path: root/src/algebra/manip.spad.pamphlet

\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra manip.spad}
\author{Manuel Bronstein, Robert Sutor}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package FACTFUNC FactoredFunctions}
<<package FACTFUNC FactoredFunctions>>=
)abbrev package FACTFUNC FactoredFunctions
++ Author: Manuel Bronstein
++ Date Created: 2 Feb 1988
++ Date Last Updated: 25 Jun 1990
++ Description: computes various functions on factored arguments.
-- not visible to the user
FactoredFunctions(M:IntegralDomain): Exports == Implementation where
  N ==> NonNegativeInteger

  Exports ==> with
    nthRoot: (Factored M,N) -> Record(exponent:N,coef:M,radicand:List M)
      ++ nthRoot(f, n) returns \spad{(p, r, [r1,...,rm])} such that
      ++ the nth-root of f is equal to \spad{r * pth-root(r1 * ... * rm)},
      ++ where r1,...,rm are distinct factors of f,
      ++ each of which has an exponent smaller than p in f.
    log : Factored M -> List Record(coef:N, logand:M)
      ++ log(f) returns \spad{[(a1,b1),...,(am,bm)]} such that
      ++ the logarithm of f is equal to \spad{a1*log(b1) + ... + am*log(bm)}.

  Implementation ==> add
    nthRoot(ff, n) ==
      coeff:M       := 1
--      radi:List(M)  := (one? unit ff => empty(); [unit ff])
      radi:List(M)  := (((unit ff) = 1) => empty(); [unit ff])
      lf            := factors ff
      d:N :=
        empty? radi => gcd(concat(n, [t.exponent::N for t in lf]))::N
        1
      n             := n quo d
      for term in lf repeat
        qr    := divide(term.exponent::N quo d, n)
        coeff := coeff * term.factor ** qr.quotient
        not zero?(qr.remainder) =>
          radi := concat_!(radi, term.factor ** qr.remainder)
      [n, coeff, radi]

    log ff ==
      ans := unit ff
      concat([1, unit ff],
             [[term.exponent::N, term.factor] for term in factors ff])

@
\section{package POLYROOT PolynomialRoots}
<<package POLYROOT PolynomialRoots>>=
)abbrev package POLYROOT PolynomialRoots
++ Author: Manuel Bronstein
++ Date Created: 15 July 1988
++ Date Last Updated: 10 November 1993
++ Description: computes n-th roots of quotients of
++ multivariate polynomials
-- not visible to the user
PolynomialRoots(E, V, R, P, F):Exports == Implementation where
  E: OrderedAbelianMonoidSup
  V: OrderedSet
  R: IntegralDomain
  P: PolynomialCategory(R, E, V)
  F: Field with
    coerce: P -> %
    numer : $ -> P
	++ numer(x) \undocumented
    denom : $ -> P
	++ denom(x) \undocumented

  N   ==> NonNegativeInteger
  Z   ==> Integer
  Q   ==> Fraction Z
  REC ==> Record(exponent:N, coef:F, radicand:F)

  Exports ==> with
    rroot: (R, N) -> REC
      ++ rroot(f, n) returns \spad{[m,c,r]} such
      ++ that \spad{f**(1/n) = c * r**(1/m)}.
    qroot : (Q, N) -> REC
      ++ qroot(f, n) returns \spad{[m,c,r]} such
      ++ that \spad{f**(1/n) = c * r**(1/m)}.
    if R has GcdDomain then froot: (F, N) -> REC
      ++ froot(f, n) returns \spad{[m,c,r]} such
      ++ that \spad{f**(1/n) = c * r**(1/m)}.
    nthr: (P, N) -> Record(exponent:N,coef:P,radicand:List P)
	++ nthr(p,n) should be local but conditional

  Implementation ==> add
    import FactoredFunctions Z
    import FactoredFunctions P

    rsplit: List P -> Record(coef:R, poly:P)
    zroot : (Z, N) -> Record(exponent:N, coef:Z, radicand:Z)

    zroot(x, n) ==
--      zero? x or one? x => [1, x, 1]
      zero? x or (x = 1) => [1, x, 1]
      s := nthRoot(squareFree x, n)
      [s.exponent, s.coef, */s.radicand]

    if R has imaginary: () -> R then
      czroot: (Z, N) -> REC

      czroot(x, n) ==
        rec := zroot(x, n)
        rec.exponent = 2 and rec.radicand < 0 =>
          [rec.exponent, rec.coef * imaginary()::P::F, (-rec.radicand)::F]
        [rec.exponent, rec.coef::F, rec.radicand::F]

