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--All rights reserved.
--Copyright (C) 2007-2012, Gabriel Dos Reis.
--All rights reversed.
--
--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 category BINOPC BinaryOperatorCategory
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This is the category of all domains that implement binary operations.
BinaryOperatorCategory(T: Type): Category == MappingCategory(T,T,T)
)abbrev domain BINOP BinaryOperation
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This domain implements binary operations.
BinaryOperation(T: Type): Join(BinaryOperatorCategory T, SetCategory) with
binaryOperation: ((T,T) -> T) -> %
++ \spad{binaryOperation f} constructs a binary operation value
++ out of any homogeneous mapping of arity 2.
== (T,T) -> T add
binaryOperation f == per f
elt(f,x,y) == rep(f)(x,y)
)abbrev category IDEMOPC IdempotentOperatorCategory
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This is the category of all domains that implement idempotent operations.
IdempotentOperatorCategory(T: BasicType): Category ==
BinaryOperatorCategory T with
assume idempotence ==
forall(f: %, x: T) . f(x,x) = x
)abbrev category SGPOPC SemiGroupOperatorCategory
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This is the category of all domains that implement semigroup operations
SemiGroupOperatorCategory(T: BasicType): Category ==
BinaryOperatorCategory T with
assume associativity ==
forall(f: %, x: T, y:T, z: T) . f(f(x,y),z) = f(x,f(y,z))
)abbrev domain SGPOP SemiGroupOperation
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This domain implements semigroup operations.
SemiGroupOperation(T: BasicType): Public == Private where
Public == Join(SemiGroupOperatorCategory T,SetCategory) with
semiGroupOperation: ((T,T) -> T) -> %
++ \spad{semiGroupOperation f} constructs a semigroup operation
++ out of a binary homogeneous mapping known to be associative.
Private == BinaryOperation T
)abbrev category MONOPC MonoidOperatorCategory
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This is the category of all domains that implement monoid operations
MonoidOperatorCategory(T: BasicType): Category ==
SemiGroupOperatorCategory T with
neutralValue: % -> T
++ \spad{neutralValue f} returns the neutral value of the
++ monoid operation \spad{f}.
assume neutrality ==
forall(f: %, x: T) .
f(x, neutralValue f) = x
f(neutralValue f, x) = x
)abbrev domain MONOP MonoidOperation
++ Author: Gabriel Dos Reis
++ Date Created: February 24, 2012
++ Date Last Modified: February 24, 2012
++ Description:
++ This domain implements monoid operations.
MonoidOperation(T: BasicType): Public == Private where
Public == Join(MonoidOperatorCategory T,SetCategory,_
CoercibleTo SemiGroupOperation T) with
monoidOperation: ((T,T) -> T, T) -> %
++ \spad{monoidOperation(f,e)} constructs a operation from
++ the binary mapping \spad{f} with neutral value \spad{e}.
Private == Pair(SemiGroupOperation T,T) add
monoidOperation(f,e) == per pair(semiGroupOperation f,e)
elt(op,x,y) == first(rep op)(x,y)
neutralValue op == second rep op
coerce(f: %): SemiGroupOperation T == first rep f
import Boolean
)abbrev category BASTYPE BasicType
--% BasicType
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{BasicType} is the basic category for describing a collection
++ of elements with \spadop{=} (equality).
BasicType(): Category == Type with
=: (%,%) -> Boolean ++ x=y tests if x and y are equal.
~=: (%,%) -> Boolean ++ x~=y tests if x and y are not equal.
before?: (%,%) -> Boolean
++ \spad{before?(x,y)} holds if the system representation
++ of \spad{x} comes before that of \spad{y} in a
++ an implementation defined manner.
add
x = y == not before?(x,y) and not before?(y,x)
x:% ~= y:% == not(x=y)
before?(x,y) == %before?(x,y)$Foreign(Builtin)
)abbrev category ORDTYPE OrderedType
++ Author: Gabriel Dos Reis
++ Date Created: June 28, 2010
++ Date Last Modified: June 28, 2010
++ See Also: OrderedSet
++ Description:
++ Category of types equipped with a total ordering.
++ Axioms:
++ forall(x,y)
++ x > y <=> y < x
++ x <= y <=> not(y > x)
++ x >= y <=> not(x < y)
++ x <= y and x >= y => x = y
OrderedType(): Category == BasicType with
< : (%,%) -> Boolean
++ \spad{x < y} holds if \spad{x} is less than \spad{y} in the
++ current domain.
> : (%,%) -> Boolean
++ \spad{x > y} holds if \spad{x} is greater than \spad{y} in the
++ current domain.
<= : (%,%) -> Boolean
++ \spad{x <= y} holds if \spad{x} is less or equal than \spad{y}
++ in the current domain.
>= : (%,%) -> Boolean
++ \spad{x <= y} holds if \spad{x} is greater or equal than \spad{y}
++ in the current domain.
max: (%,%) -> %
++ \spad{max(x,y)} returns the maximum of \spad{x} and \spad{y}
++ relative to the ordering.
min: (%,%) -> %
++ \spad{min(x,y)} returns the minimum of \spad{x} and \spad{y}
++ relative to the ordering.
add
x > y == y < x
x <= y == not(y < x)
x >= y == not(x < y)
x = y == not(x < y or y < x)
max(x,y) ==
x < y => y
x
min(x,y) ==
x < y => x
y
before?(x,y) == x < y
++ Author: Gabriel Dos Reis
++ Date Created: June 28, 2010
++ Date Last Modified: June 28, 2010
++ See Also: OrderedType
++ Description:
++ This domain turns any total ordering \spad{f} on a type \spad{T} into
++ a model of the category \spadtype{OrderedType}.
)abbrev domain ORDSTRCT OrderedStructure
OrderedStructure(T: Type,f: (T,T) -> Boolean): Public == Private where
Public == Join(OrderedType,HomotopicTo T) with
if T has CoercibleTo OutputForm then CoercibleTo OutputForm
Private == add
Rep == T
coerce(x: %): T == rep x
coerce(y: T): % == per y
x < y == f(rep x,rep y)
if T has CoercibleTo OutputForm then
coerce(x: %): OutputForm == rep(x)::OutputForm
)abbrev category SETCAT SetCategory
++ Author:
++ Date Created:
++ Date Last Updated:
++ 09/09/92 RSS added latex and hash
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ \spadtype{SetCategory} is the basic category for describing a collection
++ of elements with \spadop{=} (equality) and \spadfun{coerce} to output form.
++
++ Conditional Attributes:
++ canonical\tab{15}data structure equality is the same as \spadop{=}
SetCategory(): Category == Join(BasicType,CoercibleTo OutputForm) with
--operations
hash: % -> SingleInteger ++ hash(s) calculates a hash code for s.
latex: % -> String ++ latex(s) returns a LaTeX-printable output
++ representation of s.
add
import %hash: % -> SingleInteger from Foreign Builtin
hash(s : %): SingleInteger == %hash s
latex(s : %): String == "\mbox{\bf Unimplemented}"
)abbrev category STEP StepThrough
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A class of objects which can be 'stepped through'.
++ Repeated applications of \spadfun{nextItem} is guaranteed never to
++ return duplicate items and only return "failed" after exhausting
++ all elements of the domain.
++ This assumes that the sequence starts with \spad{init()}.
++ For non-fiinite domains, repeated application
++ of \spadfun{nextItem} is not required to reach all possible domain elements
++ starting from any initial element.
