map confuses sets.
(1) -> A:Set Integer:=set [-2,-1, 0]
B:Set Integer:=set [0,1, 4]
C:=map(x +-> x^2,A)
test(C=B)
somehow a sort is missing after applying map.
If S has OrderedSet then map_! should include 'sort':
map_!(f,s) == map_!(f,s)$Rep sort_!(s)$Rep removeDuplicates_!
See diff
See also: SandBoxSetAny for a more ambitious fix that defines an ordering for all Sets.
)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
Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7072998645375249993-25px002.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 SEC.
************* USER ERROR ********** available signatures for brace: NONE NEED brace: () -> ? ****** comp fails at level 1 with expression: ****** ((DEF (|brace|) (NIL) (|empty|))) ****** level 1 ****** $x:= (DEF (brace) (NIL) (empty)) $m:= $EmptyMode $f:= ((((|#| #) (< #) (<= #) (= #) ...)))
>> Apparent user error: unspecified error
Retest
A2:Set Integer:=set [-2,-1, 0]
B2:Set Integer:=set [0,1, 4]
C2:=map(x +-> x^2,A)
test(B2=C2)
But unfortunately the documentation lies:
++ 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.
This example shows that Set is not maintained in sorterd order. So the code for Set still appears to be broken if the Set is constructed over a domain that is not an OrderedSet.
)set message any off
showTypeInOutput true;
Set Any has OrderedSet
B3:Set Any:=B;B3
C3:Set Any:=C;C3
test(B3=C3)
But why does this example work? Is set equality implemented as an order n^2 comparison if the domain is not an OrderedSet?
s = t == #s = #t and empty? difference(s,t) ... difference(s:%, t:%) == m := copy s for x in parts t repeat remove_!(x, m) m