map confuses sets.
axiom
A:Set Integer:=set [-2,-1,0]
Type: Set(Integer)
axiom
B:Set Integer:=set [0,1,4]
Type: Set(Integer)
axiom
C:=map(x +-> x^2,A)
Type: Set(Integer)
axiom
test(C=B)
Type: Boolean
somehow a sort is missing after applying map.
Proposed Fix
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.
spad
)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
spad
Compiling FriCAS source code from file
/var/zope2/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: 0.28 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: 0.06 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$S;11| REDEFINED
Time: 0.01 SEC.
compiling exported reduce : ((S,S) -> S,$,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;$If;14| REDEFINED
Time: 0.28 SEC.
****** Domain: S already in scope
augmenting S: (OrderedSet)
compiling exported = : ($,$) -> Boolean
;;; *** |SET;=;2$B;15| REDEFINED
Time: 0.01 SEC.
compiling exported max : $ -> S
;;; *** |SET;max;$S;16| REDEFINED
Time: 0 SEC.
compiling exported min : $ -> S
;;; *** |SET;min;$S;17| REDEFINED
Time: 0 SEC.
compiling exported map! : (S -> S,$) -> $
;;; *** |SET;map!;M2$;18| REDEFINED
Time: 0.01 SEC.
compiling exported construct : List S -> $
;;; *** |SET;construct;L$;19| REDEFINED
Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $
;;; *** |SET;insert!;S2$;20| REDEFINED
Time: 0.01 SEC.
compiling exported member? : (S,$) -> Boolean
;;; *** |SET;member?;S$B;21| REDEFINED
Time: 0.01 SEC.
compiling exported remove! : (S,$) -> $
;;; *** |SET;remove!;S2$;22| REDEFINED
Time: 0.01 SEC.
compiling exported intersect : ($,$) -> $
;;; *** |SET;intersect;3$;23| REDEFINED
Time: 0 SEC.
compiling exported difference : ($,$) -> $
;;; *** |SET;difference;3$;24| REDEFINED
Time: 0 SEC.
compiling exported symmetricDifference : ($,$) -> $
;;; *** |SET;symmetricDifference;3$;25| REDEFINED
Time: 0 SEC.
compiling exported subset? : ($,$) -> Boolean
;;; *** |SET;subset?;2$B;26| REDEFINED
Time: 0.15 SEC.
compiling exported union : ($,$) -> $
;;; *** |SET;union;3$;27| REDEFINED
Time: 0.02 SEC.
compiling exported map! : (S -> S,$) -> $
Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $
Time: 0.01 SEC.
compiling exported remove! : (S,$) -> $
Time: 0.02 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: 10)
;;; *** |Set| REDEFINED
;;; *** |Set| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor Set
Time: 0.91 seconds
finalizing NRLIB SET
Processing Set for Browser database:
--------constructor---------
; compiling file "/var/zope2/var/LatexWiki/SET.NRLIB/SET.lsp" (written 01 AUG 2011 08:06:27 AM):
; /var/zope2/var/LatexWiki/SET.NRLIB/SET.fasl written
; compilation finished in 0:00:00.532
------------------------------------------------------------------------
Set is now explicitly exposed in frame initial
Set will be automatically loaded when needed from
/var/zope2/var/LatexWiki/SET.NRLIB/SET
>> System error:
The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
The ANSI Standard, Glossary entry for "bounding index designator"
The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERRORRetest
axiom
A2:Set Integer:=set [-2,-1,0]
Type: Set(Integer)
axiom
B2:Set Integer:=set [0,1,4]
Type: Set(Integer)
axiom
C2:=map(x +-> x^2,A)
Type: Set(Integer)
axiom
test(B2=C2)
Type: Boolean
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.
axiom
)set message any off
showTypeInOutput true;
Type: String
axiom
Set Any has OrderedSet
Type: Boolean
axiom
B3:Set Any:=B;B3
Type: Set(Any)
axiom
C3:Set Any:=C;C3
Type: Set(Any)
axiom
test(B3=C3)
Type: Boolean
But why does this example work? Is set equality implemented as
an order n^2 comparison if the domain is not an OrderedSet?
In FiniteSetAggregate
? we see:
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
Status: open => fix proposed
