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In Issue #347 it is shown that set equality fails after applying a map to a set:

fricas
A:Set Integer := set [-2,-1,0]

\label{eq1}\left\{ - 2, \: - 1, \: 0 \right\}(1)
Type: Set(Integer)
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B:Set Integer := set [0,1,4]

\label{eq2}\left\{ 0, \: 1, \: 4 \right\}(2)
Type: Set(Integer)
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C:=map(x +-> x^2,A)

\label{eq3}\left\{ 0, \: 1, \: 4 \right\}(3)
Type: Set(Integer)
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test(C=B)

\label{eq4} \mbox{\rm true} (4)
Type: Boolean

A possible fix is given in #347 for Sets whose members have OrderedSet.

A More Ambitious Fix

As suggested by the documentation in the code for Set domain, sets must be sorted based some ordering applicable to all Axiom object. One such order can be defined by the SXHASH value (ref). For example:

   order(x:S,y:S):Boolean == integer(SXHASH(x)$Lisp)$SExpression<integer(SXHASH(y)$Lisp)$SExpression

   map_!(f,s) ==
     map_!(f,s)$Rep
     sort_!(order,s)$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_!(order,a)

although the ordering may fail to be total because of collisions.

A better ordering is given by the lexical ordering function LEXGREATERP defined in the Axiom interpreter code ggreater.lisp :

   order(x:S,y:S):Boolean == null? LEXGREATERP(a,b)$Lisp

This ordering is compatible with the "natural" ordering in each domain if the domain has OrderedSet?