      qroot(x, n) ==
        sn := czroot(numer x, n)
        sd := czroot(denom x, n)
        m  := lcm(sn.exponent, sd.exponent)::N
        [m, sn.coef / sd.coef,
                    (sn.radicand ** (m quo sn.exponent)) /
                                (sd.radicand ** (m quo sd.exponent))]
    else
      qroot(x, n) ==
        sn := zroot(numer x, n)
        sd := zroot(denom x, n)
        m  := lcm(sn.exponent, sd.exponent)::N
        [m, sn.coef::F / sd.coef::F,
                    (sn.radicand ** (m quo sn.exponent))::F /
                                (sd.radicand ** (m quo sd.exponent))::F]

    if R has RetractableTo Fraction Z then
      rroot(x, n) ==
        (r := retractIfCan(x)@Union(Fraction Z,"failed")) case "failed"
          => [n, 1, x::P::F]
        qroot(r::Q, n)

    else
      if R has RetractableTo Z then
        rroot(x, n) ==
          (r := retractIfCan(x)@Union(Z,"failed")) case "failed"
            => [n, 1, x::P::F]
          qroot(r::Z::Q, n)
      else
        rroot(x, n) == [n, 1, x::P::F]

    rsplit l ==
      r := 1$R
      p := 1$P
      for q in l repeat
        if (u := retractIfCan(q)@Union(R, "failed")) case "failed"
          then p := p * q
          else r := r * u::R
      [r, p]

    if R has GcdDomain then
      if R has RetractableTo Z then
        nthr(x, n) ==
          (r := retractIfCan(x)@Union(Z,"failed")) case "failed"
             => nthRoot(squareFree x, n)
          rec := zroot(r::Z, n)
          [rec.exponent, rec.coef::P, [rec.radicand::P]]
      else nthr(x, n) == nthRoot(squareFree x, n)

      froot(x, n) ==
--        zero? x or one? x => [1, x, 1]
        zero? x or (x = 1) => [1, x, 1]
        sn := nthr(numer x, n)
        sd := nthr(denom x, n)
        pn := rsplit(sn.radicand)
        pd := rsplit(sd.radicand)
        rn := rroot(pn.coef, sn.exponent)
        rd := rroot(pd.coef, sd.exponent)
        m := lcm([rn.exponent, rd.exponent, sn.exponent, sd.exponent])::N
        [m, (sn.coef::F / sd.coef::F) * (rn.coef / rd.coef),
             ((rn.radicand ** (m quo rn.exponent)) /
                    (rd.radicand ** (m quo rd.exponent))) *
                           (pn.poly ** (m quo sn.exponent))::F /
                                    (pd.poly ** (m quo sd.exponent))::F]


@
\section{package ALGMANIP AlgebraicManipulations}
<<package ALGMANIP AlgebraicManipulations>>=
)abbrev package ALGMANIP AlgebraicManipulations
++ Author: Manuel Bronstein
++ Date Created: 28 Mar 1988
++ Date Last Updated: 5 August 1993
++ Description:
++ AlgebraicManipulations provides functions to simplify and expand
++ expressions involving algebraic operators.
++ Keywords: algebraic, manipulation.
AlgebraicManipulations(R, F): Exports == Implementation where
  R : IntegralDomain
  F : Join(Field, ExpressionSpace) with
    numer  : $ -> SparseMultivariatePolynomial(R, Kernel $)
	++ numer(x) \undocumented
    denom  : $ -> SparseMultivariatePolynomial(R, Kernel $)
	++ denom(x) \undocumented
    coerce : SparseMultivariatePolynomial(R, Kernel $) -> $
	++ coerce(x) \undocumented

  N  ==> NonNegativeInteger
  Z  ==> Integer
  OP ==> BasicOperator
  SY ==> Symbol
  K  ==> Kernel F
  P  ==> SparseMultivariatePolynomial(R, K)
  RF ==> Fraction P
  REC ==> Record(ker:List K, exponent: List Z)
  ALGOP ==> "%alg"
  NTHR  ==> "nthRoot"