++
StepThrough(): Category == SetCategory with
--operations
init: %
++ init() chooses an initial object for stepping.
nextItem: % -> Maybe %
++ \spad{nextItem(x)} returns the next item, or \spad{failed}
++ if domain is exhausted.
import PositiveInteger
)abbrev category SGROUP SemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ the class of all multiplicative semigroups, i.e. a set
++ with an associative operation \spadop{*}.
++
++ Axioms:
++ \spad{associative("*":(%,%)->%)}\tab{30}\spad{ (x*y)*z = x*(y*z)}
++
++ Conditional attributes:
++ \spad{commutative("*":(%,%)->%)}\tab{30}\spad{ x*y = y*x }
SemiGroup(): Category == SetCategory with
--operations
*: (%,%) -> % ++ x*y returns the product of x and y.
**: (%,PositiveInteger) -> % ++ x**n returns the repeated product
++ of x n times, i.e. exponentiation.
add
import RepeatedSquaring(%)
x:% ** n:PositiveInteger == expt(x,n)
)abbrev category MONOID Monoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative monoids, i.e. semigroups with a
++ multiplicative identity element.
++
++ Axioms:
++ \spad{leftIdentity("*":(%,%)->%,1)}\tab{30}\spad{1*x=x}
++ \spad{rightIdentity("*":(%,%)->%,1)}\tab{30}\spad{x*1=x}
++
++ Conditional attributes:
++ unitsKnown\tab{15}\spadfun{recip} only returns "failed" on non-units
Monoid(): Category == SemiGroup with
--operations
1: % ++ 1 is the multiplicative identity.
sample: % ++ sample yields a value of type %
one?: % -> Boolean ++ one?(x) tests if x is equal to 1.
**: (%,NonNegativeInteger) -> % ++ x**n returns the repeated product
++ of x n times, i.e. exponentiation.
recip: % -> Union(%,"failed")
++ recip(x) tries to compute the multiplicative inverse for x
++ or "failed" if it cannot find the inverse (see unitsKnown).
add
import RepeatedSquaring(%)
one? x == x = 1
sample() == 1
recip x ==
one? x => x
"failed"
x:% ** n:NonNegativeInteger ==
zero? n => 1
expt(x,n pretend PositiveInteger)
)abbrev category GROUP Group
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative groups, i.e. monoids with
++ multiplicative inverses.
++
++ Axioms:
++ \spad{leftInverse("*":(%,%)->%,inv)}\tab{30}\spad{ inv(x)*x = 1 }
++ \spad{rightInverse("*":(%,%)->%,inv)}\tab{30}\spad{ x*inv(x) = 1 }
Group(): Category == Monoid with
--operations
inv: % -> % ++ inv(x) returns the inverse of x.
/: (%,%) -> % ++ x/y is the same as x times the inverse of y.
**: (%,Integer) -> % ++ x**n returns x raised to the integer power n.
unitsKnown ++ unitsKnown asserts that recip only returns
++ "failed" for non-units.
conjugate: (%,%) -> %
++ conjugate(p,q) computes \spad{inv(q) * p * q}; this is 'right action
++ by conjugation'.
commutator: (%,%) -> %
++ commutator(p,q) computes \spad{inv(p) * inv(q) * p * q}.
add
import RepeatedSquaring(%)
x:% / y:% == x*inv(y)
recip(x:%) == inv(x)
x:% ** n:Integer ==
zero? n => 1
negative? n => expt(inv(x),(-n) pretend PositiveInteger)
expt(x,n pretend PositiveInteger)
conjugate(p,q) == inv(q) * p * q
commutator(p,q) == inv(p) * inv(q) * p * q
import PositiveInteger
)abbrev category ABELSG AbelianSemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ the class of all additive (commutative) semigroups, i.e.
++ a set with a commutative and associative operation \spadop{+}.
++
++ Axioms:
++ \spad{associative("+":(%,%)->%)}\tab{30}\spad{ (x+y)+z = x+(y+z) }
++ \spad{commutative("+":(%,%)->%)}\tab{30}\spad{ x+y = y+x }
AbelianSemiGroup(): Category == SetCategory with
--operations
+: (%,%) -> % ++ x+y computes the sum of x and y.
*: (PositiveInteger,%) -> %
++ n*x computes the left-multiplication of x by the positive integer n.
++ This is equivalent to adding x to itself n times.
add
import RepeatedDoubling(%)
if not (% has Ring) then
n:PositiveInteger * x:% == double(n,x)
)abbrev category ABELMON AbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of multiplicative monoids, i.e. semigroups with an
++ additive identity element.
++
++ Axioms:
++ \spad{leftIdentity("+":(%,%)->%,0)}\tab{30}\spad{ 0+x=x }
++ \spad{rightIdentity("+":(%,%)->%,0)}\tab{30}\spad{ x+0=x }
-- following domain must be compiled with subsumption disabled
-- define SourceLevelSubset to be EQUAL
AbelianMonoid(): Category == AbelianSemiGroup with
--operations
0: %
++ 0 is the additive identity element.
sample: %
++ sample yields a value of type %
zero?: % -> Boolean
++ zero?(x) tests if x is equal to 0.
*: (NonNegativeInteger,%) -> %
++ n * x is left-multiplication by a non negative integer
add
import RepeatedDoubling(%)
zero? x == x = 0
n:PositiveInteger * x:% == (n::NonNegativeInteger) * x
sample() == 0
if not (% has Ring) then
n:NonNegativeInteger * x:% ==
zero? n => 0
double(n pretend PositiveInteger,x)
)abbrev category CABMON CancellationAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager I
++ Description:
++ This is an \spadtype{AbelianMonoid} with the cancellation property, i.e.
++ \spad{ a+b = a+c => b=c }.
++ This is formalised by the partial subtraction operator,
++ which satisfies the axioms listed below:
++
++ Axioms:
++ \spad{c = a+b <=> c-b = a}
CancellationAbelianMonoid(): Category == AbelianMonoid with
--operations
subtractIfCan: (%,%) -> Union(%,"failed")
++ subtractIfCan(x, y) returns an element z such that \spad{z+y=x}
++ or "failed" if no such element exists.
)abbrev category ABELGRP AbelianGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of abelian groups, i.e. additive monoids where
++ each element has an additive inverse.
++
++ Axioms:
++ \spad{-(-x) = x}
++ \spad{x+(-x) = 0}
-- following domain must be compiled with subsumption disabled
AbelianGroup(): Category == Join(CancellationAbelianMonoid, LeftLinearSet Integer) with
-: % -> % ++ \spad{-x} is the additive inverse of \spad{x}
"-": (%,%) -> % ++ \spad{x-y} is the difference of \spad{x}
++ and \spad{y} i.e. \spad{x + (-y)}.
add
(x:% - y:%):% == x+(-y)
subtractIfCan(x:%, y:%):Union(%, "failed") == (x-y) :: Union(%,"failed")
n:NonNegativeInteger * x:% == (n::Integer) * x
import RepeatedDoubling(%)
if not (% has Ring) then
n:Integer * x:% ==
zero? n => 0
positive? n => double(n pretend PositiveInteger,x)
double((-n) pretend PositiveInteger,-x)
)abbrev category RNG Rng
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of associative rings, not necessarily commutative, and not
++ necessarily with a 1. This is a combination of an abelian group
++ and a semigroup, with multiplication distributing over addition.
