\documentclass{article} \usepackage{open-axiom} \begin{document} \title{\$SPAD/src/algebra si.spad} \author{Stephen M. Watt, Michael Monagan, James Davenport, Barry Trager} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{category INS IntegerNumberSystem} <>= )abbrev category INS IntegerNumberSystem ++ Author: Stephen M. Watt ++ Date Created: ++ January 1988 ++ Change History: ++ Basic Operations: ++ addmod, base, bit?, copy, dec, even?, hash, inc, invmod, length, mask, ++ positiveRemainder, symmetricRemainder, multiplicativeValuation, mulmod, ++ odd?, powmod, random, rational, rational?, rationalIfCan, shift, submod ++ Description: An \spad{IntegerNumberSystem} is a model for the integers. IntegerNumberSystem(): Category == Join(UniqueFactorizationDomain, EuclideanDomain, OrderedIntegralDomain, DifferentialRing, ConvertibleTo Integer, RetractableTo Integer, LinearlyExplicitRingOver Integer, ConvertibleTo InputForm, ConvertibleTo Pattern Integer, PatternMatchable Integer, CombinatorialFunctionCategory, RealConstant, CharacteristicZero, StepThrough) with odd? : % -> Boolean ++ odd?(n) returns true if and only if n is odd. even? : % -> Boolean ++ even?(n) returns true if and only if n is even. multiplicativeValuation ++ euclideanSize(a*b) returns \spad{euclideanSize(a)*euclideanSize(b)}. base : () -> % ++ base() returns the base for the operations of \spad{IntegerNumberSystem}. length : % -> % ++ length(a) length of \spad{a} in digits. shift : (%, %) -> % ++ shift(a,i) shift \spad{a} by i digits. bit? : (%, %) -> Boolean ++ bit?(n,i) returns true if and only if i-th bit of n is a 1. positiveRemainder : (%, %) -> % ++ positiveRemainder(a,b) (where \spad{b > 1}) yields r ++ where \spad{0 <= r < b} and \spad{r == a rem b}. symmetricRemainder : (%, %) -> % ++ symmetricRemainder(a,b) (where \spad{b > 1}) yields r ++ where \spad{ -b/2 <= r < b/2 }. rational?: % -> Boolean ++ rational?(n) tests if n is a rational number ++ (see \spadtype{Fraction Integer}). rational : % -> Fraction Integer ++ rational(n) creates a rational number (see \spadtype{Fraction Integer}).. rationalIfCan: % -> Union(Fraction Integer, "failed") ++ rationalIfCan(n) creates a rational number, or returns "failed" if this is not possible. random : () -> % ++ random() creates a random element. random : % -> % ++ random(a) creates a random element from 0 to \spad{a-1}. copy : % -> % ++ copy(n) gives a copy of n. inc : % -> % ++ inc(x) returns \spad{x + 1}. dec : % -> % ++ dec(x) returns \spad{x - 1}. mask : % -> % ++ mask(n) returns \spad{2**n-1} (an n bit mask). addmod : (%,%,%) -> % ++ addmod(a,b,p), \spad{0<=a,b

1}, means \spad{a+b mod p}. submod : (%,%,%) -> % ++ submod(a,b,p), \spad{0<=a,b

1}, means \spad{a-b mod p}. mulmod : (%,%,%) -> % ++ mulmod(a,b,p), \spad{0<=a,b

1}, means \spad{a*b mod p}. powmod : (%,%,%) -> % ++ powmod(a,b,p), \spad{0<=a,b

