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)abbrev package FORMAN FormalManipulations
++ Author: Kurt Pagani
++ Date Created: Tue Jun 19 19:09:16 CEST 2018
++ License: BSD
++ References:
++ Description:
++   Interactive Computer Manipulation of Formal Sums
++   (Thesis format: Monograph) by Nivedita Patil
++   Graduate Program in Computer Science
++     A thesis submitted in partial fulfillment of the requirements for the 
++     degree of Master of Science
++     The School of Graduate and Postdoctoral Studies
++     The University of Western Ontario
++     London, Ontario, Canada, (C) Nivedita Patil 2010
++   http://www.csd.uwo.ca/~watt/home/students/theses/NPatil2010-msc.pdf
++
++ Abstract:
++ The goal of the thesis is to explore how computer algebra systems can 
++ be augmented to allow user-guided transformation and simplification of 
++ expressions involving symbolic summation. Mathematical expressions 
++ represented as trees are one of the data objects of computer algebra 
++ systems. By accessing and manipulating these data objects we can simplify 
++ and transform expressions involving symbolic summation. To choose what 
++ transformations to be performed is under the discretion of the user. We
++ present a conceptual framework to perform transformations on expressions 
++ involving symbolic summation. This is done by creating a set of library 
++ functions for interactive manipulation of formal sums. We base our design 
++ on the properties of summation. This idea is also extended to other 
++ associative operators such as product and definite integral.
++
++ Keywords: 
++   Formal Sums, Computer Algebra System, Associative Operators.
++
FormalManipulations(R) : Exports == Implementation where

  R:Join(Comparable,IntegralDomain)
  X ==> Expression R
  PI ==> PositiveInteger
  UK ==> Union(Kernel X,"failed")
  KX ==> Kernel X
  SX ==> Segment X
  SBX ==> SegmentBinding X
  --PINT   ==> Polynomial(Integer)
  --SBPINT ==> SegmentBinding(PINT)

  Exports ==  with

    splitOp : (X,X) -> X
    multiplyOp : (X,X) -> X
    splitFunctions : X -> X 
    takeNTermsHigh : (X,X) -> X
    takeNTermsLow : (X,X) -> X
    shiftNTerms : (X,X) -> X
    splitTermsEvenOdd : X -> X
    reverseOrder : X -> X 
    --splitSum : (X,X) -> X
    --coerce : % -> OutputForm

  Implementation ==  add 

    kernelHandler(x) ==>  -- macro
      mk:UK:=mainKernel(x)
      if mk case KX then K:KX:=mk else return x
      A:List X:=argument K 
      v:Symbol:=retract(A.3)@Symbol

    splitSum(x:X,s:X):X ==
      kernelHandler(x)
      S1:=summation(subst(A.1,A.2=A.3),equation(v,segment(A.4,s)$SX)$SBX)
      S2:=summation(subst(A.1,A.2=A.3),equation(v,segment(s+1,A.5)$SX)$SBX)
      S1+S2

    multiplySum(x:X,s:X):X ==
      kernelHandler(x) 
      summation(s*subst(A.1,A.2=A.3),equation(v,segment(A.4,A.5)$SX)$SBX)

    splitFunctionSum(x:X):X ==
      kernelHandler(x)
      B:Union(List(X),"failed"):=isPlus(A.1)
      if B case List(X) then
        r:SBX:=equation(v,segment(A.4,A.5)$SX)$SBX
        reduce(_+,[summation(subst(B.j,A.2=A.3),r) for j in 1..#B]) 
      else
        x

    takeNTermsHighSum(x:X,n:X):X ==
      kernelHandler(x) 
      summation(subst(A.1,A.2=A.3),equation(v,segment(A.5-n+1,A.5)$SX)$SBX)

    takeNTermsLowSum(x:X,n:X):X ==
      kernelHandler(x) 
      summation(subst(A.1,A.2=A.3),equation(v,segment(A.4,A.4+n-1)$SX)$SBX) 

    shiftNTermsSum(x:X,n:X):X ==
      kernelHandler(x) 
      summation(subst(A.1,A.2=A.3-n),equation(v,segment(A.4+n,A.5+n)$SX)$SBX) 

    sfloor(x:X):X ==
      f:BasicOperator := operator('floor,1)
      if R has RetractableTo(Integer) then
        rx:Union(Fraction Integer, "failed"):=retractIfCan(x)
        if rx case Fraction(Integer) then return floor(rx)::X
      f(x)