++
++ Axioms:
++ \spad{ x*(y+z) = x*y + x*z}
++ \spad{ (x+y)*z = x*z + y*z }
++
++ Conditional attributes:
++ \spadnoZeroDivisors\tab{25}\spad{ ab = 0 => a=0 or b=0}
Rng(): Category == Join(AbelianGroup,SemiGroup)
)abbrev category LLINSET LeftLinearSet
++ Author: Gabriel Dos Reis
++ Date Created: May 31, 2009
++ Date Last Modified: May 31, 2009
++ Description:
++ A set is an \spad{S}-left linear set if it is stable by left-dilation
++ by elements in the semigroup \spad{S}.
++ See Also: RightLinearSet.
LeftLinearSet(S: SemiGroup): Category == SetCategory with
*: (S,%) -> %
++ \spad{s*x} is the left-dilation of \spad{x} by \spad{s}.
)abbrev category RLINSET RightLinearSet
++ Author: Gabriel Dos Reis
++ Date Created: May 31, 2009
++ Date Last Modified: May 31, 2009
++ Description:
++ A set is an \spad{S}-right linear set if it is stable by right-dilation
++ by elements in the semigroup \spad{S}.
++ See Also: LeftLinearSet.
RightLinearSet(S: SemiGroup): Category == SetCategory with
*: (%,S) -> %
++ \spad{x*s} is the right-dilation of \spad{x} by \spad{s}.
)abbrev category LINSET LinearSet
++ Author: Gabriel Dos Reis
++ Date Created: May 31, 2009
++ Date Last Modified: May 31, 2009
++ Description:
++ A set is an \spad{S}-linear set if it is stable by dilation
++ by elements in the semigroup \spad{S}.
++ See Also: LeftLinearSet, RightLinearSet.
LinearSet(S: SemiGroup): Category == Join(LeftLinearSet S, RightLinearSet S)
)abbrev category LMODULE LeftModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of left modules over an rng (ring not necessarily with unit).
++ This is an abelian group which supports left multiplation by elements of
++ the rng.
++
++ Axioms:
++ \spad{ (a*b)*x = a*(b*x) }
++ \spad{ (a+b)*x = (a*x)+(b*x) }
++ \spad{ a*(x+y) = (a*x)+(a*y) }
LeftModule(R:Rng):Category == Join(AbelianGroup, LeftLinearSet R)
)abbrev category RMODULE RightModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of right modules over an rng (ring not necessarily with unit).
++ This is an abelian group which supports right multiplation by elements of
++ the rng.
++
++ Axioms:
++ \spad{ x*(a*b) = (x*a)*b }
++ \spad{ x*(a+b) = (x*a)+(x*b) }
++ \spad{ (x+y)*x = (x*a)+(y*a) }
RightModule(R:Rng):Category == Join(AbelianGroup, RightLinearSet R)
)abbrev category SRING SemiRing
++ Author: Gabriel Dos Reis
++ Date Created: March 7, 2011
++ Date Last Modified: March 7, 2011
++ Description:
++ The category of all semiring structures, e.g. triples (D,+,*)
++ such that (D,+) is an Abelian monoid and (D,*) is a monoid
++ with the following laws:
++ Axioms:
++ a * (b + c) = (a * b) + (a * c)
++ (b + c) * a = (b * a) + (c * a)
++ 0 * a = 0
++ a * 0 = 0
SemiRing(): Category == Join(AbelianMonoid,Monoid)
)abbrev category RING Ring
++ Author:
++ Date Created:
++ Date Last Updated: February 14, 2012
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of rings with unity, always associative, but
++ not necessarily commutative.
Ring(): Category == Join(Rng,SemiRing,LeftModule(%),CoercibleFrom Integer) with
--operations
characteristic: NonNegativeInteger
++ characteristic() returns the characteristic of the ring
++ this is the smallest positive integer n such that
++ \spad{n*x=0} for all x in the ring, or zero if no such n
++ exists.
--We can not make this a constant, since some domains are mutable
unitsKnown
++ recip truly yields
++ reciprocal or "failed" if not a unit.
++ Note: \spad{recip(0) = "failed"}.
add
n:Integer
coerce(n) == n * 1$%
)abbrev category DIOID Dioid
++ Author: Gabriel Dos Reis
++ Date Created: February 14, 2012
++ Date Last Modified: February 14, 2012
++ Description:
++ Dioid is the class of semirings where the addition operation
++ induces a canonical order relation.
Dioid(): Category == Join(OrderedAbelianMonoid,SemiRing)
)abbrev category BMODULE BiModule
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A \spadtype{BiModule} is both a left and right module with respect
++ to potentially different rings.
++
++ Axiom:
++ \spad{ r*(x*s) = (r*x)*s }
BiModule(R:Ring,S:Ring):Category ==
Join(LeftModule(R),RightModule(S)) with
leftUnitary ++ \spad{1 * x = x}
rightUnitary ++ \spad{x * 1 = x}
)abbrev category ENTIRER EntireRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Entire Rings (non-commutative Integral Domains), i.e. a ring
++ not necessarily commutative which has no zero divisors.
++
++ Axioms:
++ \spad{ab=0 => a=0 or b=0} -- known as noZeroDivisors
++ \spad{not(1=0)}
EntireRing():Category == Join(Ring,BiModule(%,%)) with
noZeroDivisors ++ if a product is zero then one of the factors
++ must be zero.
)abbrev category CHARZ CharacteristicZero
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Rings of Characteristic Zero.
CharacteristicZero():Category == Ring
)abbrev category CHARNZ CharacteristicNonZero
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Rings of Characteristic Non Zero
CharacteristicNonZero():Category == Ring with
charthRoot: % -> Maybe %
++ charthRoot(x) returns the pth root of x
++ where p is the characteristic of the ring.
)abbrev category COMRING CommutativeRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of commutative rings with unity, i.e. rings where
++ \spadop{*} is commutative, and which have a multiplicative identity.
++ element.
--CommutativeRing():Category == Join(Ring,BiModule(%:Ring,%:Ring)) with
CommutativeRing():Category == Join(Ring,BiModule(%,%)) with
commutative("*") ++ multiplication is commutative.
)abbrev category MODULE Module
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of modules over a commutative ring.
++
++ Axioms:
++ \spad{1*x = x}
++ \spad{(a*b)*x = a*(b*x)}
++ \spad{(a+b)*x = (a*x)+(b*x)}
++ \spad{a*(x+y) = (a*x)+(a*y)}
Module(R:CommutativeRing): Category == Join(BiModule(R,R), LinearSet R)
add
if not(R is %) then x:%*r:R == r*x
)abbrev category ALGEBRA Algebra
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of associative algebras (modules which are themselves rings).
++
++ Axioms:
++ \spad{(b+c)::% = (b::%) + (c::%)}
++ \spad{(b*c)::% = (b::%) * (c::%)}
++ \spad{(1::R)::% = 1::%}
++ \spad{b*x = (b::%)*x}
++ \spad{r*(a*b) = (r*a)*b = a*(r*b)}
Algebra(R:CommutativeRing): Category ==
Join(Ring, Module R, CoercibleFrom R)
add
coerce(x:R):% == x * 1$%
)abbrev category LINEXP LinearlyExplicitRingOver
++ Author:
++ Date Created:
++ Date Last Updated: June 18, 2010
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An extension of left-module with an explicit linear dependence test.
