From efd303d35dd5ecb8a9bc57c44d4aa3f6c57044d2 Mon Sep 17 00:00:00 2001 From: Tim Daly Date: Fri, 8 May 2015 19:21:55 -0400 Subject: [PATCH] books/bookvol*pamphlet add more test cases --- books/bookvol10.2.pamphlet | 3754 ++++---- books/bookvol10.3.pamphlet |15693 ++++++++++++++++++++++--------- books/bookvol10.4.pamphlet | 7847 ++++++++++++++-- books/bookvol10.pamphlet | 5 +- books/bookvol5.pamphlet | 1 + changelog | 16 + patch | 8 +- src/axiom-website/patches.html | 2 + src/input/Makefile.pamphlet | 5 + src/input/cmds.input.pamphlet | 13 +- src/interp/clam.lisp.pamphlet | 2 +- src/interp/msgdb.lisp.pamphlet | 81 +- src/interp/vmlisp.lisp.pamphlet | 25 +- src/share/algebra/browse.daase | 34 +- src/share/algebra/category.daase | 2 +- src/share/algebra/interp.daase | 44 +- src/share/algebra/operation.daase | 6341 +++++++------- src/share/algebra/users.daase/index.kaf | 12 +- 18 files changed, 23642 insertions(+), 10243 deletions(-) diff --git a/books/bookvol10.2.pamphlet b/books/bookvol10.2.pamphlet index 7cac65d..03c8a4b 100644 --- a/books/bookvol10.2.pamphlet +++ b/books/bookvol10.2.pamphlet @@ -127,7 +127,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show AdditiveValuationAttribute +--R +--R AdditiveValuationAttribute is a category constructor +--R Abbreviation for AdditiveValuationAttribute is ATADDVA +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATADDVA +--R +--R------------------------------- Operations -------------------------------- +--R AdditiveValuationAttribute is a category constructor. +--R Abbreviation for AdditiveValuationAttribute is ATADDVA +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATADDVA +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -191,7 +207,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show ApproximateAttribute +--R +--R ApproximateAttribute is a category constructor +--R Abbreviation for ApproximateAttribute is ATAPPRO +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATAPPRO +--R +--R------------------------------- Operations -------------------------------- +--R ApproximateAttribute is a category constructor. +--R Abbreviation for ApproximateAttribute is ATAPPRO +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATAPPRO +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -254,7 +286,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show ArbitraryExponentAttribute +--R +--R ArbitraryExponentAttribute is a category constructor +--R Abbreviation for ArbitraryExponentAttribute is ATARBEX +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATARBEX +--R +--R------------------------------- Operations -------------------------------- +--R ArbitraryExponentAttribute is a category constructor. +--R Abbreviation for ArbitraryExponentAttribute is ATARBEX +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATARBEX +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -317,7 +365,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show ArbitraryPrecisionAttribute +--R +--R ArbitraryPrecisionAttribute is a category constructor +--R Abbreviation for ArbitraryPrecisionAttribute is ATARBPR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATARBPR +--R +--R------------------------------- Operations -------------------------------- +--R ArbitraryPrecisionAttribute is a category constructor. +--R Abbreviation for ArbitraryPrecisionAttribute is ATARBPR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATARBPR +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -383,6 +447,7 @@ digraph pic { --S 1 of 1 )show ArcHyperbolicFunctionCategory +--R --R ArcHyperbolicFunctionCategory is a category constructor --R Abbreviation for ArcHyperbolicFunctionCategory is AHYP --R This constructor is exposed in this frame. @@ -493,6 +558,7 @@ intermediate test to check that the argument has a reciprocal values. --S 1 of 1 )show ArcTrigonometricFunctionCategory +--R --R ArcTrigonometricFunctionCategory is a category constructor --R Abbreviation for ArcTrigonometricFunctionCategory is ATRIG --R This constructor is exposed in this frame. @@ -610,6 +676,23 @@ digraph pic { --S 1 of 1 )show AttributeRegistry +--R +--R AttributeRegistry is a category constructor +--R Abbreviation for AttributeRegistry is ATTREG +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATTREG +--R +--R------------------------------- Operations -------------------------------- +--R AttributeRegistry is a category constructor. +--R Abbreviation for AttributeRegistry is ATTREG +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATTREG +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R +--R --E 1 )spool @@ -807,6 +890,7 @@ digraph pic { --S 1 of 1 )show BasicType +--R --R BasicType is a category constructor --R Abbreviation for BasicType is BASTYPE --R This constructor is exposed in this frame. @@ -906,7 +990,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show CanonicalAttribute +--R +--R CanonicalAttribute is a category constructor +--R Abbreviation for CanonicalAttribute is ATCANON +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCANON +--R +--R------------------------------- Operations -------------------------------- +--R CanonicalAttribute is a category constructor. +--R Abbreviation for CanonicalAttribute is ATCANON +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCANON +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -971,7 +1071,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show CanonicalClosedAttribute +--R +--R CanonicalClosedAttribute is a category constructor +--R Abbreviation for CanonicalClosedAttribute is ATCANCL +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCANCL +--R +--R------------------------------- Operations -------------------------------- +--R CanonicalClosedAttribute is a category constructor. +--R Abbreviation for CanonicalClosedAttribute is ATCANCL +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCANCL +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -1035,7 +1151,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show CanonicalUnitNormalAttribute +--R +--R CanonicalUnitNormalAttribute is a category constructor +--R Abbreviation for CanonicalUnitNormalAttribute is ATCUNOR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCUNOR +--R +--R------------------------------- Operations -------------------------------- +--R CanonicalUnitNormalAttribute is a category constructor. +--R Abbreviation for CanonicalUnitNormalAttribute is ATCUNOR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCUNOR +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -1104,7 +1236,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show CentralAttribute +--R +--R CentralAttribute is a category constructor +--R Abbreviation for CentralAttribute is ATCENRL +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCENRL +--R +--R------------------------------- Operations -------------------------------- +--R CentralAttribute is a category constructor. +--R Abbreviation for CentralAttribute is ATCENRL +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCENRL +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -1283,6 +1431,7 @@ digraph pic { --S 1 of 1 )show CombinatorialFunctionCategory +--R --R CombinatorialFunctionCategory is a category constructor --R Abbreviation for CombinatorialFunctionCategory is CFCAT --R This constructor is exposed in this frame. @@ -1390,7 +1539,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show CommutativeStarAttribute +--R +--R CommutativeStarAttribute is a category constructor +--R Abbreviation for CommutativeStarAttribute is ATCS +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCS +--R +--R------------------------------- Operations -------------------------------- +--R CommutativeStarAttribute is a category constructor. +--R Abbreviation for CommutativeStarAttribute is ATCS +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATCS +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -1619,6 +1784,7 @@ digraph pic { --S 1 of 1 )show ElementaryFunctionCategory +--R --R ElementaryFunctionCategory is a category constructor --R Abbreviation for ElementaryFunctionCategory is ELEMFUN --R This constructor is exposed in this frame. @@ -1724,6 +1890,7 @@ digraph pic { --S 1 of 1 )show Eltable +--R --R Eltable(S: SetCategory,Index: Type) is a category constructor --R Abbreviation for Eltable is ELTAB --R This constructor is exposed in this frame. @@ -1837,7 +2004,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show FiniteAggregateAttribute +--R +--R FiniteAggregateAttribute is a category constructor +--R Abbreviation for FiniteAggregateAttribute is ATFINAG +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATFINAG +--R +--R------------------------------- Operations -------------------------------- +--R FiniteAggregateAttribute is a category constructor. +--R Abbreviation for FiniteAggregateAttribute is ATFINAG +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATFINAG +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -1905,6 +2088,7 @@ intermediate test to check that the argument has a reciprocal values. --S 1 of 1 )show HyperbolicFunctionCategory +--R --R HyperbolicFunctionCategory is a category constructor --R Abbreviation for HyperbolicFunctionCategory is HYPCAT --R This constructor is exposed in this frame. @@ -2164,7 +2348,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show JacobiIdentityAttribute +--R +--R JacobiIdentityAttribute is a category constructor +--R Abbreviation for JacobiIdentityAttribute is ATJACID +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATJACID +--R +--R------------------------------- Operations -------------------------------- +--R JacobiIdentityAttribute is a category constructor. +--R Abbreviation for JacobiIdentityAttribute is ATJACID +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATJACID +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2229,7 +2429,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show LazyRepresentationAttribute +--R +--R LazyRepresentationAttribute is a category constructor +--R Abbreviation for LazyRepresentationAttribute is ATLR +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATLR +--R +--R------------------------------- Operations -------------------------------- +--R LazyRepresentationAttribute is a category constructor. +--R Abbreviation for LazyRepresentationAttribute is ATLR +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATLR +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2292,7 +2508,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show LeftUnitaryAttribute +--R +--R LeftUnitaryAttribute is a category constructor +--R Abbreviation for LeftUnitaryAttribute is ATLUNIT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATLUNIT +--R +--R------------------------------- Operations -------------------------------- +--R LeftUnitaryAttribute is a category constructor. +--R Abbreviation for LeftUnitaryAttribute is ATLUNIT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATLUNIT +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2357,6 +2589,7 @@ digraph pic { --S 1 of 1 )show ModularAlgebraicGcdOperations +--R --R ModularAlgebraicGcdOperations(MPT: Type,MD: Type) is a category constructor --R Abbreviation for ModularAlgebraicGcdOperations is MAGCDOC --R This constructor is exposed in this frame. @@ -2503,7 +2736,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show MultiplicativeValuationAttribute +--R +--R MultiplicativeValuationAttribute is a category constructor +--R Abbreviation for MultiplicativeValuationAttribute is ATMULVA +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATMULVA +--R +--R------------------------------- Operations -------------------------------- +--R MultiplicativeValuationAttribute is a category constructor. +--R Abbreviation for MultiplicativeValuationAttribute is ATMULVA +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATMULVA +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2568,7 +2817,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show NotherianAttribute +--R +--R NotherianAttribute is a category constructor +--R Abbreviation for NotherianAttribute is ATNOTHR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNOTHR +--R +--R------------------------------- Operations -------------------------------- +--R NotherianAttribute is a category constructor. +--R Abbreviation for NotherianAttribute is ATNOTHR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNOTHR +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2631,7 +2896,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show NoZeroDivisorsAttribute +--R +--R NoZeroDivisorsAttribute is a category constructor +--R Abbreviation for NoZeroDivisorsAttribute is ATNZDIV +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNZDIV +--R +--R------------------------------- Operations -------------------------------- +--R NoZeroDivisorsAttribute is a category constructor. +--R Abbreviation for NoZeroDivisorsAttribute is ATNZDIV +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNZDIV +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2696,7 +2977,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show NullSquareAttribute +--R +--R NullSquareAttribute is a category constructor +--R Abbreviation for NullSquareAttribute is ATNULSQ +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNULSQ +--R +--R------------------------------- Operations -------------------------------- +--R NullSquareAttribute is a category constructor. +--R Abbreviation for NullSquareAttribute is ATNULSQ +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATNULSQ +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2761,6 +3058,7 @@ digraph pic { --S 1 of 1 )show OpenMath +--R --R OpenMath is a category constructor --R Abbreviation for OpenMath is OM --R This constructor is exposed in this frame. @@ -2868,7 +3166,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show PartiallyOrderedSetAttribute +--R +--R PartiallyOrderedSetAttribute is a category constructor +--R Abbreviation for PartiallyOrderedSetAttribute is ATPOSET +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATPOSET +--R +--R------------------------------- Operations -------------------------------- +--R PartiallyOrderedSetAttribute is a category constructor. +--R Abbreviation for PartiallyOrderedSetAttribute is ATPOSET +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATPOSET +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -2942,9 +3256,12 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for PTRANFN --R --R------------------------------- Operations -------------------------------- ---R acosIfCan : K -> Union(K,"failed") acotIfCan : K -> Union(K,"failed") ---R acscIfCan : K -> Union(K,"failed") asecIfCan : K -> Union(K,"failed") ---R asinIfCan : K -> Union(K,"failed") atanIfCan : K -> Union(K,"failed") +--R acosIfCan : K -> Union(K,"failed") acoshIfCan : K -> Union(K,"failed") +--R acotIfCan : K -> Union(K,"failed") acothIfCan : K -> Union(K,"failed") +--R acscIfCan : K -> Union(K,"failed") acschIfCan : K -> Union(K,"failed") +--R asecIfCan : K -> Union(K,"failed") asechIfCan : K -> Union(K,"failed") +--R asinIfCan : K -> Union(K,"failed") asinhIfCan : K -> Union(K,"failed") +--R atanIfCan : K -> Union(K,"failed") atanhIfCan : K -> Union(K,"failed") --R cosIfCan : K -> Union(K,"failed") coshIfCan : K -> Union(K,"failed") --R cotIfCan : K -> Union(K,"failed") cothIfCan : K -> Union(K,"failed") --R cscIfCan : K -> Union(K,"failed") cschIfCan : K -> Union(K,"failed") @@ -2952,12 +3269,6 @@ digraph pic { --R secIfCan : K -> Union(K,"failed") sechIfCan : K -> Union(K,"failed") --R sinIfCan : K -> Union(K,"failed") sinhIfCan : K -> Union(K,"failed") --R tanIfCan : K -> Union(K,"failed") tanhIfCan : K -> Union(K,"failed") ---R acoshIfCan : K -> Union(K,"failed") ---R acothIfCan : K -> Union(K,"failed") ---R acschIfCan : K -> Union(K,"failed") ---R asechIfCan : K -> Union(K,"failed") ---R asinhIfCan : K -> Union(K,"failed") ---R atanhIfCan : K -> Union(K,"failed") --R nthRootIfCan : (K,NonNegativeInteger) -> Union(K,"failed") --R --E 1 @@ -3400,8 +3711,8 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for RADCAT --R --R------------------------------- Operations -------------------------------- ---R nthRoot : (%,Integer) -> % sqrt : % -> % ---R ?**? : (%,Fraction(Integer)) -> % +--R ?