/usr/share/axiom-20170501/src/algebra/FMONOID.spad is in axiom-source 20170501-3.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 | )abbrev domain FMONOID FreeMonoid
++ Author: Stephen M. Watt
++ Date Last Updated: 6 June 1991
++ Description:
++ Free monoid on any set of generators
++ The free monoid on a set S is the monoid of finite products of
++ the form \spad{reduce(*,[si ** ni])} where the si's are in S, and the ni's
++ are nonnegative integers. The multiplication is not commutative.
FreeMonoid(S) : SIG == CODE where
S : SetCategory
NNI ==> NonNegativeInteger
REC ==> Record(gen: S, exp: NonNegativeInteger)
Ex ==> OutputForm
SIG ==> Join(Monoid, RetractableTo S) with
"*" : (S, $) -> $
++ s * x returns the product of x by s on the left.
"*" : ($, S) -> $
++ x * s returns the product of x by s on the right.
"**" : (S, NonNegativeInteger) -> $
++ s ** n returns the product of s by itself n times.
hclf : ($, $) -> $
++ hclf(x, y) returns the highest common left factor of x and y,
++ the largest d such that \spad{x = d a} and \spad{y = d b}.
hcrf : ($, $) -> $
++ hcrf(x, y) returns the highest common right factor of x and y,
++ the largest d such that \spad{x = a d} and \spad{y = b d}.
lquo : ($, $) -> Union($, "failed")
++ lquo(x, y) returns the exact left quotient of x by y
++ q such that \spad{x = y * q},
++ "failed" if x is not of the form \spad{y * q}.
rquo : ($, $) -> Union($, "failed")
++ rquo(x, y) returns the exact right quotient of x by y
++ q such that \spad{x = q * y},
++ "failed" if x is not of the form \spad{q * y}.
divide : ($, $) -> Union(Record(lm: $, rm: $), "failed")
++ divide(x, y) returns the left and right exact quotients of
++ x by y, \spad{[l, r]} such that \spad{x = l * y * r},
++ "failed" if x is not of the form \spad{l * y * r}.
overlap : ($, $) -> Record(lm: $, mm: $, rm: $)
++ overlap(x, y) returns \spad{[l, m, r]} such that
++ \spad{x = l * m}, \spad{y = m * r} and l and r have no overlap,
++ \spad{overlap(l, r) = [l, 1, r]}.
size : $ -> NNI
++ size(x) returns the number of monomials in x.
factors : $ -> List Record(gen: S, exp: NonNegativeInteger)
++ factors(a1\^e1,...,an\^en) returns \spad{[[a1, e1],...,[an, en]]}.
nthExpon : ($, Integer) -> NonNegativeInteger
++ nthExpon(x, n) returns the exponent of the n^th monomial of x.
nthFactor : ($, Integer) -> S
++ nthFactor(x, n) returns the factor of the n^th monomial of x.
mapExpon : (NNI -> NNI, $) -> $
++ mapExpon(f, a1\^e1 ... an\^en)
++ returns \spad{a1\^f(e1) ... an\^f(en)}.
mapGen : (S -> S, $) -> $
++ mapGen(f, a1\^e1 ... an\^en) returns
++\spad{f(a1)\^e1 ... f(an)\^en}.
