------------------------------------------------
   initializing NRLIB TENSALG for TensorAlgebra 
   compiling into NRLIB TENSALG 
   compiling exported product : (%,%) -> %
Time: 0.02 SEC.

   compiling exported convert : FreeMonoid B -> OutputForm
Time: 0 SEC.

   compiling exported coerce : % -> OutputForm
Time: 0 SEC.

****** Domain: R already in scope
augmenting R: (Comparable)
(time taken in buildFunctor:  2378)

;;;     ***       |TensorAlgebra| REDEFINED

;;;     ***       |TensorAlgebra| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor TensorAlgebra
      Time: 0.03 seconds

--------------non extending category----------------------
.. TensorAlgebra(#1,#2,#3) of cat 
(|Join| (|FreeModuleCategory| |#2| (|FreeMonoid| |#3|))
        (|XFreeAlgebra| |#3| |#2|)
        (|GradedAlgebra| |#2| (|NonNegativeInteger|))
        (CATEGORY |domain| (SIGNATURE |coerce| (% |#3|))
         (SIGNATURE |convert| ((|OutputForm|) (|FreeMonoid| |#3|)))))   has no 
(|XPolynomialsCat| |#3| |#2|)    finalizing NRLIB TENSALG 
   Processing TensorAlgebra for Browser database:
--->-->TensorAlgebra(constructor): Not documented!!!!
--->-->TensorAlgebra((coerce (% B))): Not documented!!!!
--->-->TensorAlgebra((convert ((OutputForm) (FreeMonoid B)))): Not documented!!!!
--->-->TensorAlgebra(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.lsp" (written 07 DEC 2024 06:18:53 AM):

; wrote /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.fasl
; compilation finished in 0:00:00.024
------------------------------------------------------------------------
   TensorAlgebra is now explicitly exposed in frame initial 
   TensorAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG

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B:=OrderedVariableList [e[i] for i in 1..5]

$\label{eq1}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]}\right)$

(1)

Type: Type

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e:=enumerate()$B

$\label{eq2}\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]$

(2)

Type: List(OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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R:=Expression Integer

$\label{eq3}\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)$

(3)

Type: Type

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R has CommutativeRing

$\label{eq4} \mbox{\rm true}$

(4)

Type: Boolean

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M:=FreeModule(R, B)

$\label{eq5}\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]}\right)}}\right)$

(5)

Type: Type

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-- This is the object of interest
TA:=TensorAlgebra(M,R,B)

$\label{eq6}\hbox{\axiomType{TensorAlgebra}\ } \left({{\hbox{\axiomType{FreeModule}\ } \left({{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]}\right)}}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]}\right)}}\right)$

(6)

Type: Type

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-- coerce basis to TA
b:=[a::TA for a in e]

$\label{eq7}\left[{\ {e_{1}}}, \:{\ {e_{2}}}, \:{\ {e_{3}}}, \:{\ {e_{4}}}, \:{\ {e_{5}}}\right]$

(7)

Type: List(TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])))

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v1:=x*b.1+y*b.2-z*b.3

$\label{eq8}{x \ {e_{1}}}+{y \ {e_{2}}}-{z \ {e_{3}}}$

(8)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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v2:=y^n*b.1-cos(x)*b.2

$\label{eq9}{{{y}^{n}}\ {e_{1}}}-{{\cos \left({x}\right)}\ {e_{2}}}$

(9)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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v3:=sin(x+y+z)*b.3

$\label{eq10}{\sin \left({z + y + x}\right)}\ {e_{3}}$

(10)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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t0:=exp(-x-y-z)*1$TA

$\label{eq11}{{e}^{- z - y - x}}\$

(11)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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t1:=b.1*b.2*b.3*b.4+v1

$\label{eq12}{\ {{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}+{x \ {e_{1}}}+{y \ {e_{2}}}-{z \ {e_{3}}}$

(12)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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t2:=v1*v2+v1*v3*b.5+t0*b.4*b.3

