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SandBoxTensorAlgebra2
  
last edited 4 years ago by pagani

Edit detail for SandBoxTensorAlgebra2 revision 1 of 3
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Editor: pagani
Time: 2020/02/01 17:40:20 GMT+0
Note: 
changed:
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\begin{spad}
)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
      
   
\end{spad}

\begin{axiom}
B:=OrderedVariableList [e[i] for i in 1..5]
e:=enumerate()$B

R:=Expression Integer
R has CommutativeRing

M:=FreeModule(R, B)

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

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


v1:=x*b.1+y*b.2-z*b.3
v2:=y^n*b.1-cos(x)*b.2
v3:=sin(x+y+z)*b.3

t0:=exp(-x-y-z)*1$TA
t1:=b.1*b.2*b.3*b.4+v1

p1:=product(v1,v2)
p2:=product(product(v1,v3),t1)
p3:=tan(x)*1$TA

s1:=p1+p2+p3

---
degree(v1*v2*v3*t1)
listOfTerms (v1*v1*v2*v3)
degree (1$TA)

v1+1$TA
listOfTerms %

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

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

\end{axiom}

spad
)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
   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: (OrderedAbelianMonoidSup)
(time taken in buildFunctor:  20)

;;;     ***       |TensorAlgebra| REDEFINED

;;;     ***       |TensorAlgebra| REDEFINED
Time: 0.03 SEC.

   Cumulative Statistics for Constructor TensorAlgebra
      Time: 0.05 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 01 FEB 2020 05:40:20 PM):

; /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG.fasl written
; compilation finished in 0:00:00.030
------------------------------------------------------------------------
   TensorAlgebra is now explicitly exposed in frame initial 
   TensorAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TENSALG.NRLIB/TENSALG

fricas
B:=OrderedVariableList [e[i] for i in 1..5]
\label{eq1}\hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] , e [ 4 ] , e [ 5 ] ])(1)
Type: Type
fricas
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]]))
fricas
R:=Expression Integer
\label{eq3}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })(3)
Type: Type
fricas
R has CommutativeRing
\label{eq4} \mbox{\rm true} (4)
Type: Boolean
fricas
M:=FreeModule(R, B)
\label{eq5}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] , e [ 4 ] , e [ 5 ] ]))(5)
Type: Type
fricas
-- This is the object of interest
TA:=TensorAlgebra(M,R,B)
\label{eq6}\hbox{\axiomType{TensorAlgebra}\ } (\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] , e [ 4 ] , e [ 5 ] ])) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{OrderedVariableList}\ } ([ e [ 1 ] , e [ 2 ] , e [ 3 ] , e [ 4 ] , e [ 5 ] ]))(6)
Type: Type
fricas
-- 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]])))
fricas
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]]))
fricas
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]]))
fricas
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]]))
fricas
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]]))
fricas
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]]))
fricas
p1:=product(v1,v2)
\label{eq13}\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}}}} (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]]))
fricas
p2:=product(product(v1,v3),t1)
\label{eq14}\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}}}}} (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]]))
fricas
p3:=tan(x)*1$TA
\label{eq15}{\tan \left({x}\right)}\ (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]]))
fricas
s1:=p1+p2+p3
\label{eq16}\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)}\ } (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]]))
fricas
---
degree(v1*v2*v3*t1)
\label{eq17}7(17)
Type: PositiveInteger?
fricas
listOfTerms (v1*v1*v2*v3)
\label{eq18}\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] (18)
Type: List(Record(k: FreeMonoid?(OrderedVariableList?([e[1],e[2],e[3],e[4],e[5]])),c: Expression(Integer)))
fricas
degree (1$TA)
\label{eq19}0(19)
Type: NonNegativeInteger?
fricas
v1+1$TA
\label{eq20}{x \ {e_{1}}}+{y \ {e_{2}}}-{z \ {e_{3}}}+ 1(20)
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]]))
fricas
listOfTerms %
\label{eq21}\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] (21)
Type: List(Record(k: FreeMonoid?(OrderedVariableList?([e[1],e[2],e[3],e[4],e[5]])),c: Expression(Integer)))