    sceiling(x:X):X ==
      c:BasicOperator := operator('ceiling,1)
      if R has RetractableTo(Integer) then
        rx:Union(Fraction Integer, "failed"):=retractIfCan(x)
        if rx case Fraction(Integer) then return ceiling(rx)::X
      c(x)

    startsplit(x:X):X ==
      ss:BasicOperator := operator('startSplit,1)
      if R has RetractableTo(Integer) then
        rx:Union(Integer, "failed"):=retractIfCan(x)
        if rx case Integer then
          q:Fraction Integer:=rx/2
          if odd? rx then
            return ceiling(q)::X
          else
            return (1+ceiling(q))::X
      ss(x)

    splitTermsEvenOddSum(x:X):X ==
      kernelHandler(x)
      s1:SX:=segment(sceiling(A.4/2::X),sfloor(A.5/2::X))$SX
      s2:SX:=segment(startsplit(A.4), sceiling(A.5/2::X))$SX
      S1:=summation(subst(A.1,A.2=2*A.3),equation(v,s1)$SBX)
      S2:=summation(subst(A.1,A.2=2*A.3-1),equation(v,s2)$SBX)
      S1+S2
      -- TODO: complete this
      -- DONE: see ref

    reverseOrderSum(x:X):X ==
      kernelHandler(x)
      s:SX:=segment(A.4-A.4,A.5-A.4)$SX
      summation(subst(A.1,A.2=A.5-A.3),equation(v,s)$SBX)

    --
    -- Exports
    --
    splitOp(x:X,s:X):X ==
      is?(x,'%defsum) => splitSum(x,s)
      return x

    multiplyOp(x:X,s:X):X ==
      is?(x,'%defsum) => multiplySum(x,s)
      return x

    splitFunctions(x:X):X ==
      is?(x,'%defsum) => splitFunctionSum(x)
      return x      

    takeNTermsHigh(x:X,n:X):X ==
      is?(x,'%defsum) => takeNTermsHighSum(x,n)
      return x  

    takeNTermsLow(x:X,n:X):X ==
      is?(x,'%defsum) => takeNTermsLowSum(x,n)
      return x  

    shiftNTerms(x:X,n:X):X ==
      is?(x,'%defsum) => shiftNTermsSum(x,n)
      return x        

    splitTermsEvenOdd(x:X):X ==
      is?(x,'%defsum) => splitTermsEvenOddSum(x)
      return x       

    reverseOrder(x:X):X ==
      is?(x,'%defsum) => reverseOrderSum(x)
      return x      

--  X := operator 'X ; T:=summation(X(s),s=a..b)
--  P:=summation(X(s)*x^s,s=0..b)
--  f(n)==eval(P,b=n)
--  D(P,x,2)
--  name mainKernel(P) ->> %defsum

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5640122448356627221-25px001.spad
      using old system compiler.
   FORMAN abbreviates package FormalManipulations 
------------------------------------------------------------------------
   initializing NRLIB FORMAN for FormalManipulations 
   compiling into NRLIB FORMAN 
****** Domain: R already in scope
   processing macro definition kernelHandler x ==> SEQ(:=(mk: Union(Kernel Expression R,failed),mainKernel x),IF(case(mk,Kernel Expression R),:=(K: Kernel Expression R,mk),return x),:=(A: List Expression R,argument K),exit(1,:=(v: Symbol,@(retract A 3,Symbol)))) 
   compiling local splitSum : (Expression R,Expression R) -> Expression R
Time: 0.06 SEC.

   compiling local multiplySum : (Expression R,Expression R) -> Expression R
Time: 0.01 SEC.

   compiling local splitFunctionSum : Expression R -> Expression R
Time: 0.02 SEC.

   compiling local takeNTermsHighSum : (Expression R,Expression R) -> Expression R
Time: 0.01 SEC.

   compiling local takeNTermsLowSum : (Expression R,Expression R) -> Expression R
Time: 0.01 SEC.

   compiling local shiftNTermsSum : (Expression R,Expression R) -> Expression R
Time: 0.01 SEC.

   compiling local sfloor : Expression R -> Expression R
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
Time: 0.01 SEC.

   compiling local sceiling : Expression R -> Expression R
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
Time: 0.01 SEC.

   compiling local startsplit : Expression R -> Expression R
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
Time: 0.01 SEC.