  Exports ==> with
    rootSplit: F -> F
      ++ rootSplit(f) transforms every radical of the form
      ++ \spad{(a/b)**(1/n)} appearing in f into \spad{a**(1/n) / b**(1/n)}.
      ++ This transformation is not in general valid for all
      ++ complex numbers \spad{a} and b.
    ratDenom  : F -> F
      ++ ratDenom(f) rationalizes the denominators appearing in f
      ++ by moving all the algebraic quantities into the numerators.
    ratDenom  : (F, F) -> F
      ++ ratDenom(f, a) removes \spad{a} from the denominators in f
      ++ if \spad{a} is an algebraic kernel.
    ratDenom  : (F, List F) -> F
      ++ ratDenom(f, [a1,...,an]) removes the ai's which are
      ++ algebraic kernels from the denominators in f.
    ratDenom  : (F, List K) -> F
      ++ ratDenom(f, [a1,...,an]) removes the ai's which are
      ++ algebraic from the denominators in f.
    ratPoly  : F -> SparseUnivariatePolynomial F
      ++ ratPoly(f) returns a polynomial p such that p has no
      ++ algebraic coefficients, and \spad{p(f) = 0}.
    if R has Join(OrderedSet, GcdDomain, RetractableTo Integer)
      and F has FunctionSpace(R) then
        rootPower  : F -> F
          ++ rootPower(f) transforms every radical power of the form
          ++ \spad{(a**(1/n))**m} into a simpler form if \spad{m} and
          ++ \spad{n} have a common factor.
        rootProduct: F -> F
          ++ rootProduct(f) combines every product of the form
          ++ \spad{(a**(1/n))**m * (a**(1/s))**t} into a single power
          ++ of a root of \spad{a}, and transforms every radical power
          ++ of the form \spad{(a**(1/n))**m} into a simpler form.
        rootSimp   : F -> F
          ++ rootSimp(f) transforms every radical of the form
          ++ \spad{(a * b**(q*n+r))**(1/n)} appearing in f into
          ++ \spad{b**q * (a * b**r)**(1/n)}.
          ++ This transformation is not in general valid for all
          ++ complex numbers b.
        rootKerSimp: (OP, F, N) -> F
          ++ rootKerSimp(op,f,n) should be local but conditional.

  Implementation ==> add
    import PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F)

    innerRF    : (F, List K) -> F
    rootExpand : K -> F
    algkernels : List K -> List K
    rootkernels: List K -> List K

    dummy := kernel(new()$SY)$K

    ratDenom x                == innerRF(x, algkernels tower x)
    ratDenom(x:F, l:List K):F == innerRF(x, algkernels l)
    ratDenom(x:F, y:F)        == ratDenom(x, [y])
    ratDenom(x:F, l:List F)   == ratDenom(x, [retract(y)@K for y in l]$List(K))
    algkernels l              == select_!(has?(operator #1, ALGOP), l)
    rootkernels l             == select_!(is?(operator #1, NTHR::SY), l)

    ratPoly x ==
      numer univariate(denom(ratDenom inv(dummy::P::F - x))::F, dummy)

    rootSplit x ==
      lk := rootkernels tower x
      eval(x, lk, [rootExpand k for k in lk])

    rootExpand k ==
      x  := first argument k
      n  := second argument k
      op := operator k
      op(numer(x)::F, n) / op(denom(x)::F, n)

-- all the kernels in ll must be algebraic
    innerRF(x, ll) ==
      empty?(l := sort_!(#1 > #2, kernels x)$List(K)) or
        empty? setIntersection(ll, tower x) => x
      lk := empty()$List(K)
      k : K
      while not member?(k := first l, ll) repeat
        lk := concat(k, lk)
        empty?(l := rest l) =>
          return eval(x, lk, [map(innerRF(#1, ll), kk) for kk in lk])
      q := univariate(eval(x, lk,
                 [map(innerRF(#1, ll), kk) for kk in lk]), k, minPoly k)
      map(innerRF(#1, ll), q) (map(innerRF(#1, ll), k))

    if R has Join(OrderedSet, GcdDomain, RetractableTo Integer)
     and F has FunctionSpace(R) then
      import PolynomialRoots(IndexedExponents K, K, R, P, F)

      sroot  : K -> F
      inroot : (OP, F, N) -> F
      radeval: (P, K) -> F
      breakup: List K -> List REC

      if R has RadicalCategory then
        rootKerSimp(op, x, n) ==
          (r := retractIfCan(x)@Union(R, "failed")) case R =>
             nthRoot(r::R, n)::F
          inroot(op, x, n)
      else
        rootKerSimp(op, x, n) == inroot(op, x, n)

-- l is a list of nth-roots, returns a list of records of the form
-- [a**(1/n1),a**(1/n2),...], [n1,n2,...]]
-- such that the whole list covers l exactly
      breakup l ==
        empty? l => empty()
        k := first l
        a := first(arg := argument(k := first l))
        n := retract(second arg)@Z
        expo := empty()$List(Z)
        others := same := empty()$List(K)
        for kk in rest l repeat
          if (a = first(arg := argument kk)) then
            same := concat(kk, same)
            expo := concat(retract(second arg)@Z, expo)
          else others := concat(kk, others)
        ll := breakup others
        concat([concat(k, same), concat(n, expo)], ll)

      rootProduct x ==
        for rec in breakup rootkernels tower x repeat
          k0 := first(l := rec.ker)
          nx := numer x; dx := denom x
          if empty? rest l then x := radeval(nx, k0) / radeval(dx, k0)
          else
            n  := lcm(rec.exponent)
            k  := kernel(operator k0, [first argument k0, n::F], height k0)$K
            lv := [monomial(1, k, (n quo m)::N) for m in rec.exponent]$List(P)
            x  := radeval(eval(nx, l, lv), k) / radeval(eval(dx, l, lv), k)
        x

      rootPower x ==
        for k in rootkernels tower x repeat
          x := radeval(numer x, k) / radeval(denom x, k)
        x