**? : (%,Fraction(Integer)) -> % nthRoot : (%,Integer) -> % +--R sqrt : % -> % --R --E 1 @@ -3687,7 +3998,22 @@ digraph pic { )clear all --S 1 of 1 -1 +--R +--R RightUnitaryAttribute is a category constructor +--R Abbreviation for RightUnitaryAttribute is ATRUNIT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATRUNIT +--R +--R------------------------------- Operations -------------------------------- +--R RightUnitaryAttribute is a category constructor. +--R Abbreviation for RightUnitaryAttribute is ATRUNIT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATRUNIT +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -3750,7 +4076,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show ShallowlyMutableAttribute +--R +--R ShallowlyMutableAttribute is a category constructor +--R Abbreviation for ShallowlyMutableAttribute is ATSHMUT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATSHMUT +--R +--R------------------------------- Operations -------------------------------- +--R ShallowlyMutableAttribute is a category constructor. +--R Abbreviation for ShallowlyMutableAttribute is ATSHMUT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATSHMUT +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -3819,6 +4161,7 @@ digraph pic { --S 1 of 1 )show SpecialFunctionCategory +--R --R SpecialFunctionCategory is a category constructor --R Abbreviation for SpecialFunctionCategory is SPFCAT --R This constructor is exposed in this frame. @@ -3962,6 +4305,7 @@ intermediate test to check that the argument has a reciprocal values. --S 1 of 1 )show TrigonometricFunctionCategory +--R --R TrigonometricFunctionCategory is a category constructor --R Abbreviation for TrigonometricFunctionCategory is TRIGCAT --R This constructor is exposed in this frame. @@ -4082,6 +4426,22 @@ digraph pic { --S 1 of 1 )show Type +--R +--R Type is a category constructor +--R Abbreviation for Type is TYPE +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for TYPE +--R +--R------------------------------- Operations -------------------------------- +--R Type is a category constructor. +--R Abbreviation for Type is TYPE +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for TYPE +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool @@ -4159,7 +4519,23 @@ digraph pic { )clear all --S 1 of 1 -1 +)show UnitsKnownAttribute +--R +--R UnitsKnownAttribute is a category constructor +--R Abbreviation for UnitsKnownAttribute is ATUNIKN +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATUNIKN +--R +--R------------------------------- Operations -------------------------------- +--R UnitsKnownAttribute is a category constructor. +--R Abbreviation for UnitsKnownAttribute is ATUNIKN +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ATUNIKN +--R +--R------------------------------- Operations -------------------------------- +--R +--R +--R --E 1 )spool )lisp (bye) @@ -4229,6 +4605,7 @@ digraph pic { --S 1 of 1 )show Aggregate +--R --R Aggregate is a category constructor --R Abbreviation for Aggregate is AGG --R This constructor is exposed in this frame. @@ -4642,6 +5019,7 @@ digraph pic { --S 1 of 1 )show EltableAggregate +--R --R EltableAggregate(Dom: SetCategory,Im: Type) is a category constructor --R Abbreviation for EltableAggregate is ELTAGG --R This constructor is exposed in this frame. @@ -4807,9 +5185,8 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for EVALAB --R --R------------------------------- Operations -------------------------------- ---R eval : (%,Equation(R)) -> % eval : (%,R,R) -> % ---R eval : (%,List(R),List(R)) -> % ---R eval : (%,List(Equation(R))) -> % +--R eval : (%,List(Equation(R))) -> % eval : (%,Equation(R)) -> % +--R eval : (%,R,R) -> % eval : (%,List(R),List(R)) -> % --R --E 1 @@ -4935,6 +5312,7 @@ digraph pic { --S 1 of 1 )show FortranProgramCategory +--R --R FortranProgramCategory is a category constructor --R Abbreviation for FortranProgramCategory is FORTCAT --R This constructor is exposed in this frame. @@ -5391,6 +5769,7 @@ digraph pic { --S 1 of 1 )show Logic +--R --R Logic is a category constructor --R Abbreviation for Logic is LOGIC --R This constructor is exposed in this frame. @@ -5515,10 +5894,9 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for PPCURVE --R --R------------------------------- Operations -------------------------------- ---R coerce : % -> OutputForm +--R coerce : % -> OutputForm xRange : % -> Segment(DoubleFloat) +--R yRange : % -> Segment(DoubleFloat) --R listBranches : % -> List(List(Point(DoubleFloat))) ---R xRange : % -> Segment(DoubleFloat) ---R yRange : % -> Segment(DoubleFloat) --R --E 1 @@ -5643,11 +6021,9 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for PSCURVE --R --R------------------------------- Operations -------------------------------- ---R coerce : % -> OutputForm +--R coerce : % -> OutputForm xRange : % -> Segment(DoubleFloat) +--R yRange : % -> Segment(DoubleFloat) zRange : % -> Segment(DoubleFloat) --R listBranches : % -> List(List(Point(DoubleFloat))) ---R xRange : % -> Segment(DoubleFloat) ---R yRange : % -> Segment(DoubleFloat) ---R zRange : % -> Segment(DoubleFloat) --R --E 1 @@ -5775,6 +6151,7 @@ digraph pic { --S 1 of 1 )show RealConstant +--R --R RealConstant is a category constructor --R Abbreviation for RealConstant is REAL --R This constructor is exposed in this frame. @@ -6031,6 +6408,7 @@ digraph pic { --S 1 of 1 )show SetCategory +--R --R SetCategory is a category constructor --R Abbreviation for SetCategory is SETCAT --R This constructor is exposed in this frame. @@ -6188,6 +6566,7 @@ digraph pic { --S 1 of 1 )show TranscendentalFunctionCategory +--R --R TranscendentalFunctionCategory is a category constructor --R Abbreviation for TranscendentalFunctionCategory is TRANFUN --R This constructor is exposed in this frame. @@ -6437,6 +6816,7 @@ digraph pic { --S 1 of 1 )show AbelianSemiGroup +--R --R AbelianSemiGroup is a category constructor --R Abbreviation for AbelianSemiGroup is ABELSG --R This constructor is exposed in this frame. @@ -6754,11 +7134,11 @@ digraph pic { --R cyclic? : % -> Boolean distance : (%,%) -> Integer --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R map : ((S -> S),%) -> % nodes : % -> List(%) ---R sample : () -> % tree : List(S) -> % ---R tree : S -> % tree : (S,List(%)) -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) map : ((S -> S),%) -> % +--R nodes : % -> List(%) sample : () -> % +--R tree : List(S) -> % tree : S -> % +--R tree : (S,List(%)) -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -6772,7 +7152,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -6993,12 +7372,11 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -7187,9 +7565,9 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Matrix(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Matrix(MachineFloat) -> % --R --E 1 @@ -7520,9 +7898,9 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Vector(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Vector(MachineFloat) -> % --R --E 1 @@ -8016,6 +8394,7 @@ digraph pic { --S 1 of 1 )show FileCategory +--R --R FileCategory(Name: SetCategory,S: SetCategory) is a category constructor --R Abbreviation for FileCategory is FILECAT --R This constructor is exposed in this frame. @@ -8196,6 +8575,7 @@ digraph pic { --S 1 of 1 )show Finite +--R --R Finite is a category constructor --R Abbreviation for Finite is FINITE --R This constructor is exposed in this frame. @@ -8364,6 +8744,7 @@ digraph pic { --S 1 of 1 )show FileNameCategory +--R --R FileNameCategory is a category constructor --R Abbreviation for FileNameCategory is FNCAT --R This constructor is exposed in this frame. @@ -8537,6 +8918,7 @@ digraph pic { --S 1 of 1 )show GradedModule +--R --R GradedModule(R: CommutativeRing,E: AbelianMonoid) is a category constructor --R Abbreviation for GradedModule is GRMOD --R This constructor is exposed in this frame. @@ -8711,6 +9093,45 @@ digraph pic { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagehead{LeftOreRing}{LORER} \pagepic{ps/v102leftorering.eps}{LORER}{1.00} +\begin{chunk}{LeftOreRing.input} +)set break resume +)sys rm -f LeftOreRing.output +)spool LeftOreRing.output +)set message test on +)set message auto off +)clear all + +--S 1 of 1 +)show LeftOreRing +--R +--R LeftOreRing is a category constructor +--R Abbreviation for LeftOreRing is LORER +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for LORER +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R characteristic : () -> NonNegativeInteger +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R +--E 1 + +)spool +)lisp (bye) +\end{chunk} + \begin{chunk}{LeftOreRing.help} ==================================================================== LeftOreRing @@ -8788,7 +9209,8 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R copy : % -> % empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R map : ((S -> S),%) -> % sample : () -> % +--R latex : % -> String if S has SETCAT map : ((S -> S),%) -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -8801,7 +9223,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -9082,6 +9503,7 @@ digraph pic { --S 1 of 1 )show IndexedDirectProductCategory +--R --R IndexedDirectProductCategory(A: SetCategory,S: OrderedSet) is a category constructor --R Abbreviation for IndexedDirectProductCategory is IDPC --R This constructor is exposed in this frame. @@ -9490,6 +9912,7 @@ digraph pic { --S 1 of 1 )show Monad +--R --R Monad is a category constructor --R Abbreviation for Monad is MONAD --R This constructor is exposed in this frame. @@ -10239,6 +10662,7 @@ digraph pic { --S 1 of 1 )show OrderedSet +--R --R OrderedSet is a category constructor --R Abbreviation for OrderedSet is ORDSET --R This constructor is exposed in this frame. @@ -10778,11 +11202,10 @@ digraph pic { --R ?=? : (%,%) -> Boolean allRootsOf : ThePols -> List(%) --R coerce : % -> OutputForm definingPolynomial : % -> ThePols --R hash : % -> SingleInteger latex : % -> String +--R negative? : (ThePols,%) -> Boolean positive? : (ThePols,%) -> Boolean --R sign : (ThePols,%) -> Integer zero? : (ThePols,%) -> Boolean --R ?~=? : (%,%) -> Boolean --R approximate : (ThePols,%,TheField) -> TheField ---R negative? : (ThePols,%) -> Boolean ---R positive? : (ThePols,%) -> Boolean --R recip : (ThePols,%) -> Union(ThePols,"failed") --R relativeApprox : (ThePols,%,TheField) -> TheField --R rootOf : (ThePols,PositiveInteger) -> Union(%,"failed") @@ -11154,6 +11577,7 @@ operator ``*''. A Semigroup $G(S,*)$ is: --S 1 of 1 )show SemiGroup +--R --R SemiGroup is a category constructor --R Abbreviation for SemiGroup is SGROUP --R This constructor is exposed in this frame. @@ -11322,6 +11746,7 @@ digraph pic { --S 1 of 1 )show SetCategoryWithDegree +--R --R SetCategoryWithDegree is a category constructor --R Abbreviation for SetCategoryWithDegree is SETCATD --R This constructor is exposed in this frame. @@ -11694,6 +12119,7 @@ digraph pic { --S 1 of 1 )show StepThrough +--R --R StepThrough is a category constructor --R Abbreviation for StepThrough is STEP --R This constructor is exposed in this frame. @@ -11702,8 +12128,8 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?=? : (%,%) -> Boolean coerce : % -> OutputForm --R hash : % -> SingleInteger init : () -> % ---R latex : % -> String ?~=? : (%,%) -> Boolean ---R nextItem : % -> Union(%,"failed") +--R latex : % -> String nextItem : % -> Union(%,"failed") +--R ?~=? : (%,%) -> Boolean --R --E 1 @@ -11864,8 +12290,9 @@ digraph pic { --R point : Point(R) -> % point : (%,List(R)) -> % --R point : (%,Point(R)) -> % point? : % -> Boolean --R polygon : % -> List(Point(R)) polygon : List(Point(R)) -> % ---R polygon : (%,List(List(R))) -> % polygon? : % -> Boolean ---R subspace : % -> SubSpace(3,R) ?~=? : (%,%) -> Boolean +--R polygon : (%,List(List(R))) -> % polygon : (%,List(Point(R))) -> % +--R polygon? : % -> Boolean subspace : % -> SubSpace(3,R) +--R ?~=? : (%,%) -> Boolean --R closedCurve : (%,List(List(R))) -> % --R closedCurve : (%,List(Point(R))) -> % --R enterPointData : (%,List(Point(R))) -> NonNegativeInteger @@ -11883,7 +12310,6 @@ digraph pic { --R numberOfComposites : % -> NonNegativeInteger --R objects : % -> Record(points: NonNegativeInteger,curves: NonNegativeInteger,polygons: NonNegativeInteger,constructs: NonNegativeInteger) --R point : (%,NonNegativeInteger) -> % ---R polygon : (%,List(Point(R))) -> % --R --E 1 @@ -12405,18 +12831,19 @@ digraph pic { --S 1 of 1 )show AbelianMonoid +--R --R AbelianMonoid is a category constructor --R Abbreviation for AbelianMonoid is ABELMON --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for ABELMON --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R --E 1 @@ -12825,7 +13252,8 @@ digraph pic { --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean extract! : % -> S --R insert! : (S,%) -> % inspect : % -> S ---R map : ((S -> S),%) -> % sample : () -> % +--R latex : % -> String if S has SETCAT map : ((S -> S),%) -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -12838,7 +13266,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -13047,6 +13474,7 @@ digraph pic { --S 1 of 1 )show CachableSet +--R --R CachableSet is a category constructor --R Abbreviation for CachableSet is CACHSET --R This constructor is exposed in this frame. @@ -13058,8 +13486,7 @@ digraph pic { --R ?>=? : (%,%) -> Boolean coerce : % -> OutputForm --R hash : % -> SingleInteger latex : % -> String --R max : (%,%) -> % min : (%,%) -> % ---R ?~=? : (%,%) -> Boolean ---R position : % -> NonNegativeInteger +--R position : % -> NonNegativeInteger ?~=? : (%,%) -> Boolean --R setPosition : (%,NonNegativeInteger) -> Void --R --E 1 @@ -13202,8 +13629,8 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R construct : List(S) -> % copy : % -> % --R empty : () -> % empty? : % -> Boolean ---R eq? : (%,%) -> Boolean map : ((S -> S),%) -> % ---R sample : () -> % +--R eq? : (%,%) -> Boolean latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -13218,7 +13645,6 @@ digraph pic { --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R find : ((S -> Boolean),%) -> Union(S,"failed") --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -13845,37 +14271,33 @@ digraph pic { --R box : List(%) -> % box : % -> % --R coerce : Kernel(%) -> % coerce : % -> OutputForm --R distribute : (%,%) -> % distribute : % -> % +--R elt : (BasicOperator,List(%)) -> % elt : (BasicOperator,%,%,%) -> % --R elt : (BasicOperator,%,%) -> % elt : (BasicOperator,%) -> % +--R eval : (%,Symbol,(% -> %)) -> % eval : (%,List(%),List(%)) -> % --R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % ---R eval : (%,Kernel(%),%) -> % freeOf? : (%,Symbol) -> Boolean ---R freeOf? : (%,%) -> Boolean hash : % -> SingleInteger ---R height : % -> NonNegativeInteger is? : (%,Symbol) -> Boolean +--R eval : (%,List(Equation(%))) -> % eval : (%,Kernel(%),%) -> % +--R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean +--R hash : % -> SingleInteger height : % -> NonNegativeInteger +--R is? : (%,Symbol) -> Boolean is? : (%,BasicOperator) -> Boolean --R kernel : (BasicOperator,%) -> % kernels : % -> List(Kernel(%)) ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % paren : List(%) -> % ---R paren : % -> % retract : % -> Kernel(%) ---R subst : (%,Equation(%)) -> % tower : % -> List(Kernel(%)) ---R ?~=? : (%,%) -> Boolean +--R latex : % -> String map : ((% -> %),Kernel(%)) -> % +--R max : (%,%) -> % min : (%,%) -> % +--R paren : List(%) -> % paren : % -> % +--R retract : % -> Kernel(%) subst : (%,Equation(%)) -> % +--R tower : % -> List(Kernel(%)) ?~=? : (%,%) -> Boolean --R definingPolynomial : % -> % if $ has RING ---R elt : (BasicOperator,List(%)) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,%,%,%) -> % --R eval : (%,BasicOperator,(% -> %)) -> % --R eval : (%,BasicOperator,(List(%) -> %)) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R eval : (%,List(Kernel(%)),List(%)) -> % --R even? : % -> Boolean if $ has RETRACT(INT) ---R is? : (%,BasicOperator) -> Boolean --R kernel : (BasicOperator,List(%)) -> % --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R odd? : % -> Boolean if $ has RETRACT(INT) --R operator : BasicOperator -> BasicOperator @@ -14438,6 +14860,7 @@ digraph pic { --S 1 of 1 )show GradedAlgebra +--R --R GradedAlgebra(R: CommutativeRing,E: AbelianMonoid) is a category constructor --R Abbreviation for GradedAlgebra is GRALG --R This constructor is exposed in this frame. @@ -14990,21 +15413,21 @@ digraph pic { --S 1 of 1 )show MonadWithUnit +--R --R MonadWithUnit is a category constructor --R Abbreviation for MonadWithUnit is MONADWU --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for MONADWU --R --R------------------------------- Operations -------------------------------- ---R ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean +--R ?