1}, means \spad{a**b mod p}. invmod : (%,%) -> % ++ invmod(a,b), \spad{0<=a1}, \spad{(a,b)=1} means \spad{1/a mod b}. canonicalUnitNormal -- commutative("*") -- follows from the above add characteristic == 0 differentiate x == 0 even? x == not odd? x copy x == x bit?(x, i) == odd? shift(x, -i) mask n == dec shift(1, n) rational? x == true euclideanSize(x) == x=0 => error "euclideanSize called on zero" negative? x => (-convert(x)@Integer)::NonNegativeInteger convert(x)@Integer::NonNegativeInteger convert(x:%):Float == (convert(x)@Integer)::Float convert(x:%):DoubleFloat == (convert(x)@Integer)::DoubleFloat convert(x:%):InputForm == convert(convert(x)@Integer) retract(x:%):Integer == convert(x)@Integer convert(x:%):Pattern(Integer)== convert(x)@Integer ::Pattern(Integer) factor x == factor(x)$IntegerFactorizationPackage(%) squareFree x == squareFree(x)$IntegerFactorizationPackage(%) prime? x == prime?(x)$IntegerPrimesPackage(%) factorial x == factorial(x)$IntegerCombinatoricFunctions(%) binomial(n, m) == binomial(n, m)$IntegerCombinatoricFunctions(%) permutation(n, m) == permutation(n,m)$IntegerCombinatoricFunctions(%) retractIfCan(x:%):Union(Integer, "failed") == convert(x)@Integer init() == 0 -- iterates in order 0,1,-1,2,-2,3,-3,... nextItem n == zero? n => just 1 positive? n => just(-n) just(1-n) patternMatch(x, p, l) == patternMatch(x, p, l)$PatternMatchIntegerNumberSystem(%) rational(x:%):Fraction(Integer) == (convert(x)@Integer)::Fraction(Integer) rationalIfCan(x:%):Union(Fraction Integer, "failed") == (convert(x)@Integer)::Fraction(Integer) symmetricRemainder(x, n) == r := x rem n r = 0 => r if negative? n then n:=-n positive? r => 2 * r > n => r - n r 2*r + n <= 0 => r + n r invmod(a, b) == if negative? a then a := positiveRemainder(a, b) c := a; c1:% := 1 d := b; d1:% := 0 while not zero? d repeat q := c quo d r := c-q*d r1 := c1-q*d1 c := d; c1 := d1 d := r; d1 := r1 not one? c => error "inverse does not exist" negative? c1 => c1 + b c1 powmod(x, n, p) == if negative? x then x := positiveRemainder(x, p) zero? x => 0 zero? n => 1 y:% := 1 z := x repeat if odd? n then y := mulmod(y, z, p) zero?(n := shift(n, -1)) => return y z := mulmod(z, z, p) @ \section{domain SINT SingleInteger} <>= )abbrev domain SINT SingleInteger ++ Author: Michael Monagan ++ Date Created: ++ January 1988 ++ Change History: ++ Basic Operations: max, min, ++ not, and, or, xor, Not, And, Or ++ Related Constructors: ++ Keywords: single integer ++ Description: SingleInteger is intended to support machine integer ++ arithmetic. -- MAXINT, BASE (machine integer constants) -- MODULUS, MULTIPLIER (random number generator constants) SingleInteger(): Join(IntegerNumberSystem,OrderedFinite,BooleanLogic) with canonical ++ \spad{canonical} means that mathematical equality is implied by data structure equality. xor: (%, %) -> % ++ xor(n,m) returns the bit-by-bit logical {\em xor} of ++ the single integers n and m. == SubDomain(Integer, %ismall?