/usr/share/axiom-20170501/src/algebra/SET.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 | )abbrev domain SET Set
++ Author: Michael Monagan; revised by Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: May 1991
++ 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\br
++ \tab{5}\spad{s = t} is \spad{O(min(n,m))}\br
++ \tab{5}\spad{s < t} is \spad{O(max(n,m))}\br
++ \tab{5}\spad{union(s,t)}, \spad{intersect(s,t)}, \spad{minus(s,t)},\br
++ \tab{10 \spad{symmetricDifference(s,t)} is \spad{O(max(n,m))}\br
++ \tab{5}\spad{member(x,t)} is \spad{O(n log n)}\br
++ \tab{5}\spad{insert(x,t)} and \spad{remove(x,t)} is \spad{O(n)}
Set(S) : SIG == CODE where
S : SetCategory
SIG ==> FiniteSetAggregate S
CODE ==> 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)$Rep
map(f, s) == map_!(f,copy s)
map_!(f,s) ==
map_!(f,s)$Rep
removeDuplicates_! 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))
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
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
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