   compiling local splitTermsEvenOddSum : Expression R -> Expression R
Time: 0.06 SEC.

   compiling local reverseOrderSum : Expression R -> Expression R
Time: 0.02 SEC.

   compiling exported splitOp : (Expression R,Expression R) -> Expression R
Time: 0 SEC.

   compiling exported multiplyOp : (Expression R,Expression R) -> Expression R
Time: 0 SEC.

   compiling exported splitFunctions : Expression R -> Expression R
Time: 0 SEC.

   compiling exported takeNTermsHigh : (Expression R,Expression R) -> Expression R
Time: 0 SEC.

   compiling exported takeNTermsLow : (Expression R,Expression R) -> Expression R
Time: 0 SEC.

   compiling exported shiftNTerms : (Expression R,Expression R) -> Expression R
Time: 0 SEC.

   compiling exported splitTermsEvenOdd : Expression R -> Expression R
Time: 0 SEC.

   compiling exported reverseOrder : Expression R -> Expression R
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |FormalManipulations| REDEFINED

;;;     ***       |FormalManipulations| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor FormalManipulations
      Time: 0.24 seconds

   finalizing NRLIB FORMAN 
   Processing FormalManipulations for Browser database:
--------constructor---------
--->-->FormalManipulations((splitOp ((Expression R) (Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((multiplyOp ((Expression R) (Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((splitFunctions ((Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((takeNTermsHigh ((Expression R) (Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((takeNTermsLow ((Expression R) (Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((shiftNTerms ((Expression R) (Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((splitTermsEvenOdd ((Expression R) (Expression R)))): Not documented!!!!
--->-->FormalManipulations((reverseOrder ((Expression R) (Expression R)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/FORMAN.NRLIB/FORMAN.lsp" (written 16 JUL 2018 09:33:07 PM):

; /var/aw/var/LatexWiki/FORMAN.NRLIB/FORMAN.fasl written
; compilation finished in 0:00:00.136
------------------------------------------------------------------------
   FormalManipulations is now explicitly exposed in frame initial 
   FormalManipulations will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/FORMAN.NRLIB/FORMAN

Tests (changes fricas 1.2 -> 1.3.+)

fricas

X := operator 'X

$\label{eq1}X$

(1)

Type: BasicOperator?

fricas

T:=summation(X(s),s=a..b)

$\label{eq2}\sum_{ \displaystyle {s = a}}^{ \displaystyle b}{X \left({s}\right)}$

(2)

Type: Expression(Integer)

fricas

T2:=multiplyOp(T,x^s)

$\label{eq3}\sum_{ \displaystyle {s = a}}^{ \displaystyle b}{{X \left({s}\right)}\ {{x}^{s}}}$

(3)

Type: Expression(Integer)

fricas

T3:= D(T2,x)

$\label{eq4}\sum_{ \displaystyle {s = a}}^{ \displaystyle b}{s \ {X \left({s}\right)}\ {{x}^{s - 1}}}$

(4)

Type: Expression(Integer)

fricas

T4:=takeNTermsHigh(T3,m)

$\label{eq5}\sum_{ \displaystyle {s ={- m + b + 1}}}^{ \displaystyle b}{s \ {X \left({s}\right)}\ {{x}^{s - 1}}}$

(5)

Type: Expression(Integer)

fricas

T5:=shiftNTerms(T4,m)

$\label{eq6}\sum_{ \displaystyle {s ={b + 1}}}^{ \displaystyle {m + b}}{{\left(s - m \right)}\ {X \left({s - m}\right)}\ {{x}^{s - m - 1}}}$

(6)

Type: Expression(Integer)

fricas

T6:=splitOp(T5,k)

$\label{eq7}\begin{array}{@{}l} \displaystyle {\sum_{ \displaystyle {s ={b + 1}}}^{ \displaystyle k}{{\left(s - m \right)}\ {X \left({s - m}\right)}\ {{x}^{s - m - 1}}}}+ \ \ \displaystyle {\sum_{ \displaystyle {s ={k + 1}}}^{ \displaystyle {m + b}}{{\left(s - m \right)}\ {X \left({s - m}\right)}\ {{x}^{s - m - 1}}}}$

(7)

Type: Expression(Integer)

OK now; missing functions: CombineSplitOp?,GetCoefficient?,TakeTermsValue?,CombineSplitFunctions?,GetNonVariantsOutside?,PutMultiplicandsInside?

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