LinearlyExplicitRingOver(R:Ring): Category == LeftModule R with
leftReducedSystem: Vector % -> Matrix R
++ \spad{leftReducedSystem [v1,...,vn]} returns a matrix \spad{M}
++ with coefficients in \spad{R} such that the system of equations
++ \spad{c1*v1 + ... + cn*vn = 0$%} has the same solution as
++ \spad{c * M = 0} where \spad{c} is the row vector \spad{[c1,...cn]}.
leftReducedSystem: (Vector %,%) -> Record(mat: Matrix R,vec: Vector R)
++ \spad{reducedSystem([v1,...,vn],u)} returns a matrix \spad{M}
++ with coefficients in \spad{R} and a vector \spad{w} such
++ that the system of equations \spad{c1*v1 + ... + cn*vn = u}
++ has the same solution as \spad{c * M = w} where \spad{c}
++ is the row vector \spad{[c1,...cn]}.
reducedSystem: Matrix % -> Matrix R
++ reducedSystem(A) returns a matrix B such that \spad{A x = 0} and \spad{B x = 0}
++ have the same solutions in R.
reducedSystem: (Matrix %,Vector %) -> Record(mat:Matrix R,vec:Vector R)
++ reducedSystem(A, v) returns a matrix B and a vector w such that
++ \spad{A x = v} and \spad{B x = w} have the same solutions in R.
)abbrev category FLINEXP FullyLinearlyExplicitRingOver
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ S is \spadtype{FullyLinearlyExplicitRingOver R} means that S is a
++ \spadtype{LinearlyExplicitRingOver R} and, in addition, if R is a
++ \spadtype{LinearlyExplicitRingOver Integer}, then so is S
FullyLinearlyExplicitRingOver(R:Ring):Category ==
LinearlyExplicitRingOver R with
if (R has LinearlyExplicitRingOver Integer) then
LinearlyExplicitRingOver Integer
add
if not(R is Integer) then
if (R has LinearlyExplicitRingOver Integer) then
reducedSystem(m:Matrix %):Matrix(Integer) ==
reducedSystem(reducedSystem(m)@Matrix(R))
reducedSystem(m:Matrix %, v:Vector %):
Record(mat:Matrix(Integer), vec:Vector(Integer)) ==
rec := reducedSystem(m, v)@Record(mat:Matrix R, vec:Vector R)
reducedSystem(rec.mat, rec.vec)
)abbrev category INTDOM IntegralDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager I
++ Description:
++ The category of commutative integral domains, i.e. commutative
++ rings with no zero divisors.
++
++ Conditional attributes:
++ canonicalUnitNormal\tab{20}the canonical field is the same for all associates
++ canonicalsClosed\tab{20}the product of two canonicals is itself canonical
IntegralDomain(): Category ==
-- Join(CommutativeRing, Algebra(%:CommutativeRing), EntireRing) with
Join(CommutativeRing, Algebra(%), EntireRing) with
--operations
exquo: (%,%) -> Union(%,"failed")
++ exquo(a,b) either returns an element c such that
++ \spad{c*b=a} or "failed" if no such element can be found.
unitNormal: % -> Record(unit:%,canonical:%,associate:%)
++ unitNormal(x) tries to choose a canonical element
++ from the associate class of x.
++ The attribute canonicalUnitNormal, if asserted, means that
++ the "canonical" element is the same across all associates of x
++ if \spad{unitNormal(x) = [u,c,a]} then
++ \spad{u*c = x}, \spad{a*u = 1}.
unitCanonical: % -> %
++ \spad{unitCanonical(x)} returns \spad{unitNormal(x).canonical}.
associates?: (%,%) -> Boolean
++ associates?(x,y) tests whether x and y are associates, i.e.
++ differ by a unit factor.
unit?: % -> Boolean
++ unit?(x) tests whether x is a unit, i.e. is invertible.
add
-- declaration
x,y: %
-- definitions
UCA ==> Record(unit:%,canonical:%,associate:%)
if not (% has Field) then
unitNormal(x) == [1$%,x,1$%]$UCA -- the non-canonical definition
unitCanonical(x) == unitNormal(x).canonical -- always true
recip(x) == if zero? x then "failed" else 1$% exquo x
unit?(x) == (recip x case "failed" => false; true)
if % has canonicalUnitNormal then
associates?(x,y) ==
(unitNormal x).canonical = (unitNormal y).canonical
else
associates?(x,y) ==
zero? x => zero? y
zero? y => false
x exquo y case "failed" => false
y exquo x case "failed" => false
true
)abbrev category GCDDOM GcdDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References: Davenport & Trager 1
++ Description:
++ This category describes domains where
++ \spadfun{gcd} can be computed but where there is no guarantee
++ of the existence of \spadfun{factor} operation for factorisation into irreducibles.
++ However, if such a \spadfun{factor} operation exist, factorization will be
++ unique up to order and units.
GcdDomain(): Category == IntegralDomain with
--operations
gcd: (%,%) -> %
++ gcd(x,y) returns the greatest common divisor of x and y.
-- gcd(x,y) = gcd(y,x) in the presence of canonicalUnitNormal,
-- but not necessarily elsewhere
gcd: List(%) -> %
++ gcd(l) returns the common gcd of the elements in the list l.
lcm: (%,%) -> %
++ lcm(x,y) returns the least common multiple of x and y.
-- lcm(x,y) = lcm(y,x) in the presence of canonicalUnitNormal,
-- but not necessarily elsewhere
lcm: List(%) -> %
++ lcm(l) returns the least common multiple of the elements of the list l.
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) ->
SparseUnivariatePolynomial %
++ gcdPolynomial(p,q) returns the greatest common divisor (gcd) of
++ univariate polynomials over the domain
add
lcm(x: %,y: %) ==
zero? y => 0
zero? x => 0
LCM : Union(%,"failed") := y exquo gcd(x,y)
LCM case % => x * LCM
error "bad gcd in lcm computation"
lcm(l:List %) == reduce(lcm,l,1,0)
gcd(l:List %) == reduce(gcd,l,0,1)
SUP ==> SparseUnivariatePolynomial
gcdPolynomial(p1,p2) ==
zero? p1 => unitCanonical p2
zero? p2 => unitCanonical p1
c1:= content(p1); c2:= content(p2)
p1:= (p1 exquo c1)::SUP %
p2:= (p2 exquo c2)::SUP %
if positive?(e1:=minimumDegree p1) then
p1:=(p1 exquo monomial(1,e1))::SUP %
if positive?(e2:=minimumDegree p2) then
p2:=(p2 exquo monomial(1,e2))::SUP %
e1:=min(e1,e2); c1:=gcd(c1,c2)
p1:=
zero? degree p1 or zero? degree p2 => monomial(c1,0)
p:= subResultantGcd(p1,p2)
zero? degree p => monomial(c1,0)
c2:= gcd(leadingCoefficient p1,leadingCoefficient p2)
unitCanonical(c1 * primitivePart(((c2*p) exquo leadingCoefficient p)::SUP %))
zero? e1 => p1
monomial(1,e1)*p1
)abbrev category UFD UniqueFactorizationDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A constructive unique factorization domain, i.e. where
++ we can constructively factor members into a product of
++ a finite number of irreducible elements.