*? : (%,%) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leftRecip : % -> Union(%,"failed") one? : % -> Boolean --R recip : % -> Union(%,"failed") ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % --R leftPower : (%,NonNegativeInteger) -> % --R leftPower : (%,PositiveInteger) -> % ---R leftRecip : % -> Union(%,"failed") --R rightPower : (%,NonNegativeInteger) -> % --R rightPower : (%,PositiveInteger) -> % --R rightRecip : % -> Union(%,"failed") @@ -15231,20 +15654,20 @@ digraph pic { --S 1 of 1 )show Monoid +--R --R Monoid is a category constructor --R Abbreviation for Monoid is MONOID --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for MONOID --R --R------------------------------- Operations -------------------------------- ---R ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % +--R ?*? : (%,%) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coerce : % -> OutputForm --R hash : % -> SingleInteger latex : % -> String --R one? : % -> Boolean recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R --E 1 @@ -15429,6 +15852,7 @@ digraph pic { --S 1 of 1 )show OrderedFinite +--R --R OrderedFinite is a category constructor --R Abbreviation for OrderedFinite is ORDFIN --R This constructor is exposed in this frame. @@ -16061,9 +16485,10 @@ digraph pic { --R cyclic? : % -> Boolean distance : (%,%) -> Integer --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R map : ((S -> S),%) -> % nodes : % -> List(%) ---R sample : () -> % value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) map : ((S -> S),%) -> % +--R nodes : % -> List(%) sample : () -> % +--R value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -16077,7 +16502,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -16360,14 +16784,15 @@ first column in an array and vice versa. --R elt : (%,Integer,Integer,R) -> R elt : (%,Integer,Integer) -> R --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean fill! : (%,R) -> % ---R map : (((R,R) -> R),%,%,R) -> % map : (((R,R) -> R),%,%) -> % ---R map : ((R -> R),%) -> % map! : ((R -> R),%) -> % ---R maxColIndex : % -> Integer maxRowIndex : % -> Integer ---R minColIndex : % -> Integer minRowIndex : % -> Integer ---R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger ---R parts : % -> List(R) qelt : (%,Integer,Integer) -> R ---R row : (%,Integer) -> Row sample : () -> % ---R setRow! : (%,Integer,Row) -> % +--R latex : % -> String if R has SETCAT map : (((R,R) -> R),%,%,R) -> % +--R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R map! : ((R -> R),%) -> % maxColIndex : % -> Integer +--R maxRowIndex : % -> Integer minColIndex : % -> Integer +--R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger +--R nrows : % -> NonNegativeInteger parts : % -> List(R) +--R qelt : (%,Integer,Integer) -> R row : (%,Integer) -> Row +--R sample : () -> % setColumn! : (%,Integer,Col) -> % +--R setRow! : (%,Integer,Row) -> % setelt : (%,Integer,Integer,R) -> R --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if R has SETCAT --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -16380,15 +16805,12 @@ first column in an array and vice versa. --R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) and R has SETCAT --R every? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R member? : (R,%) -> Boolean if R has SETCAT and $ has finiteAggregate --R members : % -> List(R) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R new : (NonNegativeInteger,NonNegativeInteger,R) -> % --R qsetelt! : (%,Integer,Integer,R) -> R ---R setColumn! : (%,Integer,Col) -> % ---R setelt : (%,Integer,Integer,R) -> R --R size? : (%,NonNegativeInteger) -> Boolean --R ?~=? : (%,%) -> Boolean if R has SETCAT --R @@ -16971,10 +17393,11 @@ digraph pic { --R ?.right : (%,right) -> % ?.left : (%,left) -> % --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R left : % -> % map : ((S -> S),%) -> % ---R nodes : % -> List(%) right : % -> % ---R sample : () -> % value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) left : % -> % +--R map : ((S -> S),%) -> % nodes : % -> List(%) +--R right : % -> % sample : () -> % +--R value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -16988,7 +17411,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -17322,18 +17744,19 @@ digraph pic { --S 1 of 1 )show CancellationAbelianMonoid +--R --R CancellationAbelianMonoid is a category constructor --R Abbreviation for CancellationAbelianMonoid is CABMON --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for CABMON --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -17494,8 +17917,8 @@ digraph pic { --R dictionary : () -> % empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean --R extract! : % -> S insert! : (S,%) -> % ---R inspect : % -> S map : ((S -> S),%) -> % ---R sample : () -> % +--R inspect : % -> S latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -17510,7 +17933,6 @@ digraph pic { --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R find : ((S -> Boolean),%) -> Union(S,"failed") --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -17801,11 +18223,11 @@ digraph pic { --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean --R head : % -> % last : % -> S ---R leaf? : % -> Boolean leaves : % -> List(S) ---R map : ((S -> S),%) -> % next : % -> % ---R nodes : % -> List(%) previous : % -> % ---R sample : () -> % tail : % -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) map : ((S -> S),%) -> % +--R next : % -> % nodes : % -> List(%) +--R previous : % -> % sample : () -> % +--R tail : % -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -17820,7 +18242,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -18077,6 +18498,7 @@ digraph pic { --S 1 of 1 )show Group +--R --R Group is a category constructor --R Abbreviation for Group is GROUP --R This constructor is exposed in this frame. @@ -18084,16 +18506,15 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?/? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R coerce : % -> OutputForm commutator : (%,%) -> % --R conjugate : (%,%) -> % hash : % -> SingleInteger --R inv : % -> % latex : % -> String --R one? : % -> Boolean recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R --E 1 @@ -18309,9 +18730,10 @@ digraph pic { --R empty? : % -> Boolean entries : % -> List(S) --R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R insert : (S,%,Integer) -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % ---R qelt : (%,Integer) -> S sample : () -> % +--R insert : (S,%,Integer) -> % latex : % -> String if S has SETCAT +--R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R new : (NonNegativeInteger,S) -> % qelt : (%,Integer) -> S +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -18331,7 +18753,6 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R maxIndex : % -> Integer if Integer has ORDSET @@ -19503,30 +19924,31 @@ inverse matrix [[j**i for i in 0..4] for j in 1..5] --R ?*? : (Row,%) -> Row ?*? : (%,Col) -> Col --R ?*? : (Integer,%) -> % ?*? : (%,R) -> % --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % antisymmetric? : % -> Boolean +--R ?**? : (%,NonNegativeInteger) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R ?/? : (%,R) -> % if R has FIELD antisymmetric? : % -> Boolean --R coerce : Col -> % column : (%,Integer) -> Col --R copy : % -> % diagonal? : % -> Boolean --R diagonalMatrix : List(%) -> % diagonalMatrix : List(R) -> % --R elt : (%,Integer,Integer,R) -> R elt : (%,Integer,Integer) -> R --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean fill! : (%,R) -> % ---R horizConcat : (%,%) -> % listOfLists : % -> List(List(R)) ---R map : (((R,R) -> R),%,%,R) -> % map : (((R,R) -> R),%,%) -> % ---R map : ((R -> R),%) -> % map! : ((R -> R),%) -> % ---R matrix : List(List(R)) -> % maxColIndex : % -> Integer ---R maxRowIndex : % -> Integer minColIndex : % -> Integer ---R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger ---R nrows : % -> NonNegativeInteger parts : % -> List(R) +--R horizConcat : (%,%) -> % latex : % -> String if R has SETCAT +--R listOfLists : % -> List(List(R)) map : (((R,R) -> R),%,%,R) -> % +--R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R map! : ((R -> R),%) -> % matrix : List(List(R)) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R parts : % -> List(R) pfaffian : % -> R if R has COMRING --R qelt : (%,Integer,Integer) -> R row : (%,Integer) -> Row ---R sample : () -> % setRow! : (%,Integer,Row) -> % +--R sample : () -> % setColumn! : (%,Integer,Col) -> % +--R setRow! : (%,Integer,Row) -> % setelt : (%,Integer,Integer,R) -> R --R square? : % -> Boolean squareTop : % -> % --R symmetric? : % -> Boolean transpose : % -> % --R transpose : Row -> % vertConcat : (%,%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?**? : (%,Integer) -> % if R has FIELD ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ?=? : (%,%) -> Boolean if R has SETCAT --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R coerce : % -> OutputForm if R has SETCAT @@ -19543,7 +19965,6 @@ inverse matrix [[j**i for i in 0..4] for j in 1..5] --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R hash : % -> SingleInteger if R has SETCAT --R inverse : % -> Union(%,"failed") if R has FIELD ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R matrix : (NonNegativeInteger,NonNegativeInteger,((Integer,Integer) -> R)) -> % --R member? : (R,%) -> Boolean if R has SETCAT and $ has finiteAggregate @@ -19553,14 +19974,11 @@ inverse matrix [[j**i for i in 0..4] for j in 1..5] --R new : (NonNegativeInteger,NonNegativeInteger,R) -> % --R nullSpace : % -> List(Col) if R has INTDOM --R nullity : % -> NonNegativeInteger if R has INTDOM ---R pfaffian : % -> R if R has COMRING --R qsetelt! : (%,Integer,Integer,R) -> R --R rank : % -> NonNegativeInteger if R has INTDOM --R rowEchelon : % -> % if R has EUCDOM --R scalarMatrix : (NonNegativeInteger,R) -> % ---R setColumn! : (%,Integer,Col) -> % --R setelt : (%,List(Integer),List(Integer),%) -> % ---R setelt : (%,Integer,Integer,R) -> R --R setsubMatrix! : (%,Integer,Integer,%) -> % --R size? : (%,NonNegativeInteger) -> Boolean --R subMatrix : (%,Integer,Integer,Integer,Integer) -> % @@ -21093,6 +21511,7 @@ digraph pic { --S 1 of 1 )show OrderedMonoid +--R --R OrderedMonoid is a category constructor --R Abbreviation for OrderedMonoid is ORDMON --R This constructor is exposed in this frame. @@ -21100,16 +21519,15 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 1 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : % -> OutputForm +--R ?**? : (%,NonNegativeInteger) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,PositiveInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % coerce : % -> OutputForm --R hash : % -> SingleInteger latex : % -> String --R max : (%,%) -> % min : (%,%) -> % --R one? : % -> Boolean recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R --E 1 @@ -21976,9 +22394,9 @@ digraph pic { --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean extract! : % -> S --R insert! : (S,%) -> % inspect : % -> S ---R map : ((S -> S),%) -> % max : % -> S ---R merge : (%,%) -> % merge! : (%,%) -> % ---R sample : () -> % +--R latex : % -> String if S has SETCAT map : ((S -> S),%) -> % +--R max : % -> S merge : (%,%) -> % +--R merge! : (%,%) -> % sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -21991,7 +22409,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -22212,8 +22629,9 @@ digraph pic { --R enqueue! : (S,%) -> S eq? : (%,%) -> Boolean --R extract! : % -> S front : % -> S --R insert! : (S,%) -> % inspect : % -> S ---R length : % -> NonNegativeInteger map : ((S -> S),%) -> % ---R rotate! : % -> % sample : () -> % +--R latex : % -> String if S has SETCAT length : % -> NonNegativeInteger +--R map : ((S -> S),%) -> % rotate! : % -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -22226,7 +22644,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -22463,8 +22880,9 @@ digraph pic { --R latex : % -> String map : ((S -> S),%) -> % --R sample : () -> % set : List(S) -> % --R set : () -> % subset? : (%,%) -> Boolean ---R union : (S,%) -> % union : (%,S) -> % ---R union : (%,%) -> % ?~=? : (%,%) -> Boolean +--R symmetricDifference : (%,%) -> % union : (S,%) -> % +--R union : (%,S) -> % union : (%,%) -> % +--R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R convert : % -> InputForm if S has KONVERT(INFORM) @@ -22490,7 +22908,6 @@ digraph pic { --R removeDuplicates : % -> % if S has SETCAT and $ has finiteAggregate --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean ---R symmetricDifference : (%,%) -> % --R --E 1 @@ -22817,9 +23234,10 @@ digraph pic { --R depth : % -> NonNegativeInteger empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean --R extract! : % -> S insert! : (S,%) -> % ---R inspect : % -> S map : ((S -> S),%) -> % ---R pop! : % -> S push! : (S,%) -> S ---R sample : () -> % top : % -> S +--R inspect : % -> S latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % pop! : % -> S +--R push! : (S,%) -> S sample : () -> % +--R top : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -22832,7 +23250,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -23073,12 +23490,13 @@ digraph pic { --R ?.first : (%,first) -> S ?.value : (%,value) -> S --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean first : % -> S ---R last : % -> S leaf? : % -> Boolean +--R last : (%,NonNegativeInteger) -> % last : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean --R leaves : % -> List(S) map : ((S -> S),%) -> % ---R nodes : % -> List(%) rest : % -> % ---R sample : () -> % second : % -> S ---R tail : % -> % third : % -> S ---R value : % -> S +--R nodes : % -> List(%) rest : (%,NonNegativeInteger) -> % +--R rest : % -> % sample : () -> % +--R second : % -> S tail : % -> % +--R third : % -> S value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -23097,8 +23515,6 @@ digraph pic { --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -23106,7 +23522,6 @@ digraph pic { --R more? : (%,NonNegativeInteger) -> Boolean --R node? : (%,%) -> Boolean if S has SETCAT --R parts : % -> List(S) if $ has finiteAggregate ---R rest : (%,NonNegativeInteger) -> % --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,last,S) -> S if $ has shallowlyMutable --R setelt : (%,rest,%) -> % if $ has shallowlyMutable @@ -23621,20 +24036,20 @@ digraph pic { --S 1 of 1 )show AbelianGroup +--R --R AbelianGroup is a category constructor --R Abbreviation for AbelianGroup is ABELGRP --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for ABELGRP --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -23820,11 +24235,11 @@ digraph pic { --R ?.right : (%,right) -> % ?.left : (%,left) -> % --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R left : % -> % map : ((S -> S),%) -> % ---R node : (%,S,%) -> % nodes : % -> List(%) ---R right : % -> % sample : () -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) left : % -> % +--R map : ((S -> S),%) -> % node : (%,S,%) -> % +--R nodes : % -> List(%) right : % -> % +--R sample : () -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -23838,7 +24253,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -24116,8 +24530,8 @@ digraph pic { --R dictionary : () -> % empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean --R extract! : % -> S insert! : (S,%) -> % ---R inspect : % -> S map : ((S -> S),%) -> % ---R sample : () -> % +--R inspect : % -> S latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -24132,7 +24546,6 @@ digraph pic { --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R find : ((S -> Boolean),%) -> Union(S,"failed") --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -24424,11 +24837,12 @@ digraph pic { --R extractTop! : % -> S front : % -> S --R height : % -> NonNegativeInteger insert! : (S,%) -> % --R insertBottom! : (S,%) -> S insertTop! : (S,%) -> S ---R inspect : % -> S length : % -> NonNegativeInteger ---R map : ((S -> S),%) -> % pop! : % -> S ---R push! : (S,%) -> S reverse! : % -> % ---R rotate! : % -> % sample : () -> % ---R top : % -> S top! : % -> S +--R inspect : % -> S latex : % -> String if S has SETCAT +--R length : % -> NonNegativeInteger map : ((S -> S),%) -> % +--R pop! : % -> S push! : (S,%) -> S +--R reverse! : % -> % rotate! : % -> % +--R sample : () -> % top : % -> S +--R top! : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -24441,7 +24855,6 @@ digraph pic { --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -24727,9 +25140,10 @@ digraph pic { --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) --R insert : (%,%,Integer) -> % insert : (S,%,Integer) -> % --R insert! : (%,%,Integer) -> % insert! : (S,%,Integer) -> % ---R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % ---R new : (NonNegativeInteger,S) -> % qelt : (%,Integer) -> S ---R sample : () -> % +--R latex : % -> String if S has SETCAT map : (((S,S) -> S),%,%) -> % +--R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % +--R qelt : (%,Integer) -> S remove! : ((S -> Boolean),%) -> % +--R sample : () -> % select! : ((S -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -24750,7 +25164,6 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R maxIndex : % -> Integer if Integer has ORDSET @@ -24768,11 +25181,9 @@ digraph pic { --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate --R remove : (S,%) -> % if S has SETCAT and $ has finiteAggregate --R remove! : (S,%) -> % if S has SETCAT ---R remove! : ((S -> Boolean),%) -> % --R removeDuplicates : % -> % if S has SETCAT and $ has finiteAggregate --R removeDuplicates! : % -> % if S has SETCAT --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R select! : ((S -> Boolean),%) -> % --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean @@ -25118,10 +25529,12 @@ digraph pic { --R empty? : % -> Boolean entries : % -> List(S) --R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R insert : (S,%,Integer) -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % ---R qelt : (%,Integer) -> S reverse : % -> % ---R sample : () -> % +--R insert : (S,%,Integer) -> % latex : % -> String if S has SETCAT +--R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET +--R new : (NonNegativeInteger,S) -> % qelt : (%,Integer) -> S +--R reverse : % -> % sample : () -> % +--R sort : % -> % if S has ORDSET sort : (((S,S) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -25146,16 +25559,13 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate --R members : % -> List(S) if $ has finiteAggregate --R merge : (%,%) -> % if S has ORDSET --R merge : (((S,S) -> Boolean),%,%) -> % ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(S) if $ has finiteAggregate @@ -25174,8 +25584,6 @@ digraph pic { --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if S has ORDSET ---R sort : (((S,S) -> Boolean),%) -> % --R sort! : % -> % if S has ORDSET and $ has shallowlyMutable --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if S has ORDSET @@ -25561,17 +25969,17 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for FAMONC --R --R------------------------------- Operations -------------------------------- ---R ?*? : (E,S) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (S,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (S,%) -> E coerce : S -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String mapCoef : ((E -> E),%) -> % ---R mapGen : ((S -> S),%) -> % nthCoef : (%,Integer) -> E ---R nthFactor : (%,Integer) -> S retract : % -> S ---R sample : () -> % size : % -> NonNegativeInteger ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (E,S) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (S,%) -> E +--R coerce : S -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R mapCoef : ((E -> E),%) -> % mapGen : ((S -> S),%) -> % +--R nthCoef : (%,Integer) -> E nthFactor : (%,Integer) -> S +--R retract : % -> S sample : () -> % +--R size : % -> NonNegativeInteger zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R highCommonTerms : (%,%) -> % if E has OAMON --R retractIfCan : % -> Union(S,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -25804,8 +26212,9 @@ digraph pic { --R dictionary : () -> % empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean --R extract! : % -> S insert! : (S,%) -> % ---R inspect : % -> S map : ((S -> S),%) -> % ---R removeDuplicates! : % -> % sample : () -> % +--R inspect : % -> S latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % removeDuplicates! : % -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -25822,7 +26231,6 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R hash : % -> SingleInteger if S has SETCAT --R insert! : (S,%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate @@ -26071,21 +26479,22 @@ digraph pic { --S 1 of 1 )show OrderedAbelianMonoid +--R --R OrderedAbelianMonoid is a category constructor --R Abbreviation for OrderedAbelianMonoid is OAMON --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for OAMON --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R --E 1 @@ -26226,9 +26635,10 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?/? : (%,%) -> % ---R ? Boolean ?=? : (%,%) -> Boolean ---R 1 : () -> % ?^? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?/? : (%,%) -> % ? Boolean +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coerce : % -> OutputForm --R commutator : (%,%) -> % conjugate : (%,%) -> % --R cycle : List(S) -> % cycles : List(List(S)) -> % @@ -26237,11 +26647,9 @@ digraph pic { --R latex : % -> String one? : % -> Boolean --R orbit : (%,S) -> Set(S) recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % --R ?<=? : (%,%) -> Boolean if S has ORDSET or S has FINITE --R ?>? : (%,%) -> Boolean if S has ORDSET or S has FINITE --R ?>=? : (%,%) -> Boolean if S has ORDSET or S has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R max : (%,%) -> % if S has ORDSET or S has FINITE --R min : (%,%) -> % if S has ORDSET or S has FINITE --R @@ -26493,14 +26901,15 @@ digraph pic { --R explicitlyFinite? : % -> Boolean first : % -> S --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) --R insert : (S,%,Integer) -> % insert : (%,%,Integer) -> % ---R last : % -> S leaf? : % -> Boolean +--R last : (%,NonNegativeInteger) -> % last : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean --R leaves : % -> List(S) map : (((S,S) -> S),%,%) -> % --R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % --R nodes : % -> List(%) possiblyInfinite? : % -> Boolean ---R qelt : (%,Integer) -> S rest : % -> % ---R sample : () -> % second : % -> S ---R tail : % -> % third : % -> S ---R value : % -> S +--R qelt : (%,Integer) -> S rest : (%,NonNegativeInteger) -> % +--R rest : % -> % sample : () -> % +--R second : % -> S tail : % -> % +--R third : % -> S value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -26525,8 +26934,6 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R maxIndex : % -> Integer if Integer has ORDSET @@ -26543,7 +26950,6 @@ digraph pic { --R remove : (S,%) -> % if S has SETCAT and $ has finiteAggregate --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate --R removeDuplicates : % -> % if S has SETCAT and $ has finiteAggregate ---R rest : (%,NonNegativeInteger) -> % --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable @@ -27010,7 +27416,8 @@ digraph pic { --R normalized? : (P,%) -> Boolean reduceByQuasiMonic : (P,%) -> P --R removeZero : (P,%) -> P rest : % -> Union(%,"failed") --R retract : List(P) -> % sample : () -> % ---R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R select : (%,V) -> Union(P,"failed") stronglyReduce : (P,%) -> P +--R stronglyReduced? : % -> Boolean stronglyReduced? : (P,%) -> Boolean --R trivialIdeal? : % -> Boolean variables : % -> List(V) --R zeroSetSplit : List(P) -> List(%) ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -27055,11 +27462,9 @@ digraph pic { --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) --R @@ -27925,20 +28330,19 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for FDIVCAT --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R divisor : (R,UP,UP,UP,F) -> % divisor : (F,F,Integer) -> % ---R divisor : (F,F) -> % divisor : R -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm divisor : (R,UP,UP,UP,F) -> % +--R divisor : (F,F,Integer) -> % divisor : (F,F) -> % +--R divisor : R -> % generator : % -> Union(R,"failed") --R hash : % -> SingleInteger latex : % -> String --R principal? : % -> Boolean reduce : % -> % --R sample : () -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % --R decompose : % -> Record(id: FractionalIdeal(UP,Fraction(UP),UPUP,R),principalPart: R) --R divisor : FractionalIdeal(UP,Fraction(UP),UPUP,R) -> % ---R generator : % -> Union(R,"failed") --R ideal : % -> FractionalIdeal(UP,Fraction(UP),UPUP,R) --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -28144,6 +28548,7 @@ digraph pic { --S 1 of 1 )show FiniteSetAggregate +--R --R FiniteSetAggregate(S: SetCategory) is a category constructor --R Abbreviation for FiniteSetAggregate is FSAGG --R This constructor is exposed in this frame. @@ -28162,10 +28567,12 @@ digraph pic { --R inspect : % -> S intersect : (%,%) -> % --R latex : % -> String map : ((S -> S),%) -> % --R max : % -> S if S has ORDSET min : % -> S if S has ORDSET ---R sample : () -> % set : () -> % ---R set : List(S) -> % subset? : (%,%) -> Boolean +--R random : () -> % if S has FINITE sample : () -> % +--R set : () -> % set : List(S) -> % +--R subset? : (%,%) -> Boolean symmetricDifference : (%,%) -> % --R union : (%,%) -> % union : (%,S) -> % ---R union : (S,%) -> % ?~=? : (%,%) -> Boolean +--R union : (S,%) -> % universe : () -> % if S has FINITE +--R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R cardinality : % -> NonNegativeInteger @@ -28188,7 +28595,6 @@ digraph pic { --R members : % -> List(S) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(S) if $ has finiteAggregate ---R random : () -> % if S has FINITE --R reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate --R reduce : (((S,S) -> S),%,S) -> S if $ has finiteAggregate --R reduce : (((S,S) -> S),%,S,S) -> S if S has SETCAT and $ has finiteAggregate @@ -28201,8 +28607,6 @@ digraph pic { --R select! : ((S -> Boolean),%) -> % if $ has finiteAggregate --R size : () -> NonNegativeInteger if S has FINITE --R size? : (%,NonNegativeInteger) -> Boolean ---R symmetricDifference : (%,%) -> % ---R universe : () -> % if S has FINITE --R --E 1 @@ -28898,17 +29302,18 @@ digraph pic { --R extend : (%,Integer) -> % first : % -> S --R frst : % -> S index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (S,%,Integer) -> % ---R insert : (%,%,Integer) -> % last : % -> S +--R insert : (%,%,Integer) -> % last : (%,NonNegativeInteger) -> % +--R last : % -> S latex : % -> String if S has SETCAT --R lazy? : % -> Boolean lazyEvaluate : % -> % --R leaf? : % -> Boolean leaves : % -> List(S) --R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % --R new : (NonNegativeInteger,S) -> % nodes : % -> List(%) --R possiblyInfinite? : % -> Boolean qelt : (%,Integer) -> S ---R remove : ((S -> Boolean),%) -> % rest : % -> % ---R rst : % -> % sample : () -> % ---R second : % -> S select : ((S -> Boolean),%) -> % ---R tail : % -> % third : % -> S ---R value : % -> S +--R remove : ((S -> Boolean),%) -> % rest : (%,NonNegativeInteger) -> % +--R rest : % -> % rst : % -> % +--R sample : () -> % second : % -> S +--R select : ((S -> Boolean),%) -> % tail : % -> % +--R third : % -> S value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -28933,8 +29338,6 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R maxIndex : % -> Integer if Integer has ORDSET @@ -28951,7 +29354,6 @@ digraph pic { --R reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate --R remove : (S,%) -> % if S has SETCAT and $ has finiteAggregate --R removeDuplicates : % -> % if S has SETCAT and $ has finiteAggregate ---R rest : (%,NonNegativeInteger) -> % --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable @@ -29828,13 +30230,13 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -30003,15 +30405,20 @@ digraph pic { --R first : % -> S index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (S,%,Integer) -> % --R insert : (%,%,Integer) -> % insert! : (S,%,Integer) -> % ---R insert! : (%,%,Integer) -> % last : % -> S +--R insert! : (%,%,Integer) -> % last : (%,NonNegativeInteger) -> % +--R last : % -> S latex : % -> String if S has SETCAT --R leaf? : % -> Boolean leaves : % -> List(S) --R list : S -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % +--R map : ((S -> S),%) -> % max : (%,%) -> % if S has ORDSET +--R min : (%,%) -> % if S has ORDSET new : (NonNegativeInteger,S) -> % --R nodes : % -> List(%) possiblyInfinite? : % -> Boolean ---R qelt : (%,Integer) -> S rest : % -> % +--R qelt : (%,Integer) -> S remove! : ((S -> Boolean),%) -> % +--R rest : (%,NonNegativeInteger) -> % rest : % -> % --R reverse : % -> % sample : () -> % ---R second : % -> S tail : % -> % ---R third : % -> S value : % -> S +--R second : % -> S select! : ((S -> Boolean),%) -> % +--R sort : (((S,S) -> Boolean),%) -> % sort : % -> % if S has ORDSET +--R tail : % -> % third : % -> S +--R value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -30040,11 +30447,8 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate --R members : % -> List(S) if $ has finiteAggregate @@ -30052,7 +30456,6 @@ digraph pic { --R merge : (%,%) -> % if S has ORDSET --R merge! : (((S,S) -> Boolean),%,%) -> % --R merge! : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R node? : (%,%) -> Boolean if S has SETCAT @@ -30066,14 +30469,11 @@ digraph pic { --R reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate --R remove : (S,%) -> % if S has SETCAT and $ has finiteAggregate --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R remove! : ((S -> Boolean),%) -> % --R remove! : (S,%) -> % if S has SETCAT --R removeDuplicates : % -> % if S has SETCAT and $ has finiteAggregate --R removeDuplicates! : % -> % if S has SETCAT ---R rest : (%,NonNegativeInteger) -> % --R reverse! : % -> % if $ has shallowlyMutable --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R select! : ((S -> Boolean),%) -> % --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable @@ -30086,8 +30486,6 @@ digraph pic { --R setrest! : (%,%) -> % if $ has shallowlyMutable --R setvalue! : (%,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : (((S,S) -> Boolean),%) -> % ---R sort : % -> % if S has ORDSET --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sort! : % -> % if S has ORDSET and $ has shallowlyMutable --R sorted? : (((S,S) -> Boolean),%) -> Boolean @@ -30715,9 +31113,9 @@ digraph pic { --R latex : % -> String map : ((S -> S),%) -> % --R removeDuplicates! : % -> % sample : () -> % --R set : () -> % set : List(S) -> % ---R subset? : (%,%) -> Boolean union : (%,%) -> % ---R union : (%,S) -> % union : (S,%) -> % ---R ?