$\label{eq13}\begin{array}{@{}l} \displaystyle {x \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{5}}}}}+ \ \ \displaystyle {y \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{5}}}}}- \ \ \displaystyle {z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{5}}}}}+{x \ {{y}^{n}}\ {{e_{1}}\otimes{e_{1}}}}- \ \ \displaystyle {x \ {\cos \left({x}\right)}\ {{e_{1}}\otimes{e_{2}}}}+{y \ {{y}^{n}}\ {{e_{2}}\otimes{e_{1}}}}- \ \ \displaystyle {y \ {\cos \left({x}\right)}\ {{e_{2}}\otimes{e_{2}}}}-{z \ {{y}^{n}}\ {{e_{3}}\otimes{e_{1}}}}+ \ \ \displaystyle {z \ {\cos \left({x}\right)}\ {{e_{3}}\otimes{e_{2}}}}+{{{e}^{- z - y - x}}\ {{e_{4}}\otimes{e_{3}}}}$

(13)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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-- product(x,y) same as x*y
p1:=product(v1,v2)

$\label{eq14}\begin{array}{@{}l} \displaystyle {x \ {{y}^{n}}\ {{e_{1}}\otimes{e_{1}}}}-{x \ {\cos \left({x}\right)}\ {{e_{1}}\otimes{e_{2}}}}+{y \ {{y}^{n}}\ {{e_{2}}\otimes{e_{1}}}}- \ \ \displaystyle {y \ {\cos \left({x}\right)}\ {{e_{2}}\otimes{e_{2}}}}-{z \ {{y}^{n}}\ {{e_{3}}\otimes{e_{1}}}}+ \ \ \displaystyle {z \ {\cos \left({x}\right)}\ {{e_{3}}\otimes{e_{2}}}}$

(14)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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p2:=product(product(v1,v3),t1)

$\label{eq15}\begin{array}{@{}l} \displaystyle {x \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}+ \ \ \displaystyle {y \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}- \ \ \displaystyle {z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}+ \ \ \displaystyle {{{x}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{1}}}}}+ \ \ \displaystyle {x \ y \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{2}}}}}- \ \ \displaystyle {x \ z \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{3}}}}}+ \ \ \displaystyle {x \ y \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{1}}}}}+ \ \ \displaystyle {{{y}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{2}}}}}- \ \ \displaystyle {y \ z \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{3}}}}}- \ \ \displaystyle {x \ z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{1}}}}}- \ \ \displaystyle {y \ z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{2}}}}}+ \ \ \displaystyle {{{z}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{3}}}}}$

(15)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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p3:=tan(x)*1$TA

$\label{eq16}{\tan \left({x}\right)}\$

(16)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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s1:=p1+p2+p3

$\label{eq17}\begin{array}{@{}l} \displaystyle {x \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}+ \ \ \displaystyle {y \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}- \ \ \displaystyle {z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{{e_{1}}\otimes{{e_{2}}\otimes{{e_{3}}\otimes{e_{4}}}}}}}}+ \ \ \displaystyle {{{x}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{1}}}}}+ \ \ \displaystyle {x \ y \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{2}}}}}- \ \ \displaystyle {x \ z \ {\sin \left({z + y + x}\right)}\ {{e_{1}}\otimes{{e_{3}}\otimes{e_{3}}}}}+ \ \ \displaystyle {x \ y \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{1}}}}}+ \ \ \displaystyle {{{y}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{2}}}}}- \ \ \displaystyle {y \ z \ {\sin \left({z + y + x}\right)}\ {{e_{2}}\otimes{{e_{3}}\otimes{e_{3}}}}}- \ \ \displaystyle {x \ z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{1}}}}}- \ \ \displaystyle {y \ z \ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{2}}}}}+ \ \ \displaystyle {{{z}^{2}}\ {\sin \left({z + y + x}\right)}\ {{e_{3}}\otimes{{e_{3}}\otimes{e_{3}}}}}+{x \ {{y}^{n}}\ {{e_{1}}\otimes{e_{1}}}}- \ \ \displaystyle {x \ {\cos \left({x}\right)}\ {{e_{1}}\otimes{e_{2}}}}+{y \ {{y}^{n}}\ {{e_{2}}\otimes{e_{1}}}}- \ \ \displaystyle {y \ {\cos \left({x}\right)}\ {{e_{2}}\otimes{e_{2}}}}-{z \ {{y}^{n}}\ {{e_{3}}\otimes{e_{1}}}}+ \ \ \displaystyle {z \ {\cos \left({x}\right)}\ {{e_{3}}\otimes{e_{2}}}}+{{\tan \left({x}\right)}\ }$

(17)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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---
degree(v1*v2*v3*t1)

$\label{eq18}7$

(18)

Type: PositiveInteger?