UniqueFactorizationDomain(): Category == GcdDomain with
--operations
prime?: % -> Boolean
++ prime?(x) tests if x can never be written as the product of two
++ non-units of the ring,
++ i.e., x is an irreducible element.
squareFree : % -> Factored(%)
++ squareFree(x) returns the square-free factorization of x
++ i.e. such that the factors are pairwise relatively prime
++ and each has multiple prime factors.
squareFreePart: % -> %
++ squareFreePart(x) returns a product of prime factors of
++ x each taken with multiplicity one.
factor: % -> Factored(%)
++ factor(x) returns the factorization of x into irreducibles.
add
squareFreePart x ==
unit(s := squareFree x) * _*/[f.factor for f in factors s]
prime? x == one?(# factorList factor x)
)abbrev category PFECAT PolynomialFactorizationExplicit
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is the category of domains that know "enough" about
++ themselves in order to factor univariate polynomials over themselves.
++ This will be used in future releases for supporting factorization
++ over finitely generated coefficient fields, it is not yet available
++ in the current release of axiom.
PolynomialFactorizationExplicit(): Category == Definition where
P ==> SparseUnivariatePolynomial %
Definition ==>
UniqueFactorizationDomain with
-- operations
squareFreePolynomial: P -> Factored(P)
++ squareFreePolynomial(p) returns the
++ square-free factorization of the
++ univariate polynomial p.
factorPolynomial: P -> Factored(P)
++ factorPolynomial(p) returns the factorization
++ into irreducibles of the univariate polynomial p.
factorSquareFreePolynomial: P -> Factored(P)
++ factorSquareFreePolynomial(p) factors the
++ univariate polynomial p into irreducibles
++ where p is known to be square free
++ and primitive with respect to its main variable.
gcdPolynomial: (P, P) -> P
++ gcdPolynomial(p,q) returns the gcd of the univariate
++ polynomials p qnd q.
-- defaults to Euclidean, but should be implemented via
-- modular or p-adic methods.
solveLinearPolynomialEquation: (List P, P) -> Union(List P,"failed")
++ solveLinearPolynomialEquation([f1, ..., fn], g)
++ (where the fi are relatively prime to each other)
++ returns a list of ai such that
++ \spad{g/prod fi = sum ai/fi}
++ or returns "failed" if no such list of ai's exists.
if % has CharacteristicNonZero then
conditionP: Matrix % -> Union(Vector %,"failed")
++ conditionP(m) returns a vector of elements, not all zero,
++ whose \spad{p}-th powers (p is the characteristic of the domain)
++ are a solution of the homogenous linear system represented
++ by m, or "failed" is there is no such vector.
charthRoot: % -> Maybe %
++ charthRoot(r) returns the \spad{p}-th root of r, or
++ \spad{nothing} if none exists in the domain.
-- this is a special case of conditionP, but often the one we want
add
gcdPolynomial(f,g) ==
zero? f => g
zero? g => f
cf:=content f
if not one? cf then f:=(f exquo cf)::P
cg:=content g
if not one? cg then g:=(g exquo cg)::P
ans:=subResultantGcd(f,g)$P
gcd(cf,cg)*(ans exquo content ans)::P
if % has CharacteristicNonZero then
charthRoot f ==
-- to take p'th root of f, solve the system X-fY=0,
-- so solution is [x,y]
-- with x^p=X and y^p=Y, then (x/y)^p = f
zero? f => just 0
m:Matrix % := matrix [[1,-f]]
ans:= conditionP m
ans case "failed" => nothing
r := (ans.1) exquo (ans.2)
r case "failed" => nothing
just r
if % has Field then
solveLinearPolynomialEquation(lf,g) ==
multiEuclidean(lf,g)$P
else solveLinearPolynomialEquation(lf,g) ==
LPE ==> LinearPolynomialEquationByFractions %
solveLinearPolynomialEquationByFractions(lf,g)$LPE
)abbrev category PID PrincipalIdealDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of constructive principal ideal domains, i.e.
++ where a single generator can be constructively found for
++ any ideal given by a finite set of generators.
++ Note that this constructive definition only implies that
++ finitely generated ideals are principal. It is not clear
++ what we would mean by an infinitely generated ideal.
PrincipalIdealDomain(): Category == GcdDomain with
--operations
principalIdeal: List % -> Record(coef:List %,generator:%)
++ principalIdeal([f1,...,fn]) returns a record whose
++ generator component is a generator of the ideal
++ generated by \spad{[f1,...,fn]} whose coef component satisfies
++ \spad{generator = sum (input.i * coef.i)}
expressIdealMember: (List %,%) -> Maybe List %
++ \spad{expressIdealMember([f1,...,fn],h)} returns a representation
++ of h as a linear combination of the fi or \spad{nothing} if h
++ is not in the ideal generated by the fi.
)abbrev category EUCDOM EuclideanDomain
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A constructive euclidean domain, i.e. one can divide producing
++ a quotient and a remainder where the remainder is either zero
++ or is smaller (\spadfun{euclideanSize}) than the divisor.
++
++ Conditional attributes:
++ multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)}
++ additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)}
EuclideanDomain(): Category == PrincipalIdealDomain with
--operations
sizeLess?: (%,%) -> Boolean
++ sizeLess?(x,y) tests whether x is strictly
++ smaller than y with respect to the \spadfunFrom{euclideanSize}{EuclideanDomain}.
euclideanSize: % -> NonNegativeInteger
++ euclideanSize(x) returns the euclidean size of the element x.
++ Error: if x is zero.
divide: (%,%) -> Record(quotient:%,remainder:%)
++ divide(x,y) divides x by y producing a record containing a
++ \spad{quotient} and \spad{remainder},
++ where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain})
++ than the divisor y.
quo : (%,%) -> %
++ x quo y is the same as \spad{divide(x,y).quotient}.
++ See \spadfunFrom{divide}{EuclideanDomain}.
rem: (%,%) -> %
++ x rem y is the same as \spad{divide(x,y).remainder}.
++ See \spadfunFrom{divide}{EuclideanDomain}.
extendedEuclidean: (%,%) -> Record(coef1:%,coef2:%,generator:%)
-- formerly called princIdeal
++ extendedEuclidean(x,y) returns a record rec where
++ \spad{rec.coef1*x+rec.coef2*y = rec.generator} and
++ rec.generator is a gcd of x and y.
++ The gcd is unique only
++ up to associates if \spadatt{canonicalUnitNormal} is not asserted.
++ \spadfun{principalIdeal} provides a version of this operation
++ which accepts an arbitrary length list of arguments.
extendedEuclidean: (%,%,%) -> Union(Record(coef1:%,coef2:%),"failed")
-- formerly called expressIdealElt
++ extendedEuclidean(x,y,z) either returns a record rec
++ where \spad{rec.coef1*x+rec.coef2*y=z} or returns "failed"
++ if z cannot be expressed as a linear combination of x and y.
multiEuclidean: (List %,%) -> Union(List %,"failed")
++ multiEuclidean([f1,...,fn],z) returns a list of coefficients
++ \spad{[a1, ..., an]} such that
++ \spad{ z / prod fi = sum aj/fj}.
