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# Edit detail for sets.spad revision 1 of 1

 1 Editor: Time: 2007/11/29 04:14:21 GMT-8 Note: proposed fix for issue #347

changed:
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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra sets.spad} \author{Michael Monagan, Richard Jenks} \date{May 1991} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{domain SET Set} <<domain SET Set>>= )abbrev domain SET Set ++ Author: Michael Monagan; revised by Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: May 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A set over a domain D models the usual mathematical notion of a finite set ++ of elements from D. ++ Sets are unordered collections of distinct elements ++ (that is, order and duplication does not matter). ++ The notation \spad{set [a,b,c]} can be used to create ++ a set and the usual operations such as union and intersection are available ++ to form new sets. ++ In our implementation, \Language{} maintains the entries in ++ sorted order. Specifically, the parts function returns the entries ++ as a list in ascending order and ++ the extract operation returns the maximum entry. ++ Given two sets s and t where \spad{#s = m} and \spad{#t = n}, ++ the complexity of ++ \spad{s = t} is \spad{O(min(n,m))} ++ \spad{s < t} is \spad{O(max(n,m))} ++ \spad{union(s,t)}, \spad{intersect(s,t)}, \spad{minus(s,t)}, \spad{symmetricDifference(s,t)} is \spad{O(max(n,m))} ++ \spad{member(x,t)} is \spad{O(n log n)} ++ \spad{insert(x,t)} and \spad{remove(x,t)} is \spad{O(n)} Set(S:SetCategory): FiniteSetAggregate S == add Rep := FlexibleArray(S) # s == _#$Rep s
brace()   == empty()
set()     == empty()
empty()   == empty()$Rep copy s == copy(s)$Rep
parts s   == parts(s)$Rep inspect s == (empty? s => error "Empty set"; s(maxIndex s)) extract_! s == x := inspect s delete_!(s, maxIndex s) x find(f, s) == find(f, s)$Rep

map(f, s) == map_!(f,copy s)

reduce(f, s) == reduce(f, s)$Rep reduce(f, s, x) == reduce(f, s, x)$Rep

reduce(f, s, x, y) == reduce(f, s, x, y)$Rep if S has ConvertibleTo InputForm then convert(x:%):InputForm == convert [convert("set"::Symbol)@InputForm, convert(parts x)@InputForm] if S has OrderedSet then s = t == s =$Rep t
max s == inspect s
min s == (empty? s => error "Empty set"; s(minIndex s))

map_!(f,s) ==
map_!(f,s)$Rep sort_!(s)$Rep
removeDuplicates_! s

construct l ==
zero?(n := #l) => empty()
a := new(n, first l)
for i in minIndex(a).. for x in l repeat a.i := x
removeDuplicates_! sort_! a

insert_!(x, s) ==
n := inc maxIndex s
k := minIndex s
while k < n and x > s.k repeat k := inc k
k < n and s.k = x => s
insert_!(x, s, k)

member?(x, s) == -- binary search
empty? s => false
t := maxIndex s
b := minIndex s
while b < t repeat
m := (b+t) quo 2
if x > s.m then b := m+1 else t := m
x = s.t

remove_!(x:S, s:%) ==
n := inc maxIndex s
k := minIndex s
while k < n and x > s.k repeat k := inc k
k < n and x = s.k => delete_!(s, k)
s

-- the set operations are implemented as variations of merging
intersect(s, t) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (concat_!(r, s.i); i := i+1; j := j+1)
if s.i < t.j then i := i+1 else j := j+1
r

difference(s:%, t:%) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (i := i+1; j := j+1)
s.i < t.j => (concat_!(r, s.i); i := i+1)
j := j+1
while i <= m repeat (concat_!(r, s.i); i := i+1)
r

symmetricDifference(s, t) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i < t.j => (concat_!(r, s.i); i := i+1)
s.i > t.j => (concat_!(r, t.j); j := j+1)
i := i+1; j := j+1
while i <= m repeat (concat_!(r, s.i); i := i+1)
while j <= n repeat (concat_!(r, t.j); j := j+1)
r

subset?(s, t) ==
m := maxIndex s
n := maxIndex t
m > n => false
i := minIndex s
j := minIndex t
while i <= m and j <= n repeat
s.i = t.j => (i := i+1; j := j+1)
s.i > t.j => j := j+1
return false
i > m

union(s:%, t:%) ==
m := maxIndex s
n := maxIndex t
i := minIndex s
j := minIndex t
r := empty()
while i <= m and j <= n repeat
s.i = t.j => (concat_!(r, s.i); i := i+1; j := j+1)
s.i < t.j => (concat_!(r, s.i); i := i+1)
(concat_!(r, t.j); j := j+1)
while i <= m repeat (concat_!(r, s.i); i := i+1)
while j <= n repeat (concat_!(r, t.j); j := j+1)
r

else
map_!(f,s) ==
map_!(f,s)\$Rep
removeDuplicates_! s

insert_!(x, s) ==
for k in minIndex s .. maxIndex s repeat
s.k = x => return s
insert_!(x, s, inc maxIndex s)

remove_!(x:S, s:%) ==
n := inc maxIndex s
k := minIndex s
while k < n repeat
x = s.k => return delete_!(s, k)
k := inc k
s

@
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--
--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.
@
<<*>>=

<<domain SET Set>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document} 