\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra setorder.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package UDPO UserDefinedPartialOrdering}
<<package UDPO UserDefinedPartialOrdering>>=
)abbrev package UDPO UserDefinedPartialOrdering
++ Author: Manuel Bronstein
++ Date Created: March 1990
++ Date Last Updated: 9 April 1991
++ Description:
++  Provides functions to force a partial ordering on any set.
UserDefinedPartialOrdering(S:SetCategory): with
  setOrder    : List S -> Void
    ++ setOrder([a1,...,an]) defines a partial ordering on S given by:
    ++    (1)  \spad{a1 < a2 < ... < an}.
    ++    (2)  \spad{b < ai   for i = 1..n} and b not among the ai's.
    ++    (3)  undefined on \spad{(b, c)} if neither is among the ai's.
  setOrder    : (List S, List S) -> Void
    ++ setOrder([b1,...,bm], [a1,...,an]) defines a partial
    ++ ordering on S given by:
    ++    (1)  \spad{b1 < b2 < ... < bm < a1 < a2 < ... < an}.
    ++    (2)  \spad{bj < c < ai}  for c not among the ai's and bj's.
    ++    (3)  undefined on \spad{(c,d)} if neither is among the ai's,bj's.
  getOrder    : () -> Record(low: List S, high: List S)
    ++ getOrder() returns \spad{[[b1,...,bm], [a1,...,an]]} such that the
    ++ partial ordering on S was given by
    ++ \spad{setOrder([b1,...,bm],[a1,...,an])}.
  less?       : (S, S) -> Union(Boolean, "failed")
    ++ less?(a, b) compares \spad{a} and b in the partial ordering induced by
    ++ setOrder.
  less?       : (S, S, (S, S) -> Boolean) -> Boolean
    ++ less?(a, b, fn) compares \spad{a} and b in the partial ordering induced
    ++ by setOrder, and returns \spad{fn(a, b)} if \spad{a}
    ++ and b are not comparable
    ++ in that ordering.
  largest     : (List S, (S, S) -> Boolean) -> S
    ++ largest(l, fn) returns the largest element of l where the partial
    ++ ordering induced by setOrder is completed into a total one by fn.
  userOrdered?: () -> Boolean
    ++ userOrdered?() tests if the partial ordering induced by
    ++ \spadfunFrom{setOrder}{UserDefinedPartialOrdering} is not empty.
  if S has OrderedSet then
    largest: List S -> S
      ++ largest l returns the largest element of l where the partial
      ++ ordering induced by setOrder is completed into a total one by
      ++ the ordering on S.
    more?  : (S, S) -> Boolean
      ++ more?(a, b) compares \spad{a} and b in the partial ordering induced
      ++ by setOrder, and uses the ordering on S if \spad{a} and b are not
      ++ comparable in the partial ordering.

 == add
  llow :Reference List S := ref nil()
  lhigh:Reference List S := ref nil()

  userOrdered?() == not(empty? deref llow) or not(empty? deref lhigh)
  getOrder()     == [deref llow, deref lhigh]
  setOrder l     == setOrder(nil(), l)

  setOrder(l, h) ==
    setref(llow, removeDuplicates l)
    setref(lhigh, removeDuplicates h)
    void

  less?(a, b, f) ==
    (u := less?(a, b)) case "failed" => f(a, b)
    u::Boolean

  largest(x, f) ==
    empty? x => error "largest: empty list"
    empty? rest x => first x
    a := largest(rest x, f)
    less?(first x, a, f) => a
    first x

  less?(a, b) ==
    for x in deref llow repeat
      x = a => return(a ~= b)
      x = b => return false
    aa := bb := false$Boolean
    for x in deref lhigh repeat
      if x = a then
        bb => return false
        aa := true
      if x = b then
        aa => return(a ~= b)
        bb := true
    aa => false
    bb => true
    "failed"

  if S has OrderedSet then
    more?(a, b) == not less?(a, b, (y, z) +-> y <$S z)
    largest x   == largest(x, (y, z) +-> y <$S z)

@
\section{package UDVO UserDefinedVariableOrdering}
<<package UDVO UserDefinedVariableOrdering>>=
)abbrev package UDVO UserDefinedVariableOrdering
++ Author: Manuel Bronstein
++ Date Created: March 1990
++ Date Last Updated: 9 April 1991
++ Description:
++  This packages provides functions to allow the user to select the ordering
++  on the variables and operators for displaying polynomials,
++  fractions and expressions. The ordering affects the display
++  only and not the computations.
UserDefinedVariableOrdering(): with
  setVariableOrder  : List Symbol -> Void
    ++ setVariableOrder([a1,...,an]) defines an ordering on the
    ++ variables given by \spad{a1 > a2 > ... > an > other variables}.
  setVariableOrder  : (List Symbol, List Symbol) -> Void
    ++ setVariableOrder([b1,...,bm], [a1,...,an]) defines an ordering
    ++ on the variables given by
    ++ \spad{b1 > b2 > ... > bm >} other variables \spad{>  a1 > a2 > ... > an}.
  getVariableOrder  : () -> Record(high:List Symbol, low:List Symbol)
    ++ getVariableOrder() returns \spad{[[b1,...,bm], [a1,...,an]]} such that
    ++ the ordering on the variables was given by
    ++ \spad{setVariableOrder([b1,...,bm], [a1,...,an])}.
  resetVariableOrder: () -> Void
    ++ resetVariableOrder() cancels any previous use of
    ++ setVariableOrder and returns to the default system ordering.
 == add
  import UserDefinedPartialOrdering(Symbol)

  setVariableOrder l       == setOrder reverse l
  setVariableOrder(l1, l2) == setOrder(reverse l2, reverse l1)
  resetVariableOrder()     == setVariableOrder(nil(), nil())

  getVariableOrder() ==
    r := getOrder()
    [reverse(r.high), reverse(r.low)]

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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>>

<<package UDPO UserDefinedPartialOrdering>>
<<package UDVO UserDefinedVariableOrdering>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
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\end{document}