~=? : (%,%) -> Boolean +--R subset? : (%,%) -> Boolean symmetricDifference : (%,%) -> % +--R union : (%,%) -> % union : (%,S) -> % +--R union : (S,%) -> % ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R convert : % -> InputForm if S has KONVERT(INFORM) @@ -30748,7 +31146,6 @@ digraph pic { --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate --R select! : ((S -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean ---R symmetricDifference : (%,%) -> % --R --E 1 @@ -30987,6 +31384,7 @@ digraph pic { --S 1 of 1 )show NonAssociativeRng +--R --R NonAssociativeRng is a category constructor --R Abbreviation for NonAssociativeRng is NARNG --R This constructor is exposed in this frame. @@ -30994,15 +31392,15 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % antiCommutator : (%,%) -> % ---R associator : (%,%,%) -> % coerce : % -> OutputForm ---R commutator : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R antiCommutator : (%,%) -> % associator : (%,%,%) -> % +--R coerce : % -> OutputForm commutator : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R leftPower : (%,PositiveInteger) -> % --R rightPower : (%,PositiveInteger) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -31204,10 +31602,12 @@ digraph pic { --R empty? : % -> Boolean entries : % -> List(S) --R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R insert : (S,%,Integer) -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % ---R qelt : (%,Integer) -> S reverse : % -> % ---R sample : () -> % +--R insert : (S,%,Integer) -> % latex : % -> String if S has SETCAT +--R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET +--R new : (NonNegativeInteger,S) -> % qelt : (%,Integer) -> S +--R reverse : % -> % sample : () -> % +--R sort : % -> % if S has ORDSET sort : (((S,S) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -31232,16 +31632,13 @@ digraph pic { --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if S has SETCAT and $ has finiteAggregate --R members : % -> List(S) if $ has finiteAggregate --R merge : (%,%) -> % if S has ORDSET --R merge : (((S,S) -> Boolean),%,%) -> % ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(S) if $ has finiteAggregate @@ -31260,8 +31657,6 @@ digraph pic { --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if S has ORDSET ---R sort : (((S,S) -> Boolean),%) -> % --R sort! : % -> % if S has ORDSET and $ has shallowlyMutable --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if S has ORDSET @@ -31807,21 +32202,22 @@ digraph pic { --S 1 of 1 )show OrderedCancellationAbelianMonoid +--R --R OrderedCancellationAbelianMonoid is a category constructor --R Abbreviation for OrderedCancellationAbelianMonoid is OCAMON --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for OCAMON --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -31988,17 +32384,20 @@ digraph pic { --R headReduce : (P,%) -> P headReduced? : % -> Boolean --R headReduced? : (P,%) -> Boolean infRittWu? : (%,%) -> Boolean --R initiallyReduce : (P,%) -> P initiallyReduced? : % -> Boolean ---R initials : % -> List(P) internalAugment : (P,%) -> % ---R intersect : (P,%) -> List(%) invertible? : (P,%) -> Boolean ---R invertibleSet : (P,%) -> List(%) last : % -> Union(P,"failed") ---R latex : % -> String mainVariable? : (V,%) -> Boolean ---R mainVariables : % -> List(V) map : ((P -> P),%) -> % ---R mvar : % -> V normalized? : % -> Boolean ---R normalized? : (P,%) -> Boolean purelyAlgebraic? : % -> Boolean +--R initials : % -> List(P) internalAugment : (List(P),%) -> % +--R internalAugment : (P,%) -> % intersect : (P,List(%)) -> List(%) +--R intersect : (List(P),%) -> List(%) intersect : (P,%) -> List(%) +--R invertible? : (P,%) -> Boolean invertibleSet : (P,%) -> List(%) +--R last : % -> Union(P,"failed") latex : % -> String +--R mainVariable? : (V,%) -> Boolean mainVariables : % -> List(V) +--R map : ((P -> P),%) -> % mvar : % -> V +--R normalized? : % -> Boolean normalized? : (P,%) -> Boolean +--R purelyAlgebraic? : % -> Boolean purelyAlgebraic? : (P,%) -> Boolean --R reduceByQuasiMonic : (P,%) -> P removeZero : (P,%) -> P --R rest : % -> Union(%,"failed") retract : List(P) -> % ---R sample : () -> % stronglyReduce : (P,%) -> P ---R stronglyReduced? : % -> Boolean trivialIdeal? : % -> Boolean +--R sample : () -> % select : (%,V) -> Union(P,"failed") +--R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R stronglyReduced? : (P,%) -> Boolean trivialIdeal? : % -> Boolean --R variables : % -> List(V) zeroSetSplit : List(P) -> List(%) --R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -32022,10 +32421,7 @@ digraph pic { --R find : ((P -> Boolean),%) -> Union(P,"failed") --R headRemainder : (P,%) -> Record(num: P,den: R) if R has INTDOM --R initiallyReduced? : (P,%) -> Boolean ---R internalAugment : (List(P),%) -> % ---R intersect : (P,List(%)) -> List(%) --R intersect : (List(P),List(%)) -> List(%) ---R intersect : (List(P),%) -> List(%) --R invertible? : (P,%) -> List(Record(val: Boolean,tower: %)) --R invertibleElseSplit? : (P,%) -> Union(Boolean,List(%)) --R lastSubResultant : (P,P,%) -> List(Record(val: P,tower: %)) @@ -32036,7 +32432,6 @@ digraph pic { --R members : % -> List(P) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(P) if $ has finiteAggregate ---R purelyAlgebraic? : (P,%) -> Boolean --R purelyAlgebraicLeadingMonomial? : (P,%) -> Boolean --R purelyTranscendental? : (P,%) -> Boolean --R quasiComponent : % -> Record(close: List(P),open: List(P)) @@ -32057,12 +32452,10 @@ digraph pic { --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) --R squareFreePart : (P,%) -> List(Record(val: P,tower: %)) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplit : (List(P),Boolean) -> List(%) --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) @@ -32795,13 +33188,13 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -32945,6 +33338,7 @@ Rng is a Ring that does not necessarily have a unit. --S 1 of 1 )show Rng +--R --R Rng is a category constructor --R Abbreviation for Rng is RNG --R This constructor is exposed in this frame. @@ -32952,14 +33346,14 @@ Rng is a Ring that does not necessarily have a unit. --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R ?^? : (%,PositiveInteger) -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -33118,6 +33512,7 @@ digraph pic { --S 1 of 1 )show BiModule +--R --R BiModule(R: Ring,S: Ring) is a category constructor --R Abbreviation for BiModule is BMODULE --R This constructor is exposed in this frame. @@ -33125,14 +33520,13 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,S) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -33328,14 +33722,14 @@ digraph pic { --R empty : () -> % empty? : % -> Boolean --R entries : % -> List(Boolean) eq? : (%,%) -> Boolean --R hash : % -> SingleInteger index? : (Integer,%) -> Boolean ---R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % nand : (%,%) -> % ---R nor : (%,%) -> % not? : % -> % ---R ?or? : (%,%) -> % qelt : (%,Integer) -> Boolean ---R reverse : % -> % sample : () -> % ---R xor : (%,%) -> % ~? : % -> % ---R ?~=? : (%,%) -> Boolean +--R indices : % -> List(Integer) insert : (Boolean,%,Integer) -> % +--R insert : (%,%,Integer) -> % latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R nand : (%,%) -> % nor : (%,%) -> % +--R not? : % -> % ?or? : (%,%) -> % +--R qelt : (%,Integer) -> Boolean reverse : % -> % +--R sample : () -> % xor : (%,%) -> % +--R ~? : % -> % ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((Boolean -> Boolean),%) -> Boolean if $ has finiteAggregate --R convert : % -> InputForm if Boolean has KONVERT(INFORM) @@ -33354,7 +33748,6 @@ digraph pic { --R fill! : (%,Boolean) -> % if $ has shallowlyMutable --R find : ((Boolean -> Boolean),%) -> Union(Boolean,"failed") --R first : % -> Boolean if Integer has ORDSET ---R insert : (Boolean,%,Integer) -> % --R less? : (%,NonNegativeInteger) -> Boolean --R map : ((Boolean -> Boolean),%) -> % --R map : (((Boolean,Boolean) -> Boolean),%,%) -> % @@ -33768,6 +34161,7 @@ digraph pic { --S 1 of 1 )show NonAssociativeRing +--R --R NonAssociativeRing is a category constructor --R Abbreviation for NonAssociativeRing is NASRING --R This constructor is exposed in this frame. @@ -33775,22 +34169,21 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % --R antiCommutator : (%,%) -> % associator : (%,%,%) -> % --R coerce : Integer -> % coerce : % -> OutputForm --R commutator : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R latex : % -> String leftRecip : % -> Union(%,"failed") +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R leftPower : (%,NonNegativeInteger) -> % --R leftPower : (%,PositiveInteger) -> % ---R leftRecip : % -> Union(%,"failed") --R rightPower : (%,NonNegativeInteger) -> % --R rightPower : (%,PositiveInteger) -> % --R rightRecip : % -> Union(%,"failed") @@ -34011,17 +34404,20 @@ digraph pic { --R headReduce : (P,%) -> P headReduced? : % -> Boolean --R headReduced? : (P,%) -> Boolean infRittWu? : (%,%) -> Boolean --R initiallyReduce : (P,%) -> P initiallyReduced? : % -> Boolean ---R initials : % -> List(P) internalAugment : (P,%) -> % ---R intersect : (P,%) -> List(%) invertible? : (P,%) -> Boolean ---R invertibleSet : (P,%) -> List(%) last : % -> Union(P,"failed") ---R latex : % -> String mainVariable? : (V,%) -> Boolean ---R mainVariables : % -> List(V) map : ((P -> P),%) -> % ---R mvar : % -> V normalized? : % -> Boolean ---R normalized? : (P,%) -> Boolean purelyAlgebraic? : % -> Boolean +--R initials : % -> List(P) internalAugment : (List(P),%) -> % +--R internalAugment : (P,%) -> % intersect : (P,List(%)) -> List(%) +--R intersect : (List(P),%) -> List(%) intersect : (P,%) -> List(%) +--R invertible? : (P,%) -> Boolean invertibleSet : (P,%) -> List(%) +--R last : % -> Union(P,"failed") latex : % -> String +--R mainVariable? : (V,%) -> Boolean mainVariables : % -> List(V) +--R map : ((P -> P),%) -> % mvar : % -> V +--R normalized? : % -> Boolean normalized? : (P,%) -> Boolean +--R purelyAlgebraic? : % -> Boolean purelyAlgebraic? : (P,%) -> Boolean --R reduceByQuasiMonic : (P,%) -> P removeZero : (P,%) -> P --R rest : % -> Union(%,"failed") retract : List(P) -> % ---R sample : () -> % stronglyReduce : (P,%) -> P ---R stronglyReduced? : % -> Boolean trivialIdeal? : % -> Boolean +--R sample : () -> % select : (%,V) -> Union(P,"failed") +--R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R stronglyReduced? : (P,%) -> Boolean trivialIdeal? : % -> Boolean --R variables : % -> List(V) zeroSetSplit : List(P) -> List(%) --R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -34045,10 +34441,7 @@ digraph pic { --R find : ((P -> Boolean),%) -> Union(P,"failed") --R headRemainder : (P,%) -> Record(num: P,den: R) if R has INTDOM --R initiallyReduced? : (P,%) -> Boolean ---R internalAugment : (List(P),%) -> % ---R intersect : (P,List(%)) -> List(%) --R intersect : (List(P),List(%)) -> List(%) ---R intersect : (List(P),%) -> List(%) --R invertible? : (P,%) -> List(Record(val: Boolean,tower: %)) --R invertibleElseSplit? : (P,%) -> Union(Boolean,List(%)) --R lastSubResultant : (P,P,%) -> List(Record(val: P,tower: %)) @@ -34059,7 +34452,6 @@ digraph pic { --R members : % -> List(P) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(P) if $ has finiteAggregate ---R purelyAlgebraic? : (P,%) -> Boolean --R purelyAlgebraicLeadingMonomial? : (P,%) -> Boolean --R purelyTranscendental? : (P,%) -> Boolean --R quasiComponent : % -> Record(close: List(P),open: List(P)) @@ -34080,12 +34472,10 @@ digraph pic { --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) --R squareFreePart : (P,%) -> List(Record(val: P,tower: %)) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplit : (List(P),Boolean) -> List(%) --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) @@ -34470,23 +34860,23 @@ digraph pic { --S 1 of 1 )show OrderedAbelianGroup +--R --R OrderedAbelianGroup is a category constructor --R Abbreviation for OrderedAbelianGroup is OAGROUP --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for OAGROUP --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R max : (%,%) -> % min : (%,%) -> % ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -34627,22 +35017,22 @@ digraph pic { --S 1 of 1 )show OrderedAbelianMonoidSup +--R --R OrderedAbelianMonoidSup is a category constructor --R Abbreviation for OrderedAbelianMonoidSup is OAMONS --R This constructor is exposed in this frame. --R Issue )edit bookvol10.2.pamphlet to see algebra source code for OAMONS --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % sample : () -> % ---R sup : (%,%) -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R sample : () -> % sup : (%,%) -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -34828,9 +35218,9 @@ digraph pic { --R merge! : (%,%) -> % min : % -> S --R removeDuplicates! : % -> % sample : () -> % --R set : () -> % set : List(S) -> % ---R subset? : (%,%) -> Boolean union : (%,%) -> % ---R union : (%,S) -> % union : (S,%) -> % ---R ?~=? : (%,%) -> Boolean +--R subset? : (%,%) -> Boolean symmetricDifference : (%,%) -> % +--R union : (%,%) -> % union : (%,S) -> % +--R union : (S,%) -> % ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R convert : % -> InputForm if S has KONVERT(INFORM) @@ -34861,7 +35251,6 @@ digraph pic { --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate --R select! : ((S -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean ---R symmetricDifference : (%,%) -> % --R --E 1 @@ -35140,6 +35529,7 @@ digraph pic { --S 1 of 1 )show Ring +--R --R Ring is a category constructor --R Abbreviation for Ring is RING --R This constructor is exposed in this frame. @@ -35147,18 +35537,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -35427,17 +35816,20 @@ digraph pic { --R headReduce : (P,%) -> P headReduced? : % -> Boolean --R headReduced? : (P,%) -> Boolean infRittWu? : (%,%) -> Boolean --R initiallyReduce : (P,%) -> P initiallyReduced? : % -> Boolean ---R initials : % -> List(P) internalAugment : (P,%) -> % ---R intersect : (P,%) -> List(%) invertible? : (P,%) -> Boolean ---R invertibleSet : (P,%) -> List(%) last : % -> Union(P,"failed") ---R latex : % -> String mainVariable? : (V,%) -> Boolean ---R mainVariables : % -> List(V) map : ((P -> P),%) -> % ---R mvar : % -> V normalized? : % -> Boolean ---R normalized? : (P,%) -> Boolean purelyAlgebraic? : % -> Boolean +--R initials : % -> List(P) internalAugment : (List(P),%) -> % +--R internalAugment : (P,%) -> % intersect : (P,List(%)) -> List(%) +--R intersect : (List(P),%) -> List(%) intersect : (P,%) -> List(%) +--R invertible? : (P,%) -> Boolean invertibleSet : (P,%) -> List(%) +--R last : % -> Union(P,"failed") latex : % -> String +--R mainVariable? : (V,%) -> Boolean mainVariables : % -> List(V) +--R map : ((P -> P),%) -> % mvar : % -> V +--R normalized? : % -> Boolean normalized? : (P,%) -> Boolean +--R purelyAlgebraic? : % -> Boolean purelyAlgebraic? : (P,%) -> Boolean --R reduceByQuasiMonic : (P,%) -> P removeZero : (P,%) -> P --R rest : % -> Union(%,"failed") retract : List(P) -> % ---R sample : () -> % stronglyReduce : (P,%) -> P ---R stronglyReduced? : % -> Boolean trivialIdeal? : % -> Boolean +--R sample : () -> % select : (%,V) -> Union(P,"failed") +--R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R stronglyReduced? : (P,%) -> Boolean trivialIdeal? : % -> Boolean --R variables : % -> List(V) zeroSetSplit : List(P) -> List(%) --R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -35461,10 +35853,7 @@ digraph pic { --R find : ((P -> Boolean),%) -> Union(P,"failed") --R headRemainder : (P,%) -> Record(num: P,den: R) if R has INTDOM --R initiallyReduced? : (P,%) -> Boolean ---R internalAugment : (List(P),%) -> % ---R intersect : (P,List(%)) -> List(%) --R intersect : (List(P),List(%)) -> List(%) ---R intersect : (List(P),%) -> List(%) --R invertible? : (P,%) -> List(Record(val: Boolean,tower: %)) --R invertibleElseSplit? : (P,%) -> Union(Boolean,List(%)) --R lastSubResultant : (P,P,%) -> List(Record(val: P,tower: %)) @@ -35475,7 +35864,6 @@ digraph pic { --R members : % -> List(P) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(P) if $ has finiteAggregate ---R purelyAlgebraic? : (P,%) -> Boolean --R purelyAlgebraicLeadingMonomial? : (P,%) -> Boolean --R purelyTranscendental? : (P,%) -> Boolean --R quasiComponent : % -> Record(close: List(P),open: List(P)) @@ -35496,12 +35884,10 @@ digraph pic { --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) --R squareFreePart : (P,%) -> List(Record(val: P,tower: %)) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplit : (List(P),Boolean) -> List(%) --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) @@ -35893,14 +36279,14 @@ digraph pic { --R empty : () -> % empty? : % -> Boolean --R entries : % -> List(Character) eq? : (%,%) -> Boolean --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R insert : (%,%,Integer) -> % leftTrim : (%,Character) -> % ---R lowerCase : % -> % lowerCase! : % -> % ---R prefix? : (%,%) -> Boolean qelt : (%,Integer) -> Character ---R reverse : % -> % rightTrim : (%,Character) -> % ---R sample : () -> % split : (%,Character) -> List(%) ---R suffix? : (%,%) -> Boolean trim : (%,CharacterClass) -> % ---R trim : (%,Character) -> % upperCase : % -> % ---R upperCase! : % -> % +--R insert : (%,%,Integer) -> % leftTrim : (%,CharacterClass) -> % +--R leftTrim : (%,Character) -> % lowerCase : % -> % +--R lowerCase! : % -> % prefix? : (%,%) -> Boolean +--R qelt : (%,Integer) -> Character reverse : % -> % +--R rightTrim : (%,Character) -> % sample : () -> % +--R split : (%,Character) -> List(%) suffix? : (%,%) -> Boolean +--R trim : (%,CharacterClass) -> % trim : (%,Character) -> % +--R upperCase : % -> % upperCase! : % -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if Character has ORDSET --R ?<=? : (%,%) -> Boolean if Character has ORDSET @@ -35928,7 +36314,6 @@ digraph pic { --R hash : % -> SingleInteger if Character has SETCAT --R insert : (Character,%,Integer) -> % --R latex : % -> String if Character has SETCAT ---R leftTrim : (%,CharacterClass) -> % --R less? : (%,NonNegativeInteger) -> Boolean --R map : (((Character,Character) -> Character),%,%) -> % --R map : ((Character -> Character),%) -> % @@ -36973,24 +37358,26 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for VECTCAT --R --R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % if R has MONOID ?*? : (R,%) -> % if R has MONOID +--R ?+? : (%,%) -> % if R has ABELSG ?