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--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev domain LMOPS ListMonoidOps
++ Internal representation for monoids
++ Author: Manuel Bronstein
++ Date Created: November 1989
++ Date Last Updated: 6 June 1991
++ Description:
++ This internal package represents monoid (abelian or not, with or
++ without inverses) as lists and provides some common operations
++ to the various flavors of monoids.
ListMonoidOps(S, E, un): Exports == Implementation where
S : SetCategory
E : AbelianMonoid
un: E
REC ==> Record(gen:S, exp: E)
O ==> OutputForm
Exports ==> Join(SetCategory, RetractableTo S) with
outputForm : ($, (O, O) -> O, (O, O) -> O, Integer) -> O
++ outputForm(l, fop, fexp, unit) converts the monoid element
++ represented by l to an \spadtype{OutputForm}.
++ Argument unit is the output form
++ for the \spadignore{unit} of the monoid (e.g. 0 or 1),
++ \spad{fop(a, b)} is the
++ output form for the monoid operation applied to \spad{a} and b
++ (e.g. \spad{a + b}, \spad{a * b}, \spad{ab}),
++ and \spad{fexp(a, n)} is the output form
++ for the exponentiation operation applied to \spad{a} and n
++ (e.g. \spad{n a}, \spad{n * a}, \spad{a ** n}, \spad{a\^n}).
listOfMonoms : $ -> List REC
++ listOfMonoms(l) returns the list of the monomials forming l.
makeTerm : (S, E) -> $
++ makeTerm(s, e) returns the monomial s exponentiated by e
++ (e.g. s^e or e * s).
makeMulti : List REC -> $
++ makeMulti(l) returns the element whose list of monomials is l.
nthExpon : ($, Integer) -> E
++ nthExpon(l, n) returns the exponent of the n^th monomial of l.
nthFactor : ($, Integer) -> S
++ nthFactor(l, n) returns the factor of the n^th monomial of l.
reverse : $ -> $
++ reverse(l) reverses the list of monomials forming l. This
++ has some effect if the monoid is non-abelian, i.e.
++ \spad{reverse(a1\^e1 ... an\^en) = an\^en ... a1\^e1} which is different.
reverse! : $ -> $
++ reverse!(l) reverses the list of monomials forming l, destroying
++ the element l.
size : $ -> NonNegativeInteger
++ size(l) returns the number of monomials forming l.
makeUnit : () -> $
++ makeUnit() returns the unit element of the monomial.
rightMult : ($, S) -> $
++ rightMult(a, s) returns \spad{a * s} where \spad{*}
++ is the monoid operation,
++ which is assumed non-commutative.
leftMult : (S, $) -> $
++ leftMult(s, a) returns \spad{s * a} where
++ \spad{*} is the monoid operation,
++ which is assumed non-commutative.
plus : (S, E, $) -> $
++ plus(s, e, x) returns \spad{e * s + x} where \spad{+}
++ is the monoid operation,
++ which is assumed commutative.
plus : ($, $) -> $
++ plus(x, y) returns \spad{x + y} where \spad{+}
++ is the monoid operation,
++ which is assumed commutative.
commutativeEquality: ($, $) -> Boolean
++ commutativeEquality(x,y) returns true if x and y are equal
++ assuming commutativity
mapExpon : (E -> E, $) -> $
++ 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}.