(#1)$Foreign(Builtin)) add import %icst0: % from Foreign Builtin import %icst1: % from Foreign Builtin import %icstmin: % from Foreign Builtin import %icstmax: % from Foreign Builtin import %iadd: (%,%) -> % from Foreign Builtin import %iaddmod: (%,%,%) -> % from Foreign Builtin import %isub: (%,%) -> % from Foreign Builtin import %isubmod: (%,%,%) -> % from Foreign Builtin import %imul: (%,%) -> % from Foreign Builtin import %imulmod: (%,%,%) -> % from Foreign Builtin import %irem: (%,%) -> % from Foreign Builtin import %iquo: (%,%) -> % from Foreign Builtin import %ilshift: (%,%) -> % from Foreign Builtin import %ineg: % -> % from Foreign Builtin import %iinc: % -> % from Foreign Builtin import %idec: % -> % from Foreign Builtin import %iabs: % -> % from Foreign Builtin import %irandom: % -> % from Foreign Builtin import %imax: (%,%) -> % from Foreign Builtin import %imin: (%,%) -> % from Foreign Builtin import %igcd: (%,%) -> % from Foreign Builtin import %hash: % -> SingleInteger from Foreign Builtin import %ilength: % -> % from Foreign Builtin import %iodd?: % -> Boolean from Foreign Builtin import %ieven?: % -> Boolean from Foreign Builtin import %ieq: (%,%) -> Boolean from Foreign Builtin import %ilt: (%,%) -> Boolean from Foreign Builtin import %ile: (%,%) -> Boolean from Foreign Builtin import %igt: (%,%) -> Boolean from Foreign Builtin import %ige: (%,%) -> Boolean from Foreign Builtin import %bitnot: % -> % from Foreign Builtin import %bitand: (%,%) -> % from Foreign Builtin import %bitior: (%,%) -> % from Foreign Builtin import %bitxor: (%,%) -> % from Foreign Builtin reducedSystem(m: Matrix %) == m pretend Matrix(Integer) coerce(x):OutputForm == rep(x)::OutputForm convert(x:%):Integer == rep x i:Integer * y:% == %imul(i::%,y) 0 == %icst0 1 == %icst1 base() == per 2 max() == %icstmax min() == %icstmin x = y == %ieq(x,y) ~ x == %bitnot x not(x) == %bitnot x x /\ y == %bitand(x,y) x \/ y == %bitior(x,y) x and y == %bitand(x,y) x or y == %bitior(x,y) xor(x,y) == %bitxor(x,y) x < y == %ilt(x,y) x > y == %igt(x,y) x <= y == %ile(x,y) x >= y == %ige(x,y) inc x == %iinc x dec x == %idec x - x == %ineg x x + y == %iadd(x,y) x:% - y:% == %isub(x,y) x:% * y:% == %imul(x,y) x:% ** n:NonNegativeInteger == (%ipow(x, n)$Foreign(Builtin) pretend Integer)::% x quo y == %iquo(x,y) x rem y == %irem(x,y) divide(x, y) == %idivide(x,y)$Foreign(Builtin) gcd(x,y) == %igcd(x,y) abs(x) == %iabs x odd?(x) == %iodd? x even?(x) == %ieven? x zero?(x) == %ieq(x,%icst0) one?(x) == %ieq(x,%icst1) max(x,y) == %imax(x,y) min(x,y) == %imin(x,y) hash(x) == %hash x length(x) == %ilength x shift(x,n) == %ilshift(x,n) mulmod(a,b,p) == %imulmod(a,b,p) addmod(a,b,p) == %iaddmod(a,b,p) submod(a,b,p) == %isubmod(a,b,p) negative?(x) == %ilt(x,%icst0) size() == (%icstmax - %icstmin + %icst1) pretend NonNegativeInteger index i == per(i + rep %icstmin - rep %icst1) lookup x == (rep x - rep %icstmin + rep %icst1) pretend PositiveInteger reducedSystem(m: Matrix %, v: Vector %) == [m pretend Matrix(Integer), v pretend Vector(Integer)] positiveRemainder(x,n) == r := %irem(x,n) negative? r => negative? n => x - n r + n r coerce(x:Integer):% == per x random() == random %icstmax random(n) == %irandom n UCA ==> Record(unit:%,canonical:%,associate:%) unitNormal x == negative? x => [-%icst1,-x,-%icst1]$UCA [%icst1,x,%icst1]$UCA positive? x == %icst0 < x @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. -- Copyright (C) 2007-2010, Gabriel Dos Reis. -- 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. @ <<*>>= <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}