++ If no such list of coefficients exists, "failed" is returned.
add
-- declarations
x,y,z: %
l: List %
-- definitions
sizeLess?(x,y) ==
zero? y => false
zero? x => true
euclideanSize(x)<euclideanSize(y)
x quo y == divide(x,y).quotient --divide must be user-supplied
x rem y == divide(x,y).remainder
x exquo y ==
zero? x => 0
zero? y => "failed"
qr:=divide(x,y)
zero?(qr.remainder) => qr.quotient
"failed"
gcd(x,y) == --Euclidean Algorithm
x:=unitCanonical x
y:=unitCanonical y
while not zero? y repeat
(x,y):= (y,x rem y)
y:=unitCanonical y -- this doesn't affect the
-- correctness of Euclid's algorithm,
-- but
-- a) may improve performance
-- b) ensures gcd(x,y)=gcd(y,x)
-- if canonicalUnitNormal
x
IdealElt ==> Record(coef1:%,coef2:%,generator:%)
unitNormalizeIdealElt(s:IdealElt):IdealElt ==
(u,c,a):=unitNormal(s.generator)
one? a => s
[a*s.coef1,a*s.coef2,c]$IdealElt
extendedEuclidean(x,y) == --Extended Euclidean Algorithm
s1:=unitNormalizeIdealElt([1$%,0$%,x]$IdealElt)
s2:=unitNormalizeIdealElt([0$%,1$%,y]$IdealElt)
zero? y => s1
zero? x => s2
while not zero?(s2.generator) repeat
qr:= divide(s1.generator, s2.generator)
s3:=[s1.coef1 - qr.quotient * s2.coef1,
s1.coef2 - qr.quotient * s2.coef2, qr.remainder]$IdealElt
s1:=s2
s2:=unitNormalizeIdealElt s3
if not(zero?(s1.coef1)) and not sizeLess?(s1.coef1,y)
then
qr:= divide(s1.coef1,y)
s1.coef1:= qr.remainder
s1.coef2:= s1.coef2 + qr.quotient * x
s1 := unitNormalizeIdealElt s1
s1
TwoCoefs ==> Record(coef1:%,coef2:%)
extendedEuclidean(x,y,z) ==
zero? z => [0,0]$TwoCoefs
s:= extendedEuclidean(x,y)
(w:= z exquo s.generator) case "failed" => "failed"
zero? y =>
[s.coef1 * w, s.coef2 * w]$TwoCoefs
qr:= divide((s.coef1 * w), y)
[qr.remainder, s.coef2 * w + qr.quotient * x]$TwoCoefs
principalIdeal l ==
l = [] => error "empty list passed to principalIdeal"
rest l = [] =>
uca:=unitNormal(first l)
[[uca.unit],uca.canonical]
rest rest l = [] =>
u:= extendedEuclidean(first l,second l)
[[u.coef1, u.coef2], u.generator]
v:=principalIdeal rest l
u:= extendedEuclidean(first l,v.generator)
[[u.coef1,:[u.coef2*vv for vv in v.coef]],u.generator]
expressIdealMember(l,z) ==
zero? z => just [0 for v in l]
pid := principalIdeal l
(q := z exquo (pid.generator)) case "failed" => nothing
just [q*v for v in pid.coef]
multiEuclidean(l,z) ==
n := #l
zero? n => error "empty list passed to multiEuclidean"
one? n => [z]
l1 := copy l
l2 := split!(l1, n quo 2)
u:= extendedEuclidean(*/l1, */l2, z)
u case "failed" => "failed"
v1 := multiEuclidean(l1,u.coef2)
v1 case "failed" => "failed"
v2 := multiEuclidean(l2,u.coef1)
v2 case "failed" => "failed"
concat(v1,v2)
)abbrev category DIVRING DivisionRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A division ring (sometimes called a skew field),
++ i.e. a not necessarily commutative ring where
++ all non-zero elements have multiplicative inverses.
DivisionRing(): Category ==
Join(EntireRing, Algebra Fraction Integer) with
**: (%,Integer) -> %
++ x**n returns x raised to the integer power n.
inv : % -> %
++ inv x returns the multiplicative inverse of x.
++ Error: if x is 0.
-- Q-algebra is a lie, should be conditional on characteristic 0,
-- but knownInfo cannot handle the following commented
-- if % has CharacteristicZero then Algebra Fraction Integer
add
n: Integer
x: %
import RepeatedSquaring(%)
x ** n: Integer ==
zero? n => 1
zero? x =>
negative? n => error "division by zero"
x
negative? n =>
expt(inv x,(-n) pretend PositiveInteger)
expt(x,n pretend PositiveInteger)
-- if % has CharacteristicZero() then
q:Fraction(Integer) * x:% == numer(q) * inv(denom(q)::%) * x
)abbrev category FIELD Field
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of commutative fields, i.e. commutative rings
++ where all non-zero elements have multiplicative inverses.
++ The \spadfun{factor} operation while trivial is useful to have defined.
++
++ Axioms:
++ \spad{a*(b/a) = b}
++ \spad{inv(a) = 1/a}
Field(): Category == Join(EuclideanDomain,UniqueFactorizationDomain,
DivisionRing) with
--operations
/: (%,%) -> %
++ x/y divides the element x by the element y.
++ Error: if y is 0.
canonicalUnitNormal ++ either 0 or 1.
canonicalsClosed ++ since \spad{0*0=0}, \spad{1*1=1}
add
--declarations
x,y: %
n: Integer
-- definitions
UCA ==> Record(unit:%,canonical:%,associate:%)
unitNormal(x) ==
if zero? x then [1$%,0$%,1$%]$UCA else [x,1$%,inv(x)]$UCA
unitCanonical(x) == if zero? x then x else 1
associates?(x,y) == if zero? x then zero? y else not(zero? y)
inv x ==((u:=recip x) case "failed" => error "not invertible"; u)
x exquo y == (zero? y => "failed"; x / y)
gcd(x,y) == 1
euclideanSize(x) == 0
prime? x == false
squareFree x == x::Factored(%)
factor x == x::Factored(%)
x / y == (zero? y => error "catdef: division by zero"; x * inv(y))
divide(x,y) == [x / y,0]
)abbrev category FINITE Finite
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The category of domains composed of a finite set of elements.
++ We include the functions \spadfun{lookup} and \spadfun{index} to give a bijection
++ between the finite set and an initial segment of positive integers.
++
++ Axioms:
++ \spad{lookup(index(n)) = n}
++ \spad{index(lookup(s)) = s}
Finite(): Category == SetCategory with
size: () -> NonNegativeInteger
++ size() returns the number of elements in the set.
index: PositiveInteger -> %
++ index(i) takes a positive integer i less than or equal
++ to \spad{size()} and
++ returns the \spad{i}-th element of the set. This operation establishs a bijection
++ between the elements of the finite set and \spad{1..size()}.
lookup: % -> PositiveInteger
++ lookup(x) returns a positive integer such that
++ \spad{x = index lookup x}.
random: () -> %
++ random() returns a random element from the set.
add
--FIXME: Tthe only purpose of this local function is to bring
--FIXME: the compiler to admission that the successor of a
--FIXME: nonnegative integer has positive value.
--FIXME: Take it out when the its logic is sufficiently advanced.
succ(x: NonNegativeInteger): PositiveInteger ==
(1 + x) : PositiveInteger
random() ==
index succ random(size())$NonNegativeInteger
)abbrev category VSPACE VectorSpace
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Vector Spaces (not necessarily finite dimensional) over a field.
VectorSpace(S:Field): Category == Module(S) with
/ : (%, S) -> %
++ x/y divides the vector x by the scalar y.
dimension: () -> CardinalNumber
++ dimension() returns the dimensionality of the vector space.
add
(v:% / s:S):% == inv(s) * v
import Boolean
)abbrev category ORDSET OrderedSet
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ The class of totally ordered sets, that is, sets such that for each pair of elements \spad{(a,b)}
++ exactly one of the following relations holds \spad{a<b or a=b or b<a}
++ and the relation is transitive, i.e. \spad{a<b and b<c => a<c}.