Implementation ==> add
Rep := List REC
localplus: ($, $) -> $
makeUnit() == empty()$Rep
size l == # listOfMonoms l
coerce(s:S):$ == [[s, un]]
coerce(l:$):O == coerce(l)$Rep
makeTerm(s, e) == (zero? e => makeUnit(); [[s, e]])
makeMulti l == l
f = g == f =$Rep g
listOfMonoms l == l pretend List(REC)
nthExpon(f, i) == f.(i-1+minIndex f).exp
nthFactor(f, i) == f.(i-1+minIndex f).gen
reverse l == reverse(l)$Rep
reverse! l == reverse!(l)$Rep
mapGen(f, l) == [[f(x.gen), x.exp] for x in l]
mapExpon(f, l) ==
ans:List(REC) := empty()
for x in l repeat
if (a := f(x.exp)) ~= 0 then ans := concat([x.gen, a], ans)
reverse! ans
outputForm(l, op, opexp, id) ==
empty? l => id::OutputForm
l:List(O) :=
[(p.exp = un => p.gen::O; opexp(p.gen::O, p.exp::O)) for p in l]
reduce(op, l)
retractIfCan(l:$):Union(S, "failed") ==
not empty? l and empty? rest l and l.first.exp = un => l.first.gen
"failed"
rightMult(f, s) ==
empty? f => s::$
s = f.last.gen => (setlast!(h := copy f, [s, f.last.exp + un]); h)
concat(f, [s, un])
leftMult(s, f) ==
empty? f => s::$
s = f.first.gen => concat([s, f.first.exp + un], rest f)
concat([s, un], f)
commutativeEquality(s1:$, s2:$):Boolean ==
#s1 ~= #s2 => false
for t1 in s1 repeat
if not member?(t1,s2) then return false
true
plus!(s:S, n:E, f:$):$ ==
h := g := concat([s, n], f)
h1 := rest h
while not empty? h1 repeat
s = h1.first.gen =>
l :=
zero?(m := n + h1.first.exp) => rest h1
concat([s, m], rest h1)
setrest!(h, l)
return rest g
h := h1
h1 := rest h1
g
plus(s, n, f) == plus!(s,n,copy f)
plus(f, g) ==
#f < #g => localplus(f, g)
localplus(g, f)
localplus(f, g) ==
g := copy g
for x in f repeat
g := plus(x.gen, x.exp, g)
g
)abbrev category FMONCAT FreeMonoidCategory
++ Free monoid on any set of generators
++ Author: Stephen M. Watt, Gabriel Dos Reis
++ Date Created: September 26, 2009
++ Date Last Updated: September 26, 2009
++ Description:
++ A 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.
FreeMonoidCategory(S: SetCategory): Category == Exports where
macro NNI == NonNegativeInteger
macro REC == Record(gen: S, exp: NonNegativeInteger)
macro Ex == OutputForm
Exports == Join(Monoid, RetractableTo S) with
*: (S, %) -> %
++ \spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
*: (%, S) -> %
++ \spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
**: (S, NonNegativeInteger) -> %
++ \spad{s ** n} returns the product of \spad{s} by itself \spad{n} times.
hclf: (%, %) -> %
++ \spad{hclf(x, y)} returns the highest common left factor of
++ \spad{x} and \spad{y},
++ i.e. 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,
++ i.e. 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 i.e.
++ 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 i.e.
++ 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, i.e. \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,
++ i.e. \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
)abbrev domain FMONOID FreeMonoid
++ Free monoid on any set of generators
++ Author: Stephen M. Watt
++ Date Created: ???
++ Date Last Updated: 6 June 1991
++ Description:
++ 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: SetCategory): FreeMonoidCategory(S) == FMdefinition where
macro NNI == NonNegativeInteger
macro REC == Record(gen: S, exp: NonNegativeInteger)
macro Ex == OutputForm
FMdefinition == 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:$ ==
one? f => g
one? g => 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) ==
one? la or one? ar => [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) ==
one? a => [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 positive? na and positive? nb 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
)abbrev domain FGROUP FreeGroup
++ Free group on any set of generators
++ Author: Stephen M. Watt
++ Date Created: ???
++ Date Last Updated: 6 June 1991
++ Description:
++ The free group on a set S is the group of finite products of
++ the form \spad{reduce(*,[si ** ni])} where the si's are in S, and the ni's
++ are integers. The multiplication is not commutative.