-? : (%,%) -> % if R has ABELGRP --R -? : % -> % if R has ABELGRP concat : List(%) -> % --R concat : (%,%) -> % concat : (R,%) -> % --R concat : (%,R) -> % construct : List(R) -> % ---R copy : % -> % delete : (%,Integer) -> % +--R copy : % -> % cross : (%,%) -> % if R has RING +--R delete : (%,Integer) -> % dot : (%,%) -> R if R has RING --R ?.? : (%,Integer) -> R elt : (%,Integer,R) -> R --R empty : () -> % empty? : % -> Boolean --R entries : % -> List(R) eq? : (%,%) -> Boolean --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) --R insert : (%,%,Integer) -> % insert : (R,%,Integer) -> % ---R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % ---R new : (NonNegativeInteger,R) -> % qelt : (%,Integer) -> R ---R reverse : % -> % sample : () -> % +--R latex : % -> String if R has SETCAT map : (((R,R) -> R),%,%) -> % +--R map : ((R -> R),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET new : (NonNegativeInteger,R) -> % +--R qelt : (%,Integer) -> R reverse : % -> % +--R sample : () -> % sort : % -> % if R has ORDSET +--R sort : (((R,R) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate ---R ?*? : (%,R) -> % if R has MONOID ---R ?*? : (R,%) -> % if R has MONOID --R ?*? : (Integer,%) -> % if R has ABELGRP ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has ABELGRP --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?=? : (%,%) -> Boolean if R has SETCAT @@ -37002,9 +37389,7 @@ digraph pic { --R copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable --R count : (R,%) -> NonNegativeInteger if R has SETCAT and $ has finiteAggregate --R count : ((R -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate ---R cross : (%,%) -> % if R has RING --R delete : (%,UniversalSegment(Integer)) -> % ---R dot : (%,%) -> R if R has RING --R ?.? : (%,UniversalSegment(Integer)) -> % --R entry? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -37016,18 +37401,15 @@ digraph pic { --R find : ((R -> Boolean),%) -> Union(R,"failed") --R first : % -> R if Integer has ORDSET --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R length : % -> R if R has RING and R has RADCAT --R less? : (%,NonNegativeInteger) -> Boolean --R magnitude : % -> R if R has RING and R has RADCAT --R map! : ((R -> R),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if R has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (R,%) -> Boolean if R has SETCAT and $ has finiteAggregate --R members : % -> List(R) if $ has finiteAggregate --R merge : (%,%) -> % if R has ORDSET --R merge : (((R,R) -> Boolean),%,%) -> % ---R min : (%,%) -> % if R has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R outerProduct : (%,%) -> Matrix(R) if R has RING @@ -37047,8 +37429,6 @@ digraph pic { --R setelt : (%,UniversalSegment(Integer),R) -> R if $ has shallowlyMutable --R setelt : (%,Integer,R) -> R if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if R has ORDSET ---R sort : (((R,R) -> Boolean),%) -> % --R sort! : % -> % if R has ORDSET and $ has shallowlyMutable --R sort! : (((R,R) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if R has ORDSET @@ -37468,9 +37848,10 @@ digraph pic { --R indices : % -> List(Integer) indices : % -> List(Key) --R insert : (%,%,Integer) -> % insert! : (%,%,Integer) -> % --R key? : (Key,%) -> Boolean keys : % -> List(Key) ---R leaf? : % -> Boolean map : ((Entry -> Entry),%) -> % ---R nodes : % -> List(%) possiblyInfinite? : % -> Boolean ---R qelt : (%,Key) -> Entry rest : % -> % +--R last : (%,NonNegativeInteger) -> % leaf? : % -> Boolean +--R map : ((Entry -> Entry),%) -> % nodes : % -> List(%) +--R possiblyInfinite? : % -> Boolean qelt : (%,Key) -> Entry +--R rest : % -> % rest : (%,NonNegativeInteger) -> % --R reverse : % -> % sample : () -> % --R setelt : (%,Key,Entry) -> Entry table : () -> % --R tail : % -> % @@ -37543,7 +37924,6 @@ digraph pic { --R insert! : (Record(key: Key,entry: Entry),%) -> % --R inspect : % -> Record(key: Key,entry: Entry) --R last : % -> Record(key: Key,entry: Entry) ---R last : (%,NonNegativeInteger) -> % --R latex : % -> String if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT or Record(key: Key,entry: Entry) has SETCAT --R leaves : % -> List(Record(key: Key,entry: Entry)) --R less? : (%,NonNegativeInteger) -> Boolean @@ -37600,7 +37980,6 @@ digraph pic { --R remove! : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate --R removeDuplicates : % -> % if Record(key: Key,entry: Entry) has SETCAT and $ has finiteAggregate or Record(key: Key,entry: Entry) has SETCAT and $ has finiteAggregate --R removeDuplicates! : % -> % if Record(key: Key,entry: Entry) has SETCAT ---R rest : (%,NonNegativeInteger) -> % --R reverse! : % -> % if $ has shallowlyMutable --R search : (Key,%) -> Union(Entry,"failed") --R second : % -> Record(key: Key,entry: Entry) @@ -38172,6 +38551,7 @@ digraph pic { --S 1 of 1 )show CharacteristicNonZero +--R --R CharacteristicNonZero is a category constructor --R Abbreviation for CharacteristicNonZero is CHARNZ --R This constructor is exposed in this frame. @@ -38179,18 +38559,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -38365,6 +38744,7 @@ digraph pic { --S 1 of 1 )show CharacteristicZero +--R --R CharacteristicZero is a category constructor --R Abbreviation for CharacteristicZero is CHARZ --R This constructor is exposed in this frame. @@ -38372,18 +38752,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -38551,6 +38930,7 @@ digraph pic { --S 1 of 1 )show CommutativeRing +--R --R CommutativeRing is a category constructor --R Abbreviation for CommutativeRing is COMRING --R This constructor is exposed in this frame. @@ -38558,18 +38938,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -38765,6 +39144,7 @@ digraph pic { --S 1 of 1 )show DifferentialRing +--R --R DifferentialRing is a category constructor --R Abbreviation for DifferentialRing is DIFRING --R This constructor is exposed in this frame. @@ -38772,20 +39152,18 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % D : % -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm differentiate : % -> % ---R hash : % -> SingleInteger latex : % -> String ---R one? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R differentiate : % -> % hash : % -> SingleInteger +--R latex : % -> String one? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R differentiate : (%,NonNegativeInteger) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -38992,6 +39370,7 @@ digraph pic { --S 1 of 1 )show EntireRing +--R --R EntireRing is a category constructor --R Abbreviation for EntireRing is ENTIRER --R This constructor is exposed in this frame. @@ -38999,18 +39378,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -39210,19 +39588,18 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,Basis) -> % ?*? : (%,R) -> % --R ?*? : (R,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (%,Basis) -> R coefficients : % -> List(R) ---R coerce : Basis -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R leadingCoefficient : % -> R leadingMonomial : % -> Basis ---R map : ((R -> R),%) -> % monom : (Basis,R) -> % ---R monomial? : % -> Boolean monomials : % -> List(%) ---R reductum : % -> % retract : % -> Basis ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (%,Basis) -> R +--R coefficients : % -> List(R) coerce : Basis -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String leadingCoefficient : % -> R +--R leadingMonomial : % -> Basis map : ((R -> R),%) -> % +--R monom : (Basis,R) -> % monomial? : % -> Boolean +--R monomials : % -> List(%) reductum : % -> % +--R retract : % -> Basis sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R leadingTerm : % -> Record(k: Basis,c: R) --R listOfTerms : % -> List(Record(k: Basis,c: R)) --R numberOfMonomials : % -> NonNegativeInteger @@ -39481,19 +39858,18 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -39673,18 +40049,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) --R reducedSystem : Matrix(%) -> Matrix(R) @@ -39880,14 +40255,13 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -40052,6 +40426,7 @@ digraph pic { --S 1 of 1 )show OrderedRing +--R --R OrderedRing is a category constructor --R Abbreviation for OrderedRing is ORDRING --R This constructor is exposed in this frame. @@ -40059,23 +40434,22 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?**? : (%,NonNegativeInteger) -> % --R ?+? : (%,%) -> % -? : % -> % --R ?-? : (%,%) -> % ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R max : (%,%) -> % min : (%,%) -> % ---R negative? : % -> Boolean one? : % -> Boolean ---R positive? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % sign : % -> Integer ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % ?^? : (%,NonNegativeInteger) -> % +--R abs : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % negative? : % -> Boolean +--R one? : % -> Boolean positive? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R sign : % -> Integer zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -40314,22 +40688,20 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,List(S)) -> % D : (%,S) -> % ---R 1 : () -> % 0 : () -> % +--R D : (%,S,NonNegativeInteger) -> % D : (%,List(S)) -> % +--R D : (%,S) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % --R coerce : % -> OutputForm differentiate : (%,List(S)) -> % --R differentiate : (%,S) -> % hash : % -> SingleInteger --R latex : % -> String one? : % -> Boolean --R recip : % -> Union(%,"failed") sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R D : (%,List(S),List(NonNegativeInteger)) -> % ---R D : (%,S,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R differentiate : (%,List(S),List(NonNegativeInteger)) -> % --R differentiate : (%,S,NonNegativeInteger) -> % @@ -40592,27 +40964,28 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for PTCAT --R --R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % if R has MONOID ?*? : (R,%) -> % if R has MONOID +--R ?+? : (%,%) -> % if R has ABELSG ?-? : (%,%) -> % if R has ABELGRP --R -? : % -> % if R has ABELGRP concat : List(%) -> % --R concat : (%,%) -> % concat : (R,%) -> % --R concat : (%,R) -> % construct : List(R) -> % --R convert : List(R) -> % copy : % -> % --R cross : (%,%) -> % delete : (%,Integer) -> % ---R dimension : % -> PositiveInteger ?.? : (%,Integer) -> R ---R elt : (%,Integer,R) -> R empty : () -> % ---R empty? : % -> Boolean entries : % -> List(R) ---R eq? : (%,%) -> Boolean extend : (%,List(R)) -> % ---R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R insert : (%,%,Integer) -> % insert : (R,%,Integer) -> % +--R dimension : % -> PositiveInteger dot : (%,%) -> R if R has RING +--R ?.? : (%,Integer) -> R elt : (%,Integer,R) -> R +--R empty : () -> % empty? : % -> Boolean +--R entries : % -> List(R) eq? : (%,%) -> Boolean +--R extend : (%,List(R)) -> % index? : (Integer,%) -> Boolean +--R indices : % -> List(Integer) insert : (%,%,Integer) -> % +--R insert : (R,%,Integer) -> % latex : % -> String if R has SETCAT --R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET --R new : (NonNegativeInteger,R) -> % point : List(R) -> % --R qelt : (%,Integer) -> R reverse : % -> % ---R sample : () -> % +--R sample : () -> % sort : % -> % if R has ORDSET +--R sort : (((R,R) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate ---R ?*? : (%,R) -> % if R has MONOID ---R ?*? : (R,%) -> % if R has MONOID --R ?*? : (Integer,%) -> % if R has ABELGRP ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has ABELGRP --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?=? : (%,%) -> Boolean if R has SETCAT @@ -40625,7 +40998,6 @@ digraph pic { --R count : (R,%) -> NonNegativeInteger if R has SETCAT and $ has finiteAggregate --R count : ((R -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate --R delete : (%,UniversalSegment(Integer)) -> % ---R dot : (%,%) -> R if R has RING --R ?.? : (%,UniversalSegment(Integer)) -> % --R entry? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -40637,18 +41009,15 @@ digraph pic { --R find : ((R -> Boolean),%) -> Union(R,"failed") --R first : % -> R if Integer has ORDSET --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R length : % -> R if R has RING and R has RADCAT --R less? : (%,NonNegativeInteger) -> Boolean --R magnitude : % -> R if R has RING and R has RADCAT --R map! : ((R -> R),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if R has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (R,%) -> Boolean if R has SETCAT and $ has finiteAggregate --R members : % -> List(R) if $ has finiteAggregate --R merge : (%,%) -> % if R has ORDSET --R merge : (((R,R) -> Boolean),%,%) -> % ---R min : (%,%) -> % if R has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R outerProduct : (%,%) -> Matrix(R) if R has RING @@ -40668,8 +41037,6 @@ digraph pic { --R setelt : (%,UniversalSegment(Integer),R) -> R if $ has shallowlyMutable --R setelt : (%,Integer,R) -> R if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if R has ORDSET ---R sort : (((R,R) -> Boolean),%) -> % --R sort! : % -> % if R has ORDSET and $ has shallowlyMutable --R sort! : (((R,R) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if R has ORDSET @@ -41004,9 +41371,10 @@ The RectangularMatrix domain is matrices of fixed dimension. --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,R) -> % if R has FIELD ?=? : (%,%) -> Boolean --R 0 : () -> % antisymmetric? : % -> Boolean --R coerce : % -> OutputForm column : (%,Integer) -> Col --R copy : % -> % diagonal? : % -> Boolean @@ -41023,8 +41391,6 @@ The RectangularMatrix domain is matrices of fixed dimension. --R square? : % -> Boolean symmetric? : % -> Boolean --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate ---R ?*? : (NonNegativeInteger,%) -> % ---R ?/? : (%,R) -> % if R has FIELD --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R count : ((R -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate --R count : (R,%) -> NonNegativeInteger if R has SETCAT and $ has finiteAggregate @@ -41486,17 +41852,20 @@ digraph pic { --R headReduce : (P,%) -> P headReduced? : % -> Boolean --R headReduced? : (P,%) -> Boolean infRittWu? : (%,%) -> Boolean --R initiallyReduce : (P,%) -> P initiallyReduced? : % -> Boolean ---R initials : % -> List(P) internalAugment : (P,%) -> % ---R intersect : (P,%) -> List(%) invertible? : (P,%) -> Boolean ---R invertibleSet : (P,%) -> List(%) last : % -> Union(P,"failed") ---R latex : % -> String mainVariable? : (V,%) -> Boolean ---R mainVariables : % -> List(V) map : ((P -> P),%) -> % ---R mvar : % -> V normalized? : % -> Boolean ---R normalized? : (P,%) -> Boolean purelyAlgebraic? : % -> Boolean +--R initials : % -> List(P) internalAugment : (List(P),%) -> % +--R internalAugment : (P,%) -> % intersect : (P,List(%)) -> List(%) +--R intersect : (List(P),%) -> List(%) intersect : (P,%) -> List(%) +--R invertible? : (P,%) -> Boolean invertibleSet : (P,%) -> List(%) +--R last : % -> Union(P,"failed") latex : % -> String +--R mainVariable? : (V,%) -> Boolean mainVariables : % -> List(V) +--R map : ((P -> P),%) -> % mvar : % -> V +--R normalized? : % -> Boolean normalized? : (P,%) -> Boolean +--R purelyAlgebraic? : % -> Boolean purelyAlgebraic? : (P,%) -> Boolean --R reduceByQuasiMonic : (P,%) -> P removeZero : (P,%) -> P --R rest : % -> Union(%,"failed") retract : List(P) -> % ---R sample : () -> % stronglyReduce : (P,%) -> P ---R stronglyReduced? : % -> Boolean trivialIdeal? : % -> Boolean +--R sample : () -> % select : (%,V) -> Union(P,"failed") +--R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R stronglyReduced? : (P,%) -> Boolean trivialIdeal? : % -> Boolean --R variables : % -> List(V) zeroSetSplit : List(P) -> List(%) --R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -41520,10 +41889,7 @@ digraph pic { --R find : ((P -> Boolean),%) -> Union(P,"failed") --R headRemainder : (P,%) -> Record(num: P,den: R) if R has INTDOM --R initiallyReduced? : (P,%) -> Boolean ---R internalAugment : (List(P),%) -> % ---R intersect : (P,List(%)) -> List(%) --R intersect : (List(P),List(%)) -> List(%) ---R intersect : (List(P),%) -> List(%) --R invertible? : (P,%) -> List(Record(val: Boolean,tower: %)) --R invertibleElseSplit? : (P,%) -> Union(Boolean,List(%)) --R lastSubResultant : (P,P,%) -> List(Record(val: P,tower: %)) @@ -41534,7 +41900,6 @@ digraph pic { --R members : % -> List(P) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(P) if $ has finiteAggregate ---R purelyAlgebraic? : (P,%) -> Boolean --R purelyAlgebraicLeadingMonomial? : (P,%) -> Boolean --R purelyTranscendental? : (P,%) -> Boolean --R quasiComponent : % -> Record(close: List(P),open: List(P)) @@ -41555,12 +41920,10 @@ digraph pic { --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) --R squareFreePart : (P,%) -> List(Record(val: P,tower: %)) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplit : (List(P),Boolean) -> List(%) --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) @@ -41927,15 +42290,15 @@ digraph pic { --R entries : % -> List(Character) eq? : (%,%) -> Boolean --R hash : % -> SingleInteger index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R latex : % -> String leftTrim : (%,Character) -> % ---R lowerCase : % -> % lowerCase! : % -> % ---R prefix? : (%,%) -> Boolean qelt : (%,Integer) -> Character ---R reverse : % -> % rightTrim : (%,Character) -> % ---R sample : () -> % split : (%,Character) -> List(%) ---R string : Integer -> % suffix? : (%,%) -> Boolean ---R trim : (%,CharacterClass) -> % trim : (%,Character) -> % ---R upperCase : % -> % upperCase! : % -> % ---R ?