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listOfTerms (v1*v1*v2*v3)

$\label{eq19}\begin{array}{@{}l} \displaystyle \left[{\left[{k ={{{e_{1}}^{3}}\ {e_{3}}}}, \:{c ={{{x}^{2}}\ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{{e_{1}}^{2}}\ {e_{2}}\ {e_{3}}}}, \:{c = -{{{x}^{2}}\ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{3}}}}, \:{c ={x \ y \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{1}}\ {{e_{2}}^{2}}\ {e_{3}}}}, \:{c = -{x \ y \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{1}}\ {e_{3}}\ {e_{1}}\ {e_{3}}}}, \:{c = -{x \ z \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{1}}\ {e_{3}}\ {e_{2}}\ {e_{3}}}}, \:{c ={x \ z \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{2}}\ {{e_{1}}^{2}}\ {e_{3}}}}, \:{c ={x \ y \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{2}}\ {e_{1}}\ {e_{2}}\ {e_{3}}}}, \:{c = -{x \ y \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{{e_{2}}^{2}}\ {e_{1}}\ {e_{3}}}}, \:{c ={{{y}^{2}}\ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{{e_{2}}^{3}}\ {e_{3}}}}, \:{c = -{{{y}^{2}}\ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{2}}\ {e_{3}}\ {e_{1}}\ {e_{3}}}}, \:{c = -{y \ z \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{2}}\ {e_{3}}\ {e_{2}}\ {e_{3}}}}, \:{c ={y \ z \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{3}}\ {{e_{1}}^{2}}\ {e_{3}}}}, \:{c = -{x \ z \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{3}}\ {e_{1}}\ {e_{2}}\ {e_{3}}}}, \:{c ={x \ z \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{3}}\ {e_{2}}\ {e_{1}}\ {e_{3}}}}, \:{c = -{y \ z \ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{e_{3}}\ {{e_{2}}^{2}}\ {e_{3}}}}, \:{c ={y \ z \ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{{e_{3}}^{2}}\ {e_{1}}\ {e_{3}}}}, \:{c ={{{z}^{2}}\ {\sin \left({z + y + x}\right)}\ {{y}^{n}}}}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k ={{{e_{3}}^{2}}\ {e_{2}}\ {e_{3}}}}, \:{c = -{{{z}^{2}}\ {\cos \left({x}\right)}\ {\sin \left({z + y + x}\right)}}}\right]}\right]$

(19)

Type: List(Record(k: FreeMonoid(OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),c: Expression(Integer)))

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degree (1$TA)

$\label{eq20}0$

(20)

Type: NonNegativeInteger?

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v1+1$TA

$\label{eq21}{x \ {e_{1}}}+{y \ {e_{2}}}-{z \ {e_{3}}}+ 1$

(21)

Type: TensorAlgebra?(FreeModule(Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),Expression(Integer),OrderedVariableList([e[1],e[2],e[3],e[4],e[5]]))

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listOfTerms %

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{\left[{k ={e_{1}}}, \:{c = x}\right]}, \:{\left[{k ={e_{2}}}, \:{c = y}\right]}, \:{\left[{k ={e_{3}}}, \:{c = - z}\right]}, \: \right. \ \ \displaystyle \left.{\left[{k = 1}, \:{c = 1}\right]}\right]$

(22)

Type: List(Record(k: FreeMonoid(OrderedVariableList([e[1],e[2],e[3],e[4],e[5]])),c: Expression(Integer)))

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-- degree (0$TA) ---> err in XDP ?? 