Modified Domain Set

spad
)abbrev domain SET Set
++ Author: Michael Monagan; revised by Richard Jenks
++ Date Created: August 87 through August 88
++ Date Previously 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]
order(x:S,y:S):Boolean == null?(LEXGREATERP(x,y)$Lisp)$SExpression -- Not as good? -- integer(SXHASH(x)$Lisp)$SExpression<integer(SXHASH(y)$Lisp)$SExpression
map!(f, s) == map!(f, s)$Rep sort!(order, s)$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!(order, a)
if S has OrderedSet then s = t == s =$Rep t max s == inspect s min s == (empty? s => error "Empty set"; s(minIndex s))
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
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
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3506829172687478687-25px002.spad
      using old system compiler.
   SET abbreviates domain Set 
------------------------------------------------------------------------
   initializing NRLIB SET for Set 
   compiling into NRLIB SET 
   compiling exported # : $ -> NonNegativeInteger
;;; *** |SET;#;$Nni;1| REDEFINED Time: 0.02 SEC.
compiling exported brace : () -> $
;;; *** |SET;brace;$;2| REDEFINED Time: 0 SEC.
compiling exported set : () -> $
;;; *** |SET;set;$;3| REDEFINED Time: 0 SEC.
compiling exported empty : () -> $
;;; *** |SET;empty;$;4| REDEFINED Time: 0 SEC.
compiling exported copy : $ -> $
;;; *** |SET;copy;2$;5| REDEFINED Time: 0 SEC.
compiling exported parts : $ -> List S
;;; *** |SET;parts;$L;6| REDEFINED Time: 0 SEC.
compiling exported inspect : $ -> S
;;; *** |SET;inspect;$S;7| REDEFINED Time: 0.02 SEC.
compiling exported extract! : $ -> S
;;; *** |SET;extract!;$S;8| REDEFINED Time: 0 SEC.
compiling exported find : (S -> Boolean,$) -> Union(S,failed)
;;; *** |SET;find;M$U;9| REDEFINED Time: 0 SEC.
compiling exported map : (S -> S,$) -> $
;;; *** |SET;map;M2$;10| REDEFINED Time: 0 SEC.
compiling exported reduce : ((S,S) -> S,$) -> S
;;; *** |SET;reduce;M$S;11| REDEFINED Time: 0 SEC.
compiling exported reduce : ((S,S) -> S,$,S) -> S
;;; *** |SET;reduce;M$2S;12| REDEFINED Time: 0 SEC.
compiling exported reduce : ((S,S) -> S,$,S,S) -> S
;;; *** |SET;reduce;M$3S;13| REDEFINED Time: 0 SEC.
****** Domain: S already in scope augmenting S: (ConvertibleTo (InputForm)) compiling exported convert : $ -> InputForm
;;; *** |SET;convert;$If;14| REDEFINED Time: 0.16 SEC.
compiling local order : (S,S) -> Boolean Time: 0 SEC.
compiling exported map! : (S -> S,$) -> $ Time: 0 SEC.
compiling exported construct : List S -> $ Time: 0.01 SEC.
****** Domain: S already in scope augmenting S: (OrderedSet) compiling exported = : ($,$) -> Boolean Time: 0 SEC.
compiling exported max : $ -> S Time: 0 SEC.
compiling exported min : $ -> S Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $ Time: 0 SEC.
compiling exported member? : (S,$) -> Boolean Time: 0 SEC.
compiling exported remove! : (S,$) -> $ Time: 0 SEC.
compiling exported intersect : ($,$) -> $ Time: 0.01 SEC.
compiling exported difference : ($,$) -> $ Time: 0.01 SEC.
compiling exported symmetricDifference : ($,$) -> $ Time: 0.01 SEC.
compiling exported subset? : ($,$) -> Boolean Time: 0.01 SEC.
compiling exported union : ($,$) -> $ Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $ Time: 0.01 SEC.
compiling exported remove! : (S,$) -> $ Time: 0 SEC.
****** Domain: S already in scope augmenting S: (Evalable S) ****** Domain: S already in scope augmenting S: (ConvertibleTo (InputForm)) ****** Domain: S already in scope augmenting S: (Finite) ****** Domain: S already in scope augmenting S: (OrderedSet) (time taken in buildFunctor: 0)
;;; *** |Set| REDEFINED
;;; *** |Set| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor Set Time: 0.28 seconds
finalizing NRLIB SET Processing Set for Browser database: --------constructor--------- ; compiling file "/var/aw/var/LatexWiki/SET.NRLIB/SET.lsp" (written 31 JUL 2013 05:25:17 PM):
; /var/aw/var/LatexWiki/SET.NRLIB/SET.fasl written ; compilation finished in 0:00:00.146 ------------------------------------------------------------------------ Set is now explicitly exposed in frame initial Set will be automatically loaded when needed from /var/aw/var/LatexWiki/SET.NRLIB/SET

Retest

fricas
A2:Set Integer := set [-2,-1,0]

\label{eq5}\left\{ - 2, \: - 1, \: 0 \right\}(5)
Type: Set(Integer)
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B2:Set Integer := set [0,1,4]

\label{eq6}\left\{ 0, \: 1, \: 4 \right\}(6)
Type: Set(Integer)
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C2:=map(x +-> x^2,A)

\label{eq7}\left\{ 0, \: 1, \: 4 \right\}(7)
Type: Set(Integer)
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test(B2=C2)

\label{eq8} \mbox{\rm true} (8)
Type: Boolean

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)set message any off
showTypeInOutput true;
Type: String

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Set Any has OrderedSet

\label{eq9} \mbox{\rm false} (9)
Type: Boolean
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B5:Set Any:=B

\label{eq10}\left\{{0 : \hbox{\axiomType{Integer}\ }}, \:{1 : \hbox{\axiomType{Integer}\ }}, \:{4 : \hbox{\axiomType{Integer}\ }}\right\}(10)
Type: Set(Any)
fricas
C5:Set Any:=C

\label{eq11}\left\{{0 : \hbox{\axiomType{Integer}\ }}, \:{1 : \hbox{\axiomType{Integer}\ }}, \:{4 : \hbox{\axiomType{Integer}\ }}\right\}(11)
Type: Set(Any)
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test(B5=C5)

\label{eq12} \mbox{\rm true} (12)
Type: Boolean




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