OrderedSet(): Category == Join(SetCategory,OrderedType)
)abbrev category ORDFIN OrderedFinite
++ Author:
++ Date Created:
++ Date Last Updated: December 27, 2008
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered finite sets.
OrderedFinite(): Category == Join(OrderedSet, Finite) with
min: % ++ \spad{min} is the minimum value of %.
max: % ++ \spad{max} is the maximum value of %.
)abbrev category OSGROUP OrderedSemiGroup
++ Author: Gabriel Dos Reis
++ Date Create May 25, 2008
++ Date Last Updated: May 25, 2008
++ Description: Semigroups with compatible ordering.
OrderedSemiGroup(): Category == Join(OrderedSet, SemiGroup)
)abbrev category ORDMON OrderedMonoid
++ Author:
++ Date Created:
++ Date Last Updated: May 28, 2008
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also monoids, such that multiplication
++ preserves the ordering.
++
++ Axioms:
++ \spad{x < y => x*z < y*z}
++ \spad{x < y => z*x < z*y}
OrderedMonoid(): Category == Join(OrderedSemiGroup, Monoid)
)abbrev category OASGP OrderedAbelianSemiGroup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian semigroups, such that the addition
++ preserves the ordering.
++ \spad{ x < y => x+z < y+z}
OrderedAbelianSemiGroup(): Category == Join(OrderedSet, AbelianSemiGroup)
)abbrev category OAMON OrderedAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian monoids, such that the addition
++ preserves the ordering.
OrderedAbelianMonoid(): Category ==
Join(OrderedAbelianSemiGroup, AbelianMonoid) with
positive?: % -> Boolean
++ \spad{positive?(x)} holds when \spad{x} is greater
++ than \spad{0}.
add
positive? x == 0 < x
)abbrev category OCAMON OrderedCancellationAbelianMonoid
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian cancellation monoids, such that the addition
++ preserves the ordering.
OrderedCancellationAbelianMonoid(): Category ==
Join(OrderedAbelianMonoid, CancellationAbelianMonoid)
)abbrev category OAGROUP OrderedAbelianGroup
++ Author:
++ Date Created:
++ Date Last Updated: March 10, 2011
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also abelian groups, such that the addition preserves
++ the ordering.
OrderedAbelianGroup(): Category ==
Join(OrderedCancellationAbelianMonoid, AbelianGroup) with
negative?: % -> Boolean
++ \spad{negative?(x)} holds when \spad{x} is less than \spad{0}.
sign: % -> Integer
++ \spad{sign(x)} is \spad{1} if \spad{x} is positive,
++ \spad{-1} if \spad{x} is negative, and \spad{0} otherwise.
abs: % -> %
++ \spad{abs(x)} returns the absolute value of \spad{x}.
add
negative? x == x < 0
sign x ==
positive? x => 1
negative? x => -1
0
abs x ==
positive? x => x
negative? x => -x
0
)abbrev category ORDRING OrderedRing
++ Author:
++ Date Created:
++ Date Last Updated: March 10, 2011
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Ordered sets which are also rings, that is, domains where the ring
++ operations are compatible with the ordering.
++
++ Axiom:
++ \spad{0<a and b<c => ab< ac}
OrderedRing(): Category == Join(OrderedAbelianGroup,Ring,Monoid)
)abbrev category OINTDOM OrderedIntegralDomain
++ Author: JH Davenport (after L Gonzalez-Vega)
++ Date Created: 30.1.96
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Description:
++ The category of ordered commutative integral domains, where ordering
++ and the arithmetic operations are compatible
++
OrderedIntegralDomain(): Category ==
Join(IntegralDomain, OrderedRing)
)abbrev category OAMONS OrderedAbelianMonoidSup
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain is an OrderedAbelianMonoid with a \spadfun{sup} operation added.
++ The purpose of the \spadfun{sup} operator in this domain is to act as a supremum
++ with respect to the partial order imposed by \spadop{-}, rather than with respect to
++ the total \spad{>} order (since that is "max").
++
++ Axioms:
++ \spad{sup(a,b)-a \~~= "failed"}
++ \spad{sup(a,b)-b \~~= "failed"}
++ \spad{x-a \~~= "failed" and x-b \~~= "failed" => x >= sup(a,b)}
OrderedAbelianMonoidSup(): Category == OrderedCancellationAbelianMonoid with
--operation
sup: (%,%) -> %
++ sup(x,y) returns the least element from which both
++ x and y can be subtracted.
)abbrev category DIFFDOM DifferentialDomain
++ Author: Gabriel Dos Reis
++ Date Created: June 13, 2010
++ Date Last Modified: June 13, 2010
++ Description:
++ This category captures the interface of domains with a distinguished
++ operation named \spad{differentiate}. Usually, additional properties
++ are wanted. For example, that it obeys the usual Leibniz identity
++ of differentiation of product, in case of differential rings. One
++ could also want \spad{differentiate} to obey the chain rule when
++ considering differential manifolds.
++ The lack of specific requirement in this category is an implicit
++ admission that currently \Language{} is not expressive enough to
++ express the most general notion of differentiation in an adequate
++ manner, suitable for computational purposes.
DifferentialDomain(T: Type): Category == Type with
differentiate: % -> T
++ \spad{differentiate x} compute the derivative of \spad{x}.
D: % -> T
++ \spad{D x} is a shorthand for \spad{differentiate x}
add
D x ==
differentiate x
)abbrev category DIFFSPC DifferentialSpace
++ Author: Gabriel Dos Reis
++ Date Created: June 13, 2010
++ Date Last Modified: June 15, 2010
++ Description:
++ This category is like \spadtype{DifferentialDomain} where the
++ target of the differentiation operator is the same as its source.
DifferentialSpace(): Category == DifferentialDomain % with
differentiate: (%, NonNegativeInteger) -> %
++ \spad{differentiate(x,n)} returns the \spad{n}-th
++ derivative of \spad{x}.
D: (%, NonNegativeInteger) -> %
++ \spad{D(x, n)} returns the \spad{n}-th derivative of \spad{x}.
add
differentiate(r, n) ==
for i in 1..n repeat r := differentiate r
r
D(r,n) ==
differentiate(r,n)
)abbrev category DIFRING DifferentialRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ An ordinary differential ring, that is, a ring with an operation
++ \spadfun{differentiate}.
++
++ Axioms:
++ \spad{differentiate(x+y) = differentiate(x)+differentiate(y)}
++ \spad{differentiate(x*y) = x*differentiate(y) + differentiate(x)*y}
DifferentialRing(): Category == Join(Ring,DifferentialSpace)
)abbrev category DIFFMOD DifferentialModule
++ Author: Gabriel Dos Reis
++ Date Created: June 14, 2010
++ Date Last Updated: Jun 16, 2010
++ Related Constructors: Module, DifferentialSpace
++ Also See:
++ Description:
++ An R-module equipped with a distinguised differential operator.