-- replaces (a**(1/n))**m in p by a power of a simpler radical of a if
-- n and m have a common factor
      radeval(p, k) ==
        a := first(arg := argument k)
        n := (retract(second arg)@Integer)::NonNegativeInteger
        ans:F := 0
        q := univariate(p, k)
        while (d := degree q) > 0 repeat
          term :=
--            one?(g := gcd(d, n)) => monomial(1, k, d)
            ((g := gcd(d, n)) = 1) => monomial(1, k, d)
            monomial(1, kernel(operator k, [a,(n quo g)::F], height k), d quo g)
          ans := ans + leadingCoefficient(q)::F * term::F
          q := reductum q
        leadingCoefficient(q)::F + ans

      inroot(op, x, n) ==
--        one? x => x
        (x = 1) => x
--        (x ~= -1) and (one?(num := numer x) or (num = -1)) =>
        (x ~= -1) and (((num := numer x) = 1) or (num = -1)) =>
          inv inroot(op, (num * denom x)::F, n)
        (u := isExpt(x, op)) case "failed" => kernel(op, [x, n::F])
        pr := u::Record(var:K, exponent:Integer)
        q := pr.exponent /$Fraction(Z)
                                (n * retract(second argument(pr.var))@Z)
        qr := divide(numer q, denom q)
        x  := first argument(pr.var)
        x ** qr.quotient * rootKerSimp(op,x,denom(q)::N) ** qr.remainder

      sroot k ==
        pr := froot(first(arg := argument k),(retract(second arg)@Z)::N)
        pr.coef * rootKerSimp(operator k, pr.radicand, pr.exponent)

      rootSimp x ==
        lk := rootkernels tower x
        eval(x, lk, [sroot k for k in lk])

@
\section{package SIMPAN SimplifyAlgebraicNumberConvertPackage}
<<package SIMPAN SimplifyAlgebraicNumberConvertPackage>>=
)abbrev package SIMPAN SimplifyAlgebraicNumberConvertPackage
++ Package to allow simplify to be called on AlgebraicNumbers
++ by converting to EXPR(INT)
SimplifyAlgebraicNumberConvertPackage(): with
  simplify: AlgebraicNumber -> Expression(Integer)
	++ simplify(an) applies simplifications to an
 == add
  simplify(a:AlgebraicNumber) ==
    simplify(a::Expression(Integer))$TranscendentalManipulations(Integer, Expression Integer)

@
\section{package TRMANIP TranscendentalManipulations}
<<package TRMANIP TranscendentalManipulations>>=
)abbrev package TRMANIP TranscendentalManipulations
++ Transformations on transcendental objects
++ Author: Bob Sutor, Manuel Bronstein
++ Date Created: Way back
++ Date Last Updated: 22 January 1996, added simplifyLog MCD.
++ Description:
++   TranscendentalManipulations provides functions to simplify and
++   expand expressions involving transcendental operators.
++ Keywords: transcendental, manipulation.
TranscendentalManipulations(R, F): Exports == Implementation where
  R : Join(OrderedSet, GcdDomain)
  F : Join(FunctionSpace R, TranscendentalFunctionCategory)

  Z       ==> Integer
  K       ==> Kernel F
  P       ==> SparseMultivariatePolynomial(R, K)
  UP      ==> SparseUnivariatePolynomial P
  POWER   ==> '%power
  POW     ==> Record(val: F,exponent: Z)
  PRODUCT ==> Record(coef : Z, var : K)
  FPR     ==> Fraction Polynomial R