~=? : (%,%) -> Boolean +--R latex : % -> String leftTrim : (%,CharacterClass) -> % +--R leftTrim : (%,Character) -> % lowerCase : % -> % +--R lowerCase! : % -> % prefix? : (%,%) -> Boolean +--R qelt : (%,Integer) -> Character reverse : % -> % +--R rightTrim : (%,Character) -> % sample : () -> % +--R split : (%,Character) -> List(%) string : Integer -> % +--R suffix? : (%,%) -> Boolean trim : (%,CharacterClass) -> % +--R trim : (%,Character) -> % upperCase : % -> % +--R upperCase! : % -> % ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if Character has ORDSET --R ?<=? : (%,%) -> Boolean if Character has ORDSET @@ -41961,7 +42324,6 @@ digraph pic { --R find : ((Character -> Boolean),%) -> Union(Character,"failed") --R first : % -> Character if Integer has ORDSET --R insert : (Character,%,Integer) -> % ---R leftTrim : (%,CharacterClass) -> % --R less? : (%,NonNegativeInteger) -> Boolean --R map : (((Character,Character) -> Character),%,%) -> % --R map : ((Character -> Character),%) -> % @@ -42384,26 +42746,24 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % apply : (%,R,R) -> R ---R coefficients : % -> List(R) coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R apply : (%,R,R) -> R coefficients : % -> List(R) +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM --R degree : % -> NonNegativeInteger hash : % -> SingleInteger --R latex : % -> String leadingCoefficient : % -> R --R one? : % -> Boolean recip : % -> Union(%,"failed") --R reductum : % -> % retract : % -> R --R sample : () -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R coefficient : (%,NonNegativeInteger) -> R --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) ---R content : % -> R if R has GCDDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R leftDivide : (%,%) -> Record(quotient: %,remainder: %) if R has FIELD --R leftExactQuotient : (%,%) -> Union(%,"failed") if R has FIELD @@ -42966,19 +43326,17 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R one? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String one? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -43189,19 +43547,17 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R one? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String one? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -43449,25 +43805,24 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R D : % -> % if R has DIFRING D : (%,(R -> R)) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R differentiate : (%,(R -> R)) -> % hash : % -> SingleInteger --R latex : % -> String one? : % -> Boolean --R recip : % -> Union(%,"failed") sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R D : (%,NonNegativeInteger) -> % if R has DIFRING --R D : (%,Symbol) -> % if R has PDRING(SYMBOL) --R D : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) --R D : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R D : (%,(R -> R),NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R differentiate : % -> % if R has DIFRING --R differentiate : (%,NonNegativeInteger) -> % if R has DIFRING @@ -43476,7 +43831,6 @@ digraph pic { --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,(R -> R),NonNegativeInteger) -> % ---R differentiate : (%,(R -> R)) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -43738,18 +44092,17 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) @@ -43975,15 +44328,14 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,R) -> % if R has FIELD ?=? : (%,%) -> Boolean --R 0 : () -> % coerce : % -> OutputForm --R construct : (%,%) -> % hash : % -> SingleInteger --R latex : % -> String sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?/? : (%,R) -> % if R has FIELD --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -44169,27 +44521,25 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (A,%) -> % ?*? : (%,A) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R D : () -> % 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R adjoint : % -> % apply : (%,A,A) -> A ---R coefficients : % -> List(A) coerce : A -> % ---R coerce : Integer -> % coerce : % -> OutputForm +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % adjoint : % -> % +--R apply : (%,A,A) -> A coefficients : % -> List(A) +--R coerce : A -> % coerce : Integer -> % +--R coerce : % -> OutputForm content : % -> A if A has GCDDOM --R degree : % -> NonNegativeInteger ?.? : (%,A) -> A --R hash : % -> SingleInteger latex : % -> String --R leadingCoefficient : % -> A one? : % -> Boolean --R recip : % -> Union(%,"failed") reductum : % -> % --R retract : % -> A sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R coefficient : (%,NonNegativeInteger) -> A --R coerce : Fraction(Integer) -> % if A has RETRACT(FRAC(INT)) ---R content : % -> A if A has GCDDOM --R directSum : (%,%) -> % if A has FIELD --R exquo : (%,A) -> Union(%,"failed") if A has INTDOM --R leftDivide : (%,%) -> Record(quotient: %,remainder: %) if A has FIELD @@ -44591,15 +44941,15 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % antiCommutator : (%,%) -> % ---R associator : (%,%,%) -> % coerce : % -> OutputForm ---R commutator : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R antiCommutator : (%,%) -> % associator : (%,%,%) -> % +--R coerce : % -> OutputForm commutator : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R leftPower : (%,PositiveInteger) -> % --R plenaryPower : (%,PositiveInteger) -> % --R rightPower : (%,PositiveInteger) -> % @@ -44813,15 +45163,14 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,S) -> % ?*? : (S,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?/? : (%,S) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm dimension : () -> CardinalNumber ---R hash : % -> SingleInteger latex : % -> String ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,S) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R dimension : () -> CardinalNumber hash : % -> SingleInteger +--R latex : % -> String sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -44992,38 +45341,35 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (vl,%) -> % ?*? : (%,R) -> % --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coef : (%,%) -> R coerce : vl -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coef : (%,%) -> R +--R coerce : vl -> % coerce : OrderedFreeMonoid(vl) -> % --R coerce : R -> % coerce : Integer -> % --R coerce : % -> OutputForm constant : % -> R --R constant? : % -> Boolean hash : % -> SingleInteger --R latex : % -> String lquo : (%,%) -> % --R lquo : (%,vl) -> % map : ((R -> R),%) -> % ---R mirror : % -> % monomial? : % -> Boolean ---R one? : % -> Boolean quasiRegular : % -> % ---R quasiRegular? : % -> Boolean recip : % -> Union(%,"failed") ---R rquo : (%,%) -> % rquo : (%,vl) -> % ---R sample : () -> % varList : % -> List(vl) +--R mindeg : % -> OrderedFreeMonoid(vl) mirror : % -> % +--R monomial? : % -> Boolean one? : % -> Boolean +--R quasiRegular : % -> % quasiRegular? : % -> Boolean +--R recip : % -> Union(%,"failed") rquo : (%,%) -> % +--R rquo : (%,vl) -> % sample : () -> % +--R sh : (%,%) -> % if R has COMRING varList : % -> List(vl) --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R coef : (%,OrderedFreeMonoid(vl)) -> R ---R coerce : OrderedFreeMonoid(vl) -> % --R lquo : (%,OrderedFreeMonoid(vl)) -> % ---R mindeg : % -> OrderedFreeMonoid(vl) --R mindegTerm : % -> Record(k: OrderedFreeMonoid(vl),c: R) --R monom : (OrderedFreeMonoid(vl),R) -> % --R retract : % -> OrderedFreeMonoid(vl) --R retractIfCan : % -> Union(OrderedFreeMonoid(vl),"failed") --R rquo : (%,OrderedFreeMonoid(vl)) -> % --R sh : (%,NonNegativeInteger) -> % if R has COMRING ---R sh : (%,%) -> % if R has COMRING --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -45375,27 +45721,28 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for DIRPCAT --R --R------------------------------- Operations -------------------------------- ---R -? : % -> % if R has RING 1 : () -> % if R has MONOID ---R 0 : () -> % if R has CABMON coerce : % -> Vector(R) +--R ?*? : (R,%) -> % if R has MONOID ?*? : (%,R) -> % if R has MONOID +--R ?*? : (%,%) -> % if R has MONOID ?+? : (%,%) -> % if R has ABELSG +--R -? : % -> % if R has RING ?-? : (%,%) -> % if R has RING +--R ?/? : (%,R) -> % if R has FIELD 1 : () -> % if R has MONOID +--R 0 : () -> % if R has CABMON abs : % -> % if R has ORDRING +--R coerce : R -> % if R has SETCAT coerce : % -> Vector(R) --R copy : % -> % directProduct : Vector(R) -> % ---R ?.? : (%,Integer) -> R elt : (%,Integer,R) -> R ---R empty : () -> % empty? : % -> Boolean ---R entries : % -> List(R) eq? : (%,%) -> Boolean ---R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R map : ((R -> R),%) -> % qelt : (%,Integer) -> R ---R sample : () -> % +--R dot : (%,%) -> R if R has RING ?.? : (%,Integer) -> R +--R elt : (%,Integer,R) -> R empty : () -> % +--R empty? : % -> Boolean entries : % -> List(R) +--R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean +--R indices : % -> List(Integer) latex : % -> String if R has SETCAT +--R map : ((R -> R),%) -> % one? : % -> Boolean if R has MONOID +--R qelt : (%,Integer) -> R random : () -> % if R has FINITE +--R retract : % -> R if R has SETCAT sample : () -> % +--R sup : (%,%) -> % if R has OAMONS --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (PositiveInteger,%) -> % if R has ABELSG --R ?*? : (NonNegativeInteger,%) -> % if R has CABMON ---R ?*? : (R,%) -> % if R has MONOID ---R ?*? : (%,R) -> % if R has MONOID ---R ?*? : (%,%) -> % if R has MONOID --R ?*? : (Integer,%) -> % if R has RING --R ?**? : (%,PositiveInteger) -> % if R has MONOID --R ?**? : (%,NonNegativeInteger) -> % if R has MONOID ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has RING ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDRING or R has OAMONS --R ?<=? : (%,%) -> Boolean if R has ORDRING or R has OAMONS --R ?=? : (%,%) -> Boolean if R has SETCAT @@ -45411,10 +45758,8 @@ digraph pic { --R D : % -> % if and(has(R,DifferentialRing),has(R,Ring)) --R ?^? : (%,PositiveInteger) -> % if R has MONOID --R ?^? : (%,NonNegativeInteger) -> % if R has MONOID ---R abs : % -> % if R has ORDRING --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger if R has RING ---R coerce : R -> % if R has SETCAT --R coerce : Fraction(Integer) -> % if and(has(R,RetractableTo(Fraction(Integer))),has(R,SetCategory)) --R coerce : Integer -> % if and(has(R,RetractableTo(Integer)),has(R,SetCategory)) or R has RING --R coerce : % -> OutputForm if R has SETCAT @@ -45429,7 +45774,6 @@ digraph pic { --R differentiate : (%,NonNegativeInteger) -> % if and(has(R,DifferentialRing),has(R,Ring)) --R differentiate : % -> % if and(has(R,DifferentialRing),has(R,Ring)) --R dimension : () -> CardinalNumber if R has FIELD ---R dot : (%,%) -> R if R has RING --R entry? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R enumerate : () -> List(%) if R has FINITE --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -45441,7 +45785,6 @@ digraph pic { --R first : % -> R if Integer has ORDSET --R hash : % -> SingleInteger if R has SETCAT --R index : PositiveInteger -> % if R has FINITE ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R lookup : % -> PositiveInteger if R has FINITE --R map! : ((R -> R),%) -> % if $ has shallowlyMutable @@ -45453,17 +45796,14 @@ digraph pic { --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R negative? : % -> Boolean if R has ORDRING ---R one? : % -> Boolean if R has MONOID --R parts : % -> List(R) if $ has finiteAggregate --R positive? : % -> Boolean if R has ORDRING --R qsetelt! : (%,Integer,R) -> R if $ has shallowlyMutable ---R random : () -> % if R has FINITE --R recip : % -> Union(%,"failed") if R has MONOID --R reducedSystem : Matrix(%) -> Matrix(R) if R has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) if R has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if and(has(R,LinearlyExplicitRingOver(Integer)),has(R,Ring)) --R reducedSystem : Matrix(%) -> Matrix(Integer) if and(has(R,LinearlyExplicitRingOver(Integer)),has(R,Ring)) ---R retract : % -> R if R has SETCAT --R retract : % -> Fraction(Integer) if and(has(R,RetractableTo(Fraction(Integer))),has(R,SetCategory)) --R retract : % -> Integer if and(has(R,RetractableTo(Integer)),has(R,SetCategory)) --R retractIfCan : % -> Union(R,"failed") if R has SETCAT @@ -45474,7 +45814,6 @@ digraph pic { --R size : () -> NonNegativeInteger if R has FINITE --R size? : (%,NonNegativeInteger) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") if R has CABMON ---R sup : (%,%) -> % if R has OAMONS --R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable --R unitVector : PositiveInteger -> % if R has RING --R zero? : % -> Boolean if R has CABMON @@ -46010,20 +46349,19 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Fraction(Integer) -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Fraction(Integer) -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R inv : % -> % latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -46253,26 +46591,26 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % alternative? : () -> Boolean ---R antiAssociative? : () -> Boolean antiCommutative? : () -> Boolean ---R antiCommutator : (%,%) -> % associative? : () -> Boolean ---R associator : (%,%,%) -> % coerce : % -> OutputForm ---R commutative? : () -> Boolean commutator : (%,%) -> % ---R flexible? : () -> Boolean hash : % -> SingleInteger ---R jacobiIdentity? : () -> Boolean jordanAdmissible? : () -> Boolean ---R jordanAlgebra? : () -> Boolean latex : % -> String ---R leftAlternative? : () -> Boolean leftDiscriminant : Vector(%) -> R ---R leftNorm : % -> R leftTrace : % -> R ---R lieAdmissible? : () -> Boolean lieAlgebra? : () -> Boolean ---R powerAssociative? : () -> Boolean rank : () -> PositiveInteger ---R rightAlternative? : () -> Boolean rightNorm : % -> R +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R alternative? : () -> Boolean antiAssociative? : () -> Boolean +--R antiCommutative? : () -> Boolean antiCommutator : (%,%) -> % +--R associative? : () -> Boolean associator : (%,%,%) -> % +--R coerce : % -> OutputForm commutative? : () -> Boolean +--R commutator : (%,%) -> % flexible? : () -> Boolean +--R hash : % -> SingleInteger jacobiIdentity? : () -> Boolean +--R jordanAdmissible? : () -> Boolean jordanAlgebra? : () -> Boolean +--R latex : % -> String leftAlternative? : () -> Boolean +--R leftDiscriminant : Vector(%) -> R leftNorm : % -> R +--R leftTrace : % -> R lieAdmissible? : () -> Boolean +--R lieAlgebra? : () -> Boolean powerAssociative? : () -> Boolean +--R rank : () -> PositiveInteger rightAlternative? : () -> Boolean +--R rightDiscriminant : Vector(%) -> R rightNorm : % -> R --R rightTrace : % -> R sample : () -> % --R someBasis : () -> Vector(%) zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % --R associatorDependence : () -> List(Vector(R)) if R has INTDOM --R conditionsForIdempotents : Vector(%) -> List(Polynomial(R)) --R coordinates : (Vector(%),Vector(%)) -> Matrix(R) @@ -46290,7 +46628,6 @@ digraph pic { --R recip : % -> Union(%,"failed") if R has INTDOM --R represents : (Vector(R),Vector(%)) -> % --R rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(R) ---R rightDiscriminant : Vector(%) -> R --R rightMinimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has INTDOM --R rightPower : (%,PositiveInteger) -> % --R rightRecip : % -> Union(%,"failed") if R has INTDOM @@ -47280,9 +47617,10 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,R) -> % if R has FIELD ?