if S has OrderedSet then OrderedSet
CODE ==> ListMonoidOps(S, NonNegativeInteger, 1) add
Rep := ListMonoidOps(S, NonNegativeInteger, 1)
1 == makeUnit()
one? f == empty? listOfMonoms f
coerce(f:$): Ex == outputForm(f, "*", "**", 1)
hcrf(f, g) == reverse_! hclf(reverse f, reverse g)
f:$ * s:S == rightMult(f, s)
s:S * f:$ == leftMult(s, f)
factors f == copy listOfMonoms f
mapExpon(f, x) == mapExpon(f, x)$Rep
mapGen(f, x) == mapGen(f, x)$Rep
s:S ** n:NonNegativeInteger == makeTerm(s, n)
f:$ * g:$ ==
(f = 1) => g
(g = 1) => f
lg := listOfMonoms g
ls := last(lf := listOfMonoms f)
ls.gen = lg.first.gen =>
setlast_!(h := copy lf,[lg.first.gen,lg.first.exp+ls.exp])
makeMulti concat(h, rest lg)
makeMulti concat(lf, lg)
overlap(la, ar) ==
(la = 1) or (ar = 1) => [la, 1, ar]
lla := la0 := listOfMonoms la
lar := listOfMonoms ar
l:List(REC) := empty()
while not empty? lla repeat
if lla.first.gen = lar.first.gen then
if lla.first.exp < lar.first.exp and empty? rest lla then
return [makeMulti l,
makeTerm(lla.first.gen, lla.first.exp),
makeMulti concat([lar.first.gen,
(lar.first.exp - lla.first.exp)::NNI],
rest lar)]
if lla.first.exp >= lar.first.exp then
if (ru:= lquo(makeMulti rest lar,
makeMulti rest lla)) case $ then
if lla.first.exp > lar.first.exp then
l := concat_!(l, [lla.first.gen,
(lla.first.exp - lar.first.exp)::NNI])
m := concat([lla.first.gen, lar.first.exp],
rest lla)
else m := lla
return [makeMulti l, makeMulti m, ru::$]
l := concat_!(l, lla.first)
lla := rest lla
[makeMulti la0, 1, makeMulti lar]
divide(lar, a) ==
(a = 1) => [lar, 1]
Na : Integer := #(la := listOfMonoms a)
Nlar : Integer := #(llar := listOfMonoms lar)
l:List(REC) := empty()
while Na <= Nlar repeat
if llar.first.gen = la.first.gen and
llar.first.exp >= la.first.exp then
-- Can match a portion of this lar factor.
-- Now match tail.
(q:=lquo(makeMulti rest llar,makeMulti rest la))case $ =>
if llar.first.exp > la.first.exp then
l := concat_!(l, [la.first.gen,
(llar.first.exp - la.first.exp)::NNI])
return [makeMulti l, q::$]
l := concat_!(l, first llar)
llar := rest llar
Nlar := Nlar - 1
"failed"
hclf(f, g) ==
h:List(REC) := empty()
for f0 in listOfMonoms f for g0 in listOfMonoms g repeat
f0.gen ^= g0.gen => return makeMulti h
h := concat_!(h, [f0.gen, min(f0.exp, g0.exp)])
f0.exp ^= g0.exp => return makeMulti h
makeMulti h
lquo(aq, a) ==
size a > #(laq := copy listOfMonoms aq) => "failed"
for a0 in listOfMonoms a repeat
a0.gen ^= laq.first.gen or a0.exp > laq.first.exp =>
return "failed"
if a0.exp = laq.first.exp then laq := rest laq
else setfirst_!(laq, [laq.first.gen,
(laq.first.exp - a0.exp)::NNI])
makeMulti laq
rquo(qa, a) ==
(u := lquo(reverse qa, reverse a)) case "failed" => "failed"
reverse_!(u::$)
if S has OrderedSet then
a < b ==
la := listOfMonoms a
lb := listOfMonoms b
na: Integer := #la
nb: Integer := #lb
while na > 0 and nb > 0 repeat
la.first.gen > lb.first.gen => return false
la.first.gen < lb.first.gen => return true
if la.first.exp = lb.first.exp then
la:=rest la
lb:=rest lb
na:=na - 1
nb:=nb - 1
else if la.first.exp > lb.first.exp then
la:=concat([la.first.gen,
(la.first.exp - lb.first.exp)::NNI], rest lb)
lb:=rest lb
nb:=nb - 1
else
lb:=concat([lb.first.gen,
(lb.first.exp-la.first.exp)::NNI], rest la)
la:=rest la
na:=na-1
empty? la and not empty? lb
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