  Exports ==> with
    expand     : F -> F
      ++ expand(f) performs the following expansions on f:\begin{items}
      ++ \item 1. logs of products are expanded into sums of logs,
      ++ \item 2. trigonometric and hyperbolic trigonometric functions
      ++ of sums are expanded into sums of products of trigonometric
      ++ and hyperbolic trigonometric functions.
      ++ \item 3. formal powers of the form \spad{(a/b)**c} are expanded into
      ++ \spad{a**c * b**(-c)}.
      ++ \end{items}
    simplify   : F -> F
      ++ simplify(f) performs the following simplifications on f:\begin{items}
      ++ \item 1. rewrites trigs and hyperbolic trigs in terms
      ++ of \spad{sin} ,\spad{cos}, \spad{sinh}, \spad{cosh}.
      ++ \item 2. rewrites \spad{sin**2} and \spad{sinh**2} in terms
      ++ of \spad{cos} and \spad{cosh},
      ++ \item 3. rewrites \spad{exp(a)*exp(b)} as \spad{exp(a+b)}.
      ++ \item 4. rewrites \spad{(a**(1/n))**m * (a**(1/s))**t} as a single
      ++ power of a single radical of \spad{a}.
      ++ \end{items}
    htrigs     : F -> F
      ++ htrigs(f) converts all the exponentials in f into
      ++ hyperbolic sines and cosines.
    simplifyExp: F -> F
      ++ simplifyExp(f) converts every product \spad{exp(a)*exp(b)}
      ++ appearing in f into \spad{exp(a+b)}.
    simplifyLog : F -> F
      ++ simplifyLog(f) converts every \spad{log(a) - log(b)} appearing in f
      ++ into \spad{log(a/b)}, every \spad{log(a) + log(b)} into \spad{log(a*b)}
      ++ and every \spad{n*log(a)} into \spad{log(a^n)}.
    expandPower: F -> F
      ++ expandPower(f) converts every power \spad{(a/b)**c} appearing
      ++ in f into \spad{a**c * b**(-c)}.
    expandLog  : F -> F
      ++ expandLog(f) converts every \spad{log(a/b)} appearing in f into
      ++ \spad{log(a) - log(b)}, and every \spad{log(a*b)} into
      ++ \spad{log(a) + log(b)}..
    cos2sec    : F -> F
      ++ cos2sec(f) converts every \spad{cos(u)} appearing in f into
      ++ \spad{1/sec(u)}.
    cosh2sech  : F -> F
      ++ cosh2sech(f) converts every \spad{cosh(u)} appearing in f into
      ++ \spad{1/sech(u)}.
    cot2trig   : F -> F
      ++ cot2trig(f) converts every \spad{cot(u)} appearing in f into
      ++ \spad{cos(u)/sin(u)}.
    coth2trigh : F -> F
      ++ coth2trigh(f) converts every \spad{coth(u)} appearing in f into
      ++ \spad{cosh(u)/sinh(u)}.
    csc2sin    : F -> F
      ++ csc2sin(f) converts every \spad{csc(u)} appearing in f into
      ++ \spad{1/sin(u)}.
    csch2sinh  : F -> F
      ++ csch2sinh(f) converts every \spad{csch(u)} appearing in f into
      ++ \spad{1/sinh(u)}.
    sec2cos    : F -> F
      ++ sec2cos(f) converts every \spad{sec(u)} appearing in f into
      ++ \spad{1/cos(u)}.
    sech2cosh  : F -> F
      ++ sech2cosh(f) converts every \spad{sech(u)} appearing in f into
      ++ \spad{1/cosh(u)}.
    sin2csc    : F -> F
      ++ sin2csc(f) converts every \spad{sin(u)} appearing in f into
      ++ \spad{1/csc(u)}.
    sinh2csch  : F -> F
      ++ sinh2csch(f) converts every \spad{sinh(u)} appearing in f into
      ++ \spad{1/csch(u)}.
    tan2trig   : F -> F
      ++ tan2trig(f) converts every \spad{tan(u)} appearing in f into
      ++ \spad{sin(u)/cos(u)}.
    tanh2trigh : F -> F
      ++ tanh2trigh(f) converts every \spad{tanh(u)} appearing in f into
      ++ \spad{sinh(u)/cosh(u)}.
    tan2cot    : F -> F
      ++ tan2cot(f) converts every \spad{tan(u)} appearing in f into
      ++ \spad{1/cot(u)}.
    tanh2coth  : F -> F
      ++ tanh2coth(f) converts every \spad{tanh(u)} appearing in f into
      ++ \spad{1/coth(u)}.
    cot2tan    : F -> F
      ++ cot2tan(f) converts every \spad{cot(u)} appearing in f into
      ++ \spad{1/tan(u)}.
    coth2tanh  : F -> F
      ++ coth2tanh(f) converts every \spad{coth(u)} appearing in f into
      ++ \spad{1/tanh(u)}.
    removeCosSq: F -> F
      ++ removeCosSq(f) converts every \spad{cos(u)**2} appearing in f into
      ++ \spad{1 - sin(x)**2}, and also reduces higher
      ++ powers of \spad{cos(u)} with that formula.
    removeSinSq: F -> F
      ++ removeSinSq(f) converts every \spad{sin(u)**2} appearing in f into
      ++ \spad{1 - cos(x)**2}, and also reduces higher powers of
      ++ \spad{sin(u)} with that formula.
    removeCoshSq:F -> F
      ++ removeCoshSq(f) converts every \spad{cosh(u)**2} appearing in f into
      ++ \spad{1 - sinh(x)**2}, and also reduces higher powers of
      ++ \spad{cosh(u)} with that formula.
    removeSinhSq:F -> F
      ++ removeSinhSq(f) converts every \spad{sinh(u)**2} appearing in f into
      ++ \spad{1 - cosh(x)**2}, and also reduces higher powers
      ++ of \spad{sinh(u)} with that formula.
    if R has PatternMatchable(R) and R has ConvertibleTo(Pattern(R)) and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then
      expandTrigProducts : F -> F
        ++ expandTrigProducts(e) replaces \axiom{sin(x)*sin(y)} by
        ++ \spad{(cos(x-y)-cos(x+y))/2}, \axiom{cos(x)*cos(y)} by
        ++ \spad{(cos(x-y)+cos(x+y))/2}, and \axiom{sin(x)*cos(y)} by
        ++ \spad{(sin(x-y)+sin(x+y))/2}.  Note that this operation uses
        ++ the pattern matcher and so is relatively expensive.  To avoid
        ++ getting into an infinite loop the transformations are applied
        ++ at most ten times.

