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)abbrev package ORDFUNS OrderingFunctions
++ Author: Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: OrderedDirectProduct
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides ordering functions on vectors which
++ are suitable parameters for OrderedDirectProduct.
OrderingFunctions(dim,S) : T == C where
dim : NonNegativeInteger
S : OrderedAbelianMonoid
VS == Vector S
T == with
pureLex : (VS,VS) -> Boolean
++ pureLex(v1,v2) return true if the vector v1 is less than the
++ vector v2 in the lexicographic ordering.
totalLex : (VS,VS) -> Boolean
++ totalLex(v1,v2) return true if the vector v1 is less than the
++ vector v2 in the ordering which is total degree refined by
++ lexicographic ordering.
reverseLex : (VS,VS) -> Boolean
++ reverseLex(v1,v2) return true if the vector v1 is less than the
++ vector v2 in the ordering which is total degree refined by
++ the reverse lexicographic ordering.
C == add
n:NonNegativeInteger:=dim
-- pure lexicographical ordering
pureLex(v1:VS,v2:VS) : Boolean ==
for i in 1..n repeat
if qelt(v1,i) < qelt(v2,i) then return true
if qelt(v2,i) < qelt(v1,i) then return false
false
-- total ordering refined with lex
totalLex(v1:VS,v2:VS) :Boolean ==
n1:S:=0
n2:S:=0
for i in 1..n repeat
n1:= n1+qelt(v1,i)
n2:=n2+qelt(v2,i)
n1<n2 => true
n2<n1 => false
for i in 1..n repeat
if qelt(v1,i) < qelt(v2,i) then return true
if qelt(v2,i) < qelt(v1,i) then return false
false
-- reverse lexicographical ordering
reverseLex(v1:VS,v2:VS) :Boolean ==
n1:S:=0
n2:S:=0
for i in 1..n repeat
n1:= n1+qelt(v1,i)
n2:=n2+qelt(v2,i)
n1<n2 => true
n2<n1 => false
for i in reverse(1..n) repeat
if qelt(v2,i) < qelt(v1,i) then return true
if qelt(v1,i) < qelt(v2,i) then return false
false
)abbrev domain ODP OrderedDirectProduct
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: Vector, DirectProduct
++ Also See: HomogeneousDirectProduct, SplitHomogeneousDirectProduct
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This type represents the finite direct or cartesian product of an
++ underlying ordered component type. The ordering on the type is determined
++ by its third argument which represents the less than function on
++ vectors. This type is a suitable third argument for
++ \spadtype{GeneralDistributedMultivariatePolynomial}.
OrderedDirectProduct(dim:NonNegativeInteger,
S:OrderedAbelianMonoidSup,
f:(Vector(S),Vector(S))->Boolean):T
== C where
T == DirectProductCategory(dim,S)
C == DirectProduct(dim,S) add
Rep:=Vector(S)
x:% < y:% == f(x::Rep,y::Rep)
)abbrev domain HDP HomogeneousDirectProduct
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: Vector, DirectProduct
++ Also See: OrderedDirectProduct, SplitHomogeneousDirectproduct
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This type represents the finite direct or cartesian product of an
++ underlying ordered component type. The vectors are ordered first
++ by the sum of their components, and then refined using a reverse
++ lexicographic ordering. This type is a suitable third argument for
++ \spadtype{GeneralDistributedMultivariatePolynomial}.
HomogeneousDirectProduct(dim,S) : T == C where
dim : NonNegativeInteger
S : OrderedAbelianMonoidSup
T == DirectProductCategory(dim,S)
C == DirectProduct(dim,S) add
Rep:=Vector(S)
v1:% < v2:% ==
-- reverse lexicographical ordering
n1:S:=0
n2:S:=0
for i in 1..dim repeat
n1:= n1+qelt(v1,i)
n2:=n2+qelt(v2,i)
n1<n2 => true
n2<n1 => false
for i in reverse(1..dim) repeat
if qelt(v2,i) < qelt(v1,i) then return true
if qelt(v1,i) < qelt(v2,i) then return false
false
)abbrev domain SHDP SplitHomogeneousDirectProduct
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors: Vector, DirectProduct
++ Also See: OrderedDirectProduct, HomogeneousDirectProduct
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This type represents the finite direct or cartesian product of an
++ underlying ordered component type. The vectors are ordered as if
++ they were split into two blocks. The dim1 parameter specifies the
++ length of the first block. The ordering is lexicographic between
++ the blocks but acts like \spadtype{HomogeneousDirectProduct}
++ within each block. This type is a suitable third argument for
++ \spadtype{GeneralDistributedMultivariatePolynomial}.
SplitHomogeneousDirectProduct(dimtot,dim1,S) : T == C where
NNI ==> NonNegativeInteger
dim1,dimtot : NNI
S : OrderedAbelianMonoidSup
T == DirectProductCategory(dimtot,S)
C == DirectProduct(dimtot,S) add
Rep:=Vector(S)
lessThanRlex(v1:%,v2:%,low:NNI,high:NNI):Boolean ==
-- reverse lexicographical ordering
n1:S:=0
n2:S:=0
for i in low..high repeat
n1:= n1+qelt(v1,i)
n2:=n2+qelt(v2,i)
n1<n2 => true
n2<n1 => false
for i in reverse(low..high) repeat
if qelt(v2,i) < qelt(v1,i) then return true
if qelt(v1,i) < qelt(v2,i) then return false
false
(v1:% < v2:%):Boolean ==
lessThanRlex(v1,v2,1,dim1) => true
for i in 1..dim1 repeat
if qelt(v1,i) ~= qelt(v2,i) then return false
lessThanRlex(v1,v2,dim1+1,dimtot)
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