FreeGroup(S: SetCategory): Join(Group, 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, Integer) -> $
++ s ** n returns the product of s by itself n times.
size : $ -> NonNegativeInteger
++ size(x) returns the number of monomials in x.
nthExpon : ($, Integer) -> Integer
++ nthExpon(x, n) returns the exponent of the n^th monomial of x.
nthFactor : ($, Integer) -> S
++ nthFactor(x, n) returns the factor of the n^th monomial of x.
mapExpon : (Integer -> Integer, $) -> $
++ 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}.
factors : $ -> List Record(gen: S, exp: Integer)
++ factors(a1\^e1,...,an\^en) returns \spad{[[a1, e1],...,[an, en]]}.
== ListMonoidOps(S, Integer, 1) add
Rep := ListMonoidOps(S, Integer, 1)
1 == makeUnit()
one? f == empty? listOfMonoms f
s:S ** n:Integer == makeTerm(s, n)
f:$ * s:S == rightMult(f, s)
s:S * f:$ == leftMult(s, f)
inv f == reverse! mapExpon("-", f)
factors f == copy listOfMonoms f
mapExpon(f, x) == mapExpon(f, x)$Rep
mapGen(f, x) == mapGen(f, x)$Rep
coerce(f:$):OutputForm == outputForm(f, "*", "**", 1)
f:$ * g:$ ==
one? f => g
one? g => f
r := reverse listOfMonoms f
q := copy listOfMonoms g
while not empty? r and not empty? q and r.first.gen = q.first.gen
and r.first.exp = -q.first.exp repeat
r := rest r
q := rest q
empty? r => makeMulti q
empty? q => makeMulti reverse! r
r.first.gen = q.first.gen =>
setlast!(h := reverse! r,
[q.first.gen, q.first.exp + r.first.exp])
makeMulti concat!(h, rest q)
makeMulti concat!(reverse! r, q)
)abbrev category FAMONC FreeAbelianMonoidCategory
++ Category for free abelian monoid on any set of generators
++ Author: Manuel Bronstein
++ Date Created: November 1989
++ Date Last Updated: 6 June 1991
++ Description:
++ A free abelian monoid on a set S is the monoid of finite sums of
++ the form \spad{reduce(+,[ni * si])} where the si's are in S, and the ni's
++ are in a given abelian monoid. The operation is commutative.
FreeAbelianMonoidCategory(S: SetCategory, E:CancellationAbelianMonoid): Category ==
Join(CancellationAbelianMonoid, RetractableTo S) with
+ : (S, $) -> $
++ s + x returns the sum of s and x.
* : (E, S) -> $
++ e * s returns e times s.
size : $ -> NonNegativeInteger
++ size(x) returns the number of terms in x.
++ mapGen(f, a1\^e1 ... an\^en) returns \spad{f(a1)\^e1 ... f(an)\^en}.
terms : $ -> List Record(gen: S, exp: E)
++ terms(e1 a1 + ... + en an) returns \spad{[[a1, e1],...,[an, en]]}.
nthCoef : ($, Integer) -> E
++ nthCoef(x, n) returns the coefficient of the n^th term of x.
nthFactor : ($, Integer) -> S
++ nthFactor(x, n) returns the factor of the n^th term of x.
coefficient: (S, $) -> E
++ coefficient(s, e1 a1 + ... + en an) returns ei such that
++ ai = s, or 0 if s is not one of the ai's.
mapCoef : (E -> E, $) -> $
++ mapCoef(f, e1 a1 +...+ en an) returns
++ \spad{f(e1) a1 +...+ f(en) an}.
mapGen : (S -> S, $) -> $
++ mapGen(f, e1 a1 +...+ en an) returns
++ \spad{e1 f(a1) +...+ en f(an)}.
if E has OrderedAbelianMonoid then
highCommonTerms: ($, $) -> $
++ highCommonTerms(e1 a1 + ... + en an, f1 b1 + ... + fm bm) returns
++ \spad{reduce(+,[max(ei, fi) ci])}
++ where ci ranges in the intersection
++ of \spad{{a1,...,an}} and \spad{{b1,...,bm}}.