  Implementation ==> add
    import FactoredFunctions(P)
    import PolynomialCategoryLifting(IndexedExponents K, K, R, P, F)
    import
      PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F)

    smpexp    : P -> F
    termexp   : P -> F
    exlog     : P -> F
    smplog    : P -> F
    smpexpand : P -> F
    smp2htrigs: P -> F
    kerexpand : K -> F
    expandpow : K -> F
    logexpand : K -> F
    sup2htrigs: (UP, F) -> F
    supexp    : (UP, F, F, Z) -> F
    ueval     : (F, String, F -> F) -> F
    ueval2    : (F, String, F -> F) -> F
    powersimp : (P, List K) -> F
    t2t       : F -> F
    c2t       : F -> F
    c2s       : F -> F
    s2c       : F -> F
    s2c2      : F -> F
    th2th     : F -> F
    ch2th     : F -> F
    ch2sh     : F -> F
    sh2ch     : F -> F
    sh2ch2    : F -> F
    simplify0 : F -> F
    simplifyLog1 : F -> F
    logArgs   : List F -> F

    import F
    import List F

    if R has PatternMatchable R and R has ConvertibleTo Pattern R and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then
      XX : F := coerce new()$Symbol
      YY : F := coerce new()$Symbol
      sinCosRule : RewriteRule(R,R,F) :=
        rule(cos(XX)*sin(YY),(sin(XX+YY)-sin(XX-YY))/2::F)
      sinSinRule : RewriteRule(R,R,F) :=
        rule(sin(XX)*sin(YY),(cos(XX-YY)-cos(XX+YY))/2::F)
      cosCosRule : RewriteRule(R,R,F) :=
        rule(cos(XX)*cos(YY),(cos(XX-YY)+cos(XX+YY))/2::F)
      expandTrigProducts(e:F):F ==
        applyRules([sinCosRule,sinSinRule,cosCosRule],e,10)$ApplyRules(R,R,F)

    logArgs(l:List F):F ==
      -- This function will take a list of Expressions (implicitly a sum) and
      -- add them up, combining log terms.  It also replaces n*log(x) by
      -- log(x^n).
      import K
      sum  : F := 0
      arg  : F := 1
      for term in l repeat
        is?(term,'log) =>
          arg := arg * simplifyLog(first(argument(first(kernels(term)))))
        -- Now look for multiples, including negative ones.
        prod : Union(PRODUCT, "failed") := isMult(term)
        (prod case PRODUCT) and is?(prod.var,'log) =>
            arg := arg * simplifyLog ((first argument(prod.var))**(prod.coef))
        sum := sum+term
      sum+log(arg)
    
    simplifyLog(e:F):F ==
      simplifyLog1(numerator e)/simplifyLog1(denominator e)

    simplifyLog1(e:F):F ==
      freeOf?(e,'log) => e

      -- Check for n*log(u)
      prod : Union(PRODUCT, "failed") := isMult(e)
      (prod case PRODUCT) and is?(prod.var,'log) =>
        log simplifyLog ((first argument(prod.var))**(prod.coef))
      