++ If R is a differential ring, then differentiation on the module
++ should extend differentiation on the differential ring R. The
++ latter can be the null operator. In that case, the differentiation
++ operator on the module is just an R-linear operator. For that
++ reason, we do not require that the ring R be a DifferentialRing;
++
++ Axioms:
++ \spad{differentiate(x + y) = differentiate(x) + differentiate(x)}
++ \spad{differentiate(r*y) = r*differentiate(y) + differentiate(r)*y}
DifferentialModule(R: Ring): Category ==
Join(BiModule(R,R), DifferentialSpace) with
if R has CommutativeRing then Module R
)abbrev category DMEXT DifferentialModuleExtension
++ Author: Gabriel Dos Reis
++ Date Created: June 16, 2010
++ Date Last Updated: Jun 16, 2010
++ Related Constructors: Module, DifferentialSpaceExtension
++ Also See:
++ DifferentialExtension
++ Description:
++ Category of modules that extend differential rings.
++
DifferentialModuleExtension(R: Ring): Category ==
Join(BiModule(R,R),DifferentialSpaceExtension R) with
if R has DifferentialSpace then DifferentialModule R
if R has PartialDifferentialSpace Symbol then
PartialDifferentialModule(R,Symbol)
)abbrev category PDDOM PartialDifferentialDomain
++ Author: Gabriel Dos Reis
++ Date Created: June 16, 2010
++ Date Last Modified: June 16, 2010
++ Description:
++ This category captures the interface of domains with a distinguished
++ operation named \spad{differentiate} for partial differentiation with
++ respect to some domain of variables.
++ See Also:
++ DifferentialDomain, PartialDifferentialSpace
PartialDifferentialDomain(T: Type, S: Type): Category == Type with
differentiate: (%,S) -> T
++ \spad{differentiate(x,v)} computes the partial derivative
++ of \spad{x} with respect to \spad{v}.
D: (%,S) -> T
++ \spad{D(x,v)} is a shorthand for \spad{differentiate(x,v)}
add
D(x,v) ==
differentiate(x,v)
)abbrev category PDSPC PartialDifferentialSpace
++ Author: Gabriel Dos Reis
++ Date Created: June 16, 2010
++ Date Last Modified: June 16, 2010
++ Description:
++ This category captures the interface of domains stable by partial
++ differentiation with respect to variables from some domain.
++ See Also:
++ PartialDifferentialDomain
PartialDifferentialSpace(S: SetCategory): Category ==
PartialDifferentialDomain(%,S) with
differentiate: (%,List S) -> %
++ \spad{differentiate(x,[s1,...sn])} computes successive
++ partial derivatives, i.e.
++ \spad{differentiate(...differentiate(x, s1)..., sn)}.
differentiate: (%,S,NonNegativeInteger) -> %
++ \spad{differentiate(x,s,n)} computes multiple partial
++ derivatives, i.e. \spad{n}-th derivative of \spad{x}
++ with respect to \spad{s}.
differentiate: (%,List S,List NonNegativeInteger) -> %
++ \spad{differentiate(x,[s1,...,sn],[n1,...,nn])} computes
++ multiple partial derivatives, i.e.
D: (%,List S) -> %
++ \spad{D(x,[s1,...sn])} is a shorthand for
++ \spad{differentiate(x,[s1,...sn])}.
D: (%,S,NonNegativeInteger) -> %
++ \spad{D(x,s,n)} is a shorthand for \spad{differentiate(x,s,n)}.
D: (%,List S,List NonNegativeInteger) -> %
++ \spad{D(x,[s1,...,sn],[n1,...,nn])} is a shorthand for
++ \spad{differentiate(x,[s1,...,sn],[n1,...,nn])}.
add
differentiate(r: %,l: List S) ==
for s in l repeat r := differentiate(r, s)
r
differentiate(r: %,s: S,n: NonNegativeInteger) ==
for i in 1..n repeat r := differentiate(r, s)
r
differentiate(r: %,ls: List S,ln: List NonNegativeInteger) ==
for s in ls for n in ln repeat r := differentiate(r, s, n)
r
D(r: %,v: S) ==
differentiate(r,v)
D(r: %,lv: List S) ==
differentiate(r,lv)
D(r: %,v: S,n: NonNegativeInteger) ==
differentiate(r,v,n)
D(r: %,lv: List S,ln: List NonNegativeInteger) ==
differentiate(r, lv, ln)
)abbrev category PDRING PartialDifferentialRing
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A partial differential ring with differentiations indexed by a parameter type S.
++
++ Axioms:
++ \spad{differentiate(x+y,e) = differentiate(x,e)+differentiate(y,e)}
++ \spad{differentiate(x*y,e) = x*differentiate(y,e) + differentiate(x,e)*y}
PartialDifferentialRing(S:SetCategory): Category ==
Join(Ring,PartialDifferentialSpace S)
)abbrev category PDMOD PartialDifferentialModule
++ Author: Gabriel Dos Reis
++ Date Created: June 16, 2010
++ Date Last Updated: June 18, 2010
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A partial differential R-module with differentiations
++ indexed by a parameter type S.
++
++ Axioms:
++ \spad{differentiate(x+y,e) = differentiate(x,e)+differentiate(y,e)}
++ \spad{differentiate(r*x,e) = r*differentiate(x,e) + differentiate(r,e)*x}
++ \spad{differentiate(x*r,e) = x*differentiate(r,e) + differentiate(x,e)*r}
PartialDifferentialModule(R: Ring,S: SetCategory): Category ==
Join(BiModule(R,R),PartialDifferentialSpace S) with
if R has CommutativeRing then Module R
)abbrev category DIFEXT DifferentialExtension
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Differential extensions of a ring R.
++ Given a differentiation on R, extend it to a differentiation on %.
DifferentialExtension(R:Ring): Category ==
Join(Ring,DifferentialSpaceExtension R) with
if R has DifferentialRing then DifferentialRing
if R has PartialDifferentialRing(Symbol) then
PartialDifferentialRing(Symbol)
)abbrev category DSEXT DifferentialSpaceExtension
++ Author: Gabriel Dos Reis
++ Date Created: June 16, 2010
++ Date Last Updated: Jun 16, 2010
++ Related Constructors: Module, DifferentialSpace
++ Also See:
++ Description:
++ Extension of a base differential space with a derivation.
++
DifferentialSpaceExtension(R: Type): Category == Type with
differentiate: (%,R -> R) -> %
++ \spad{differentiate(x,d)} computes the derivative of
++ \spad{x}, extending differentiation \spad{d} on \spad{R}.
differentiate: (%,R -> R,NonNegativeInteger) -> %
++ \spad{differentiate(x,d,n)} computes the \spad{n}-th derivative
++ of \spad{x} using a derivation extending \spad{d} on \spad{R}.
D: (%,R -> R) -> %
++ \spad{D(x,d)} is a shorthand for \spad{differentiate(x,d)}.
D: (%,R -> R,NonNegativeInteger) -> %
++ \spad{D(x,d,n)} is a shorthand for \spad{differentiate(x,d,n)}.
if R has DifferentialSpace then DifferentialSpace
if R has PartialDifferentialSpace Symbol then
PartialDifferentialSpace Symbol
add
differentiate(x: %, d: R -> R, n: NonNegativeInteger):% ==
for i in 1..n repeat x := differentiate(x,d)
x
D(x: %, d: R -> R) ==
differentiate(x, d)
D(x: %, d: R -> R, n: NonNegativeInteger) ==
differentiate(x,d,n)
if R has DifferentialSpace then
differentiate x == differentiate(x, differentiate$R)
if R has PartialDifferentialSpace Symbol then
differentiate(x:%, v: Symbol):% ==
differentiate(x, differentiate(#1, v)$R)
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