-- projection to TensorPower(n...) easy : filter degree = n

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)show TENSALG

 TensorAlgebra(M: FreeModuleCategory(R,B),R: CommutativeRing,B: OrderedSet) is a domain constructor
 Abbreviation for TensorAlgebra is TENSALG 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

 ?*? : (%, %) -> %                     ?*? : (Integer, %) -> %
 ?*? : (B, %) -> %                     ?*? : (%, R) -> %
 ?*? : (R, %) -> %                     ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     ?-? : (%, %) -> %
 -? : % -> %                           ?=? : (%, %) -> Boolean
 1 : () -> %                           0 : () -> %
 ?^? : (%, PositiveInteger) -> %       annihilate? : (%, %) -> Boolean
 antiCommutator : (%, %) -> %          associator : (%, %, %) -> %
 coef : (%, FreeMonoid(B)) -> R        coef : (%, %) -> R
 coefficients : % -> List(R)           coerce : Integer -> %
 coerce : R -> %                       coerce : FreeMonoid(B) -> %
 coerce : B -> %                       coerce : % -> OutputForm
 commutator : (%, %) -> %              constant : % -> R
 constant? : % -> Boolean              degree : % -> NonNegativeInteger
 latex : % -> String                   lquo : (%, B) -> %
 lquo : (%, FreeMonoid(B)) -> %        lquo : (%, %) -> %
 map : ((R -> R), %) -> %              mindeg : % -> FreeMonoid(B)
 mirror : % -> %                       monomial? : % -> Boolean
 monomials : % -> List(%)              one? : % -> Boolean
 opposite? : (%, %) -> Boolean         product : (%, %) -> %
 quasiRegular : % -> %                 quasiRegular? : % -> Boolean
 recip : % -> Union(%,"failed")        retract : % -> R
 retract : % -> FreeMonoid(B)          rquo : (%, B) -> %
 rquo : (%, FreeMonoid(B)) -> %        rquo : (%, %) -> %
 sample : () -> %                      varList : % -> List(B)
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (NonNegativeInteger, %) -> %
 ?^? : (%, NonNegativeInteger) -> %
 characteristic : () -> NonNegativeInteger
 coefficient : (%, FreeMonoid(B)) -> R
 construct : List(Record(k: FreeMonoid(B),c: R)) -> %
 constructOrdered : List(Record(k: FreeMonoid(B),c: R)) -> % if FreeMonoid(B) has COMPAR
 convert : FreeMonoid(B) -> OutputForm
 leadingCoefficient : % -> R if FreeMonoid(B) has COMPAR
 leadingMonomial : % -> % if FreeMonoid(B) has COMPAR
 leadingSupport : % -> FreeMonoid(B) if FreeMonoid(B) has COMPAR
 leadingTerm : % -> Record(k: FreeMonoid(B),c: R) if FreeMonoid(B) has COMPAR
 leftPower : (%, PositiveInteger) -> %
 leftPower : (%, NonNegativeInteger) -> %
 leftRecip : % -> Union(%,"failed")
 linearExtend : ((FreeMonoid(B) -> R), %) -> R if R has COMRING
 listOfTerms : % -> List(Record(k: FreeMonoid(B),c: R))
 mindegTerm : % -> Record(k: FreeMonoid(B),c: R)
 monomial : (R, FreeMonoid(B)) -> %
 numberOfMonomials : % -> NonNegativeInteger
 plenaryPower : (%, PositiveInteger) -> % if R has COMRING
 reductum : % -> % if FreeMonoid(B) has COMPAR
 retractIfCan : % -> Union(R,"failed")
 retractIfCan : % -> Union(FreeMonoid(B),"failed")
 rightPower : (%, PositiveInteger) -> %
 rightPower : (%, NonNegativeInteger) -> %
 rightRecip : % -> Union(%,"failed")
 sh : (%, %) -> % if R has COMRING
 sh : (%, NonNegativeInteger) -> % if R has COMRING
 smaller? : (%, %) -> Boolean if R has COMPAR and FreeMonoid(B) has COMPAR
 subtractIfCan : (%, %) -> Union(%,"failed")
 support : % -> List(FreeMonoid(B))

Subject: Be Bold !!

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