      termList : Union(List(F),"failed") := isTimes(e)
      -- I'm using two variables, termList and terms, to work round a
      -- bug in the old compiler.
      not (termList case "failed") =>
        -- We want to simplify each log term in the product and then multiply
        -- them together.  However, if there is a constant or arithmetic
        -- expression (i.e. somwthing which looks like a Polynomial) we would
        -- like to combine it with a log term.
        terms :List F := [simplifyLog(term) for term in termList::List(F)]
        exprs :List F := []
        nterms :List F := []
        for term in terms repeat
          if retractIfCan(term)@Union(FPR,"failed") case FPR then
            exprs := cons(term, exprs)
          else
            nterms := cons(term, nterms)
        terms := nterms
        if not empty? exprs then
          foundLog := false
          i : NonNegativeInteger := 0
          while (not(foundLog) and (i < #terms)) repeat
            i := i+1
            if is?(terms.i,'log) then
              args : List F := argument(retract(terms.i)@K)
              setelt(terms,i, log simplifyLog1(first(args)**(*/exprs)))
              foundLog := true
          -- The next line deals with a situation which shouldn't occur,
          -- since we have checked whether we are freeOf log already.
          if not foundLog then terms := append(exprs,terms)
        */terms
    
      terms : Union(List(F),"failed") := isPlus(e)
      not (terms case "failed") => logArgs(terms) 

      expt : Union(POW, "failed") := isPower(e)
--      (expt case POW) and not one? expt.exponent =>
      (expt case POW) and not (expt.exponent = 1) =>
        simplifyLog(expt.val)**(expt.exponent)
    
      kers : List K := kernels e
--      not(one?(#kers)) => e -- Have a constant
      not(((#kers) = 1)) => e -- Have a constant
      kernel(operator first kers,[simplifyLog(u) for u in argument first kers])


    if R has RetractableTo Integer then
      simplify x == rootProduct(simplify0 x)$AlgebraicManipulations(R,F)

    else simplify x == simplify0 x

    expandpow k ==
      a := expandPower first(arg := argument k)
      b := expandPower second arg
--      ne:F := (one? numer a => 1; numer(a)::F ** b)
      ne:F := (((numer a) = 1) => 1; numer(a)::F ** b)
--      de:F := (one? denom a => 1; denom(a)::F ** (-b))
      de:F := (((denom a) = 1) => 1; denom(a)::F ** (-b))
      ne * de

    termexp p ==
      exponent:F := 0
      coef := (leadingCoefficient p)::P
      lpow := select(is?(#1, POWER)$K, lk := variables p)$List(K)
      for k in lk repeat
        d := degree(p, k)
        if is?(k, 'exp) then
          exponent := exponent + d * first argument k
        else if not is?(k, POWER) then
          -- Expand arguments to functions as well ... MCD 23/1/97
          --coef := coef * monomial(1, k, d)
          coef := coef * monomial(1, kernel(operator k,[simplifyExp u for u in argument k], height k), d)
      coef::F * exp exponent * powersimp(p, lpow)

    expandPower f ==
      l := select(is?(#1, POWER)$K, kernels f)$List(K)
      eval(f, l, [expandpow k for k in l])

-- l is a list of pure powers appearing as kernels in p
    powersimp(p, l) ==
      empty? l => 1
      k := first l                           -- k = a**b
      a := first(arg := argument k)
      exponent := degree(p, k) * second arg
      empty?(lk := select(a = first argument #1, rest l)) =>
        (a ** exponent) * powersimp(p, rest l)
      for k0 in lk repeat
        exponent := exponent + degree(p, k0) * second argument k0
      (a ** exponent) * powersimp(p, setDifference(rest l, lk))

    t2t x         == sin(x) / cos(x)
    c2t x         == cos(x) / sin(x)
    c2s x         == inv sin x
    s2c x         == inv cos x
    s2c2 x        == 1 - cos(x)**2
    th2th x       == sinh(x) / cosh(x)
    ch2th x       == cosh(x) / sinh(x)
    ch2sh x       == inv sinh x
    sh2ch x       == inv cosh x
    sh2ch2 x      == cosh(x)**2 - 1
    ueval(x, s,f) == eval(x, s::Symbol, f)
    ueval2(x,s,f) == eval(x, s::Symbol, 2, f)
    cos2sec x     == ueval(x, "cos", inv sec #1)
    sin2csc x     == ueval(x, "sin", inv csc #1)
    csc2sin x     == ueval(x, "csc", c2s)
    sec2cos x     == ueval(x, "sec", s2c)
    tan2cot x     == ueval(x, "tan", inv cot #1)
    cot2tan x     == ueval(x, "cot", inv tan #1)
    tan2trig x    == ueval(x, "tan", t2t)
    cot2trig x    == ueval(x, "cot", c2t)
    cosh2sech x   == ueval(x, "cosh", inv sech #1)
    sinh2csch x   == ueval(x, "sinh", inv csch #1)
    csch2sinh x   == ueval(x, "csch", ch2sh)
    sech2cosh x   == ueval(x, "sech", sh2ch)
    tanh2coth x   == ueval(x, "tanh", inv coth #1)
    coth2tanh x   == ueval(x, "coth", inv tanh #1)
    tanh2trigh x  == ueval(x, "tanh", th2th)
    coth2trigh x  == ueval(x, "coth", ch2th)
    removeCosSq x == ueval2(x, "cos", 1 - (sin #1)**2)
    removeSinSq x == ueval2(x, "sin", s2c2)
    removeCoshSq x== ueval2(x, "cosh", 1 + (sinh #1)**2)
    removeSinhSq x== ueval2(x, "sinh", sh2ch2)
    expandLog x   == smplog(numer x) / smplog(denom x)
    simplifyExp x == (smpexp numer x) / (smpexp denom x)
    expand x      == (smpexpand numer x) / (smpexpand denom x)
    smpexpand p   == map(kerexpand, #1::F, p)
    smplog p      == map(logexpand, #1::F, p)
    smp2htrigs p  == map(htrigs(#1::F), #1::F, p)