)abbrev domain IFAMON InnerFreeAbelianMonoid
++ Internal free abelian monoid on any set of generators
++ Author: Manuel Bronstein
++ Date Created: November 1989
++ Date Last Updated: 6 June 1991
++ Description:
++ Internal implementation of a free abelian monoid.
InnerFreeAbelianMonoid(S: SetCategory, E:CancellationAbelianMonoid, un:E):
FreeAbelianMonoidCategory(S, E) == ListMonoidOps(S, E, un) add
Rep := ListMonoidOps(S, E, un)
0 == makeUnit()
zero? f == empty? listOfMonoms f
terms f == copy listOfMonoms f
nthCoef(f, i) == nthExpon(f, i)
nthFactor(f, i) == nthFactor(f, i)$Rep
s:S + f:$ == plus(s, un, f)
f:$ + g:$ == plus(f, g)
(f:$ = g:$):Boolean == commutativeEquality(f,g)
n:E * s:S == makeTerm(s, n)
n:NonNegativeInteger * f:$ == mapExpon(n * #1, f)
coerce(f:$):OutputForm == outputForm(f, "+", #2 * #1, 0)
mapCoef(f, x) == mapExpon(f, x)
mapGen(f, x) == mapGen(f, x)$Rep
coefficient(s, f) ==
for x in terms f repeat
x.gen = s => return(x.exp)
0
if E has OrderedAbelianMonoid then
highCommonTerms(f, g) ==
makeMulti [[x.gen, min(x.exp, n)] for x in listOfMonoms f
| positive?(n := coefficient(x.gen, g))]
)abbrev domain FAMONOID FreeAbelianMonoid
++ Free abelian monoid on any set of generators
++ Author: Manuel Bronstein
++ Date Created: November 1989
++ Date Last Updated: 6 June 1991
++ Description:
++ The free abelian monoid on a set S is the monoid of finite sums of
++ the form \spad{reduce(+,[ni * si])} where the si's are in S, and the ni's
++ are non-negative integers. The operation is commutative.
FreeAbelianMonoid(S: SetCategory):
FreeAbelianMonoidCategory(S, NonNegativeInteger)
== InnerFreeAbelianMonoid(S, NonNegativeInteger, 1)
)abbrev domain FAGROUP FreeAbelianGroup
++ Free abelian group on any set of generators
++ Author: Manuel Bronstein
++ Date Created: November 1989
++ Date Last Updated: 6 June 1991
++ Description:
++ The free abelian group on a set S is the monoid of finite sums of
++ the form \spad{reduce(+,[ni * si])} where the si's are in S, and the ni's
++ are integers. The operation is commutative.
FreeAbelianGroup(S:SetCategory): Exports == Implementation where
Exports ==> Join(AbelianGroup, Module Integer,
FreeAbelianMonoidCategory(S, Integer)) with
if S has OrderedSet then OrderedSet
Implementation ==> InnerFreeAbelianMonoid(S, Integer, 1) add
- f == mapCoef("-", f)
if S has OrderedSet then
inmax: List Record(gen: S, exp: Integer) -> Record(gen: S, exp:Integer)
inmax l ==
mx := first l
for t in rest l repeat
if t.gen > mx.gen then mx := t
mx
a < b ==
zero? a =>
zero? b => false
positive?((inmax terms b).exp)
ta := inmax terms a
zero? b => negative? ta.exp
ta := inmax terms a
tb := inmax terms b
ta.gen < tb.gen => positive? tb.exp
ta.gen > tb.gen => negative? ta.exp
ta.exp < tb.exp => true
ta.exp > tb.exp => false
lc := ta.exp * ta.gen
(a - lc) < (b - lc)
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