    htrigs f ==
      (m := mainKernel f) case "failed" => f
      op  := operator(k := m::K)
      arg := [htrigs x for x in argument k]$List(F)
      num := univariate(numer f, k)
      den := univariate(denom f, k)
      is?(op,'exp) =>
        g1 := cosh(a := first arg) + sinh(a)
        g2 := cosh(a) - sinh(a)
        supexp(num,g1,g2,b:= (degree num)::Z quo 2)/supexp(den,g1,g2,b)
      sup2htrigs(num, g1:= op arg) / sup2htrigs(den, g1)

    supexp(p, f1, f2, bse) ==
      ans:F := 0
      while p ~= 0 repeat
        g := htrigs(leadingCoefficient(p)::F)
        if ((d := degree(p)::Z - bse) >= 0) then
             ans := ans + g * f1 ** d
        else ans := ans + g * f2 ** (-d)
        p := reductum p
      ans

    sup2htrigs(p, f) ==
      (map(smp2htrigs, p)$SparseUnivariatePolynomialFunctions2(P, F)) f

    exlog p == +/[r.coef * log(r.logand::F) for r in log squareFree p]

    logexpand k ==
      nullary?(op := operator k) => k::F
      is?(op,'log) =>
         exlog(numer(x := expandLog first argument k)) - exlog denom x
      op [expandLog x for x in argument k]$List(F)

    kerexpand k ==
      nullary?(op := operator k) => k::F
      is?(op, POWER) => expandpow k
      arg := first argument k
      is?(op,'sec) => inv expand cos arg
      is?(op,'csc) => inv expand sin arg
      is?(op,'log) => exlog(numer(x := expand arg)) - exlog denom x
      num := numer arg
      den := denom arg
      (b := (reductum num) / den) ~= 0 =>
        a := (leadingMonomial num) / den
        is?(op,'exp) => exp(expand a) * expand(exp b)
        is?(op,'sin) =>
           sin(expand a) * expand(cos b) + cos(expand a) * expand(sin b)
        is?(op,'cos) =>
           cos(expand a) * expand(cos b) - sin(expand a) * expand(sin b)
        is?(op,'tan) =>
          ta := tan expand a
          tb := expand tan b
          (ta + tb) / (1 - ta * tb)
        is?(op,'cot) =>
          cta := cot expand a
          ctb := expand cot b
          (cta * ctb - 1) / (ctb + cta)
        op [expand x for x in argument k]$List(F)
      op [expand x for x in argument k]$List(F)

    smpexp p ==
      ans:F := 0
      while p ~= 0 repeat
        ans := ans + termexp leadingMonomial p
        p   := reductum p
      ans

    -- this now works in 3 passes over the expression:
    --   pass1 rewrites trigs and htrigs in terms of sin,cos,sinh,cosh
    --   pass2 rewrites sin**2 and sinh**2 in terms of cos and cosh.
    --   pass3 groups exponentials together
    simplify0 x ==
      simplifyExp eval(eval(x,
          ['tan,'cot,'sec,'csc,
           'tanh,'coth,'sech,'csch],
              [t2t,c2t,s2c,c2s,th2th,ch2th,sh2ch,ch2sh]),
                ['sin, 'sinh], [2, 2], [s2c2, sh2ch2])

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

-- SPAD files for the functional world should be compiled in the
-- following order:
--
--   op  kl  function  funcpkgs  MANIP  combfunc
--   algfunc  elemntry  constant  funceval  fe


<<package FACTFUNC FactoredFunctions>>
<<package POLYROOT PolynomialRoots>>
<<package ALGMANIP AlgebraicManipulations>>
<<package SIMPAN SimplifyAlgebraicNumberConvertPackage>>
<<package TRMANIP TranscendentalManipulations>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}