=? : (%,%) -> Boolean --R LiePoly : LyndonWord(VarSet) -> % 0 : () -> % --R coerce : VarSet -> % coerce : % -> OutputForm --R construct : (%,%) -> % degree : % -> NonNegativeInteger @@ -47290,8 +47628,6 @@ digraph pic { --R latex : % -> String mirror : % -> % --R sample : () -> % varList : % -> List(VarSet) --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?/? : (%,R) -> % if R has FIELD --R coef : (XRecursivePolynomial(VarSet,R),%) -> R --R coerce : % -> XRecursivePolynomial(VarSet,R) --R coerce : % -> XDistributedPolynomial(VarSet,R) @@ -47532,6 +47868,7 @@ digraph pic { --S 1 of 1 )show IntegralDomain +--R --R IntegralDomain is a category constructor --R Abbreviation for IntegralDomain is INTDOM --R This constructor is exposed in this frame. @@ -47539,20 +47876,19 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -47844,23 +48180,21 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : R -> % if R has COMRING coerce : Integer -> % --R coerce : % -> OutputForm degree : % -> NonNegativeInteger --R hash : % -> SingleInteger latex : % -> String --R leadingCoefficient : % -> R one? : % -> Boolean --R recip : % -> Union(%,"failed") reductum : % -> % --R sample : () -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R coefficient : (%,NonNegativeInteger) -> R ---R coerce : R -> % if R has COMRING --R minimumDegree : % -> NonNegativeInteger --R monomial : (R,NonNegativeInteger) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -48129,6 +48463,7 @@ digraph pic { --S 1 of 1 )show OctonionCategory +--R --R OctonionCategory(R: CommutativeRing) is a category constructor --R Abbreviation for OctonionCategory is OC --R This constructor is exposed in this frame. @@ -48137,30 +48472,30 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> R if R has RNS ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm conjugate : % -> % ---R hash : % -> SingleInteger imagE : % -> R ---R imagI : % -> R imagJ : % -> R ---R imagK : % -> R imagi : % -> R ---R imagj : % -> R imagk : % -> R ---R inv : % -> % if R has FIELD latex : % -> String ---R map : ((R -> R),%) -> % norm : % -> R ---R octon : (R,R,R,R,R,R,R,R) -> % one? : % -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> R if R has RNS coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R conjugate : % -> % hash : % -> SingleInteger +--R imagE : % -> R imagI : % -> R +--R imagJ : % -> R imagK : % -> R +--R imagi : % -> R imagj : % -> R +--R imagk : % -> R inv : % -> % if R has FIELD +--R latex : % -> String map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET +--R norm : % -> R octon : (R,R,R,R,R,R,R,R) -> % +--R one? : % -> Boolean random : () -> % if R has FINITE --R real : % -> R recip : % -> Union(%,"failed") --R retract : % -> R sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET --R ?>=? : (%,%) -> Boolean if R has ORDSET ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) @@ -48175,9 +48510,6 @@ digraph pic { --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) --R index : PositiveInteger -> % if R has FINITE --R lookup : % -> PositiveInteger if R has FINITE ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET ---R random : () -> % if R has FINITE --R rational : % -> Fraction(Integer) if R has INS --R rational? : % -> Boolean if R has INS --R rationalIfCan : % -> Union(Fraction(Integer),"failed") if R has INS @@ -48708,27 +49040,28 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R D : (%,(R -> R)) -> % D : % -> % if R has DIFRING --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> R if R has RNS ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm conjugate : % -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> R if R has RNS coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R conjugate : % -> % differentiate : (%,(R -> R)) -> % --R hash : % -> SingleInteger imagI : % -> R --R imagJ : % -> R imagK : % -> R --R inv : % -> % if R has FIELD latex : % -> String ---R map : ((R -> R),%) -> % norm : % -> R +--R map : ((R -> R),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET norm : % -> R --R one? : % -> Boolean quatern : (R,R,R,R) -> % --R real : % -> R recip : % -> Union(%,"failed") --R retract : % -> R sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has FIELD --R ?*? : (%,Fraction(Integer)) -> % if R has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if R has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET @@ -48740,12 +49073,10 @@ digraph pic { --R D : (%,Symbol) -> % if R has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if R has DIFRING --R ?^? : (%,Integer) -> % if R has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coerce : Fraction(Integer) -> % if R has FIELD or R has RETRACT(FRAC(INT)) --R convert : % -> InputForm if R has KONVERT(INFORM) ---R differentiate : (%,(R -> R)) -> % --R differentiate : (%,(R -> R),NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) @@ -48760,8 +49091,6 @@ digraph pic { --R eval : (%,Equation(R)) -> % if R has EVALAB(R) --R eval : (%,R,R) -> % if R has EVALAB(R) --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R rational : % -> Fraction(Integer) if R has INS --R rational? : % -> Boolean if R has INS --R rationalIfCan : % -> Union(Fraction(Integer),"failed") if R has INS @@ -49274,43 +49603,42 @@ The SquareMatrix domain is for square matrices of fixed dimension. --R ?*? : (Row,%) -> Row ?*? : (%,Col) -> Col --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : % -> % if R has DIFRING D : (%,(R -> R)) -> % ---R 1 : () -> % 0 : () -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : % -> % if R has DIFRING +--R D : (%,(R -> R)) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % antisymmetric? : % -> Boolean --R coerce : R -> % coerce : Integer -> % --R coerce : % -> OutputForm column : (%,Integer) -> Col --R copy : % -> % diagonal : % -> Row --R diagonal? : % -> Boolean diagonalMatrix : List(R) -> % ---R diagonalProduct : % -> R elt : (%,Integer,Integer) -> R ---R elt : (%,Integer,Integer,R) -> R empty : () -> % ---R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R hash : % -> SingleInteger latex : % -> String ---R listOfLists : % -> List(List(R)) map : ((R -> R),%) -> % ---R map : (((R,R) -> R),%,%) -> % matrix : List(List(R)) -> % ---R maxColIndex : % -> Integer maxRowIndex : % -> Integer ---R minColIndex : % -> Integer minRowIndex : % -> Integer ---R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger ---R one? : % -> Boolean qelt : (%,Integer,Integer) -> R ---R recip : % -> Union(%,"failed") retract : % -> R ---R row : (%,Integer) -> Row sample : () -> % ---R scalarMatrix : R -> % square? : % -> Boolean ---R symmetric? : % -> Boolean trace : % -> R ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R diagonalProduct : % -> R differentiate : (%,(R -> R)) -> % +--R elt : (%,Integer,Integer) -> R elt : (%,Integer,Integer,R) -> R +--R empty : () -> % empty? : % -> Boolean +--R eq? : (%,%) -> Boolean hash : % -> SingleInteger +--R latex : % -> String listOfLists : % -> List(List(R)) +--R map : ((R -> R),%) -> % map : (((R,R) -> R),%,%) -> % +--R matrix : List(List(R)) -> % maxColIndex : % -> Integer +--R maxRowIndex : % -> Integer minColIndex : % -> Integer +--R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger +--R nrows : % -> NonNegativeInteger one? : % -> Boolean +--R qelt : (%,Integer,Integer) -> R recip : % -> Union(%,"failed") +--R retract : % -> R row : (%,Integer) -> Row +--R sample : () -> % scalarMatrix : R -> % +--R square? : % -> Boolean symmetric? : % -> Boolean +--R trace : % -> R zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if R has FIELD ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R D : (%,NonNegativeInteger) -> % if R has DIFRING --R D : (%,Symbol) -> % if R has PDRING(SYMBOL) --R D : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) --R D : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R D : (%,(R -> R),NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) @@ -49324,7 +49652,6 @@ The SquareMatrix domain is for square matrices of fixed dimension. --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,(R -> R),NonNegativeInteger) -> % ---R differentiate : (%,(R -> R)) -> % --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT --R eval : (%,R,R) -> % if R has EVALAB(R) and R has SETCAT --R eval : (%,Equation(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -49898,40 +50225,36 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (vl,%) -> % ?*? : (%,R) -> % --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coef : (%,%) -> R coerce : vl -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coef : (%,%) -> R +--R coerce : vl -> % coerce : OrderedFreeMonoid(vl) -> % --R coerce : R -> % coerce : Integer -> % --R coerce : % -> OutputForm constant : % -> R --R constant? : % -> Boolean degree : % -> NonNegativeInteger --R hash : % -> SingleInteger latex : % -> String --R lquo : (%,%) -> % lquo : (%,vl) -> % ---R map : ((R -> R),%) -> % mirror : % -> % +--R map : ((R -> R),%) -> % maxdeg : % -> OrderedFreeMonoid(vl) +--R mindeg : % -> OrderedFreeMonoid(vl) mirror : % -> % --R monomial? : % -> Boolean one? : % -> Boolean --R quasiRegular : % -> % quasiRegular? : % -> Boolean --R recip : % -> Union(%,"failed") rquo : (%,%) -> % --R rquo : (%,vl) -> % sample : () -> % ---R varList : % -> List(vl) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R sh : (%,%) -> % if R has COMRING varList : % -> List(vl) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R coef : (%,OrderedFreeMonoid(vl)) -> R ---R coerce : OrderedFreeMonoid(vl) -> % --R lquo : (%,OrderedFreeMonoid(vl)) -> % ---R maxdeg : % -> OrderedFreeMonoid(vl) ---R mindeg : % -> OrderedFreeMonoid(vl) --R mindegTerm : % -> Record(k: OrderedFreeMonoid(vl),c: R) --R monom : (OrderedFreeMonoid(vl),R) -> % --R retract : % -> OrderedFreeMonoid(vl) --R retractIfCan : % -> Union(OrderedFreeMonoid(vl),"failed") --R rquo : (%,OrderedFreeMonoid(vl)) -> % --R sh : (%,NonNegativeInteger) -> % if R has COMRING ---R sh : (%,%) -> % if R has COMRING --R subtractIfCan : (%,%) -> Union(%,"failed") --R trunc : (%,NonNegativeInteger) -> % --R @@ -50222,11 +50545,14 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R +--R coerce : R -> % if R has COMRING coerce : % -> % if R has INTDOM --R coerce : Integer -> % coerce : % -> OutputForm --R degree : % -> E hash : % -> SingleInteger --R latex : % -> String leadingCoefficient : % -> R @@ -50237,15 +50563,9 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if R has CHARNZ ---R coerce : R -> % if R has COMRING ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Integer) -> % if R has ALGEBRA(FRAC(INT)) --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -50573,6 +50893,7 @@ digraph pic { --S 1 of 1 )show FortranMachineTypeCategory +--R --R FortranMachineTypeCategory is a category constructor --R Abbreviation for FortranMachineTypeCategory is FMTC --R This constructor is exposed in this frame. @@ -50580,24 +50901,23 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : Integer -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R max : (%,%) -> % min : (%,%) -> % ---R one? : % -> Boolean recip : % -> Union(%,"failed") ---R retract : % -> Integer sample : () -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : Integer -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % one? : % -> Boolean +--R recip : % -> Union(%,"failed") retract : % -> Integer +--R sample : () -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R retractIfCan : % -> Union(Integer,"failed") @@ -50852,31 +51172,31 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % alternative? : () -> Boolean ---R antiAssociative? : () -> Boolean antiCommutative? : () -> Boolean ---R antiCommutator : (%,%) -> % apply : (Matrix(R),%) -> % ---R associative? : () -> Boolean associator : (%,%,%) -> % ---R basis : () -> Vector(%) coerce : % -> OutputForm ---R commutative? : () -> Boolean commutator : (%,%) -> % ---R convert : Vector(R) -> % convert : % -> Vector(R) ---R coordinates : % -> Vector(R) ?.? : (%,Integer) -> R ---R flexible? : () -> Boolean hash : % -> SingleInteger ---R jacobiIdentity? : () -> Boolean jordanAdmissible? : () -> Boolean ---R jordanAlgebra? : () -> Boolean latex : % -> String ---R leftAlternative? : () -> Boolean leftDiscriminant : () -> R ---R leftDiscriminant : Vector(%) -> R leftNorm : % -> R ---R leftTrace : % -> R leftTraceMatrix : () -> Matrix(R) ---R lieAdmissible? : () -> Boolean lieAlgebra? : () -> Boolean ---R powerAssociative? : () -> Boolean rank : () -> PositiveInteger ---R represents : Vector(R) -> % rightAlternative? : () -> Boolean ---R rightDiscriminant : () -> R rightNorm : % -> R +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R alternative? : () -> Boolean antiAssociative? : () -> Boolean +--R antiCommutative? : () -> Boolean antiCommutator : (%,%) -> % +--R apply : (Matrix(R),%) -> % associative? : () -> Boolean +--R associator : (%,%,%) -> % basis : () -> Vector(%) +--R coerce : % -> OutputForm commutative? : () -> Boolean +--R commutator : (%,%) -> % convert : Vector(R) -> % +--R convert : % -> Vector(R) coordinates : % -> Vector(R) +--R ?.? : (%,Integer) -> R flexible? : () -> Boolean +--R hash : % -> SingleInteger jacobiIdentity? : () -> Boolean +--R jordanAdmissible? : () -> Boolean jordanAlgebra? : () -> Boolean +--R latex : % -> String leftAlternative? : () -> Boolean +--R leftDiscriminant : () -> R leftDiscriminant : Vector(%) -> R +--R leftNorm : % -> R leftTrace : % -> R +--R leftTraceMatrix : () -> Matrix(R) lieAdmissible? : () -> Boolean +--R lieAlgebra? : () -> Boolean powerAssociative? : () -> Boolean +--R rank : () -> PositiveInteger represents : Vector(R) -> % +--R rightAlternative? : () -> Boolean rightDiscriminant : () -> R +--R rightDiscriminant : Vector(%) -> R rightNorm : % -> R --R rightTrace : % -> R rightTraceMatrix : () -> Matrix(R) --R sample : () -> % someBasis : () -> Vector(%) --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % --R associatorDependence : () -> List(Vector(R)) if R has INTDOM --R conditionsForIdempotents : () -> List(Polynomial(R)) --R conditionsForIdempotents : Vector(%) -> List(Polynomial(R)) @@ -50898,7 +51218,6 @@ digraph pic { --R recip : % -> Union(%,"failed") if R has INTDOM --R represents : (Vector(R),Vector(%)) -> % --R rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(R) ---R rightDiscriminant : Vector(%) -> R --R rightMinimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has INTDOM --R rightPower : (%,PositiveInteger) -> % --R rightRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(R)) if R has FIELD @@ -51539,22 +51858,21 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) @@ -51823,6 +52141,7 @@ digraph pic { --S 1 of 1 )show OrderedIntegralDomain +--R --R OrderedIntegralDomain is a category constructor --R Abbreviation for OrderedIntegralDomain is OINTDOM --R This constructor is exposed in this frame. @@ -51830,25 +52149,24 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> % ---R associates? : (%,%) -> Boolean coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R max : (%,%) -> % min : (%,%) -> % ---R negative? : % -> Boolean one? : % -> Boolean ---R positive? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % sign : % -> Integer ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % associates? : (%,%) -> Boolean +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % negative? : % -> Boolean +--R one? : % -> Boolean positive? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R sign : % -> Integer unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -52080,13 +52398,16 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R --R coefficients : % -> List(R) coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm +--R coerce : % -> % if R has INTDOM coerce : Integer -> % +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM --R degree : % -> E ground : % -> R --R ground? : % -> Boolean hash : % -> SingleInteger --R latex : % -> String leadingCoefficient : % -> R @@ -52099,17 +52420,11 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT)) ---R coerce : % -> % if R has INTDOM ---R content : % -> R if R has GCDDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R numberOfMonomials : % -> NonNegativeInteger @@ -52528,13 +52843,15 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,%) -> % ---R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R acos : % -> % acosh : % -> % --R acot : % -> % acoth : % -> % --R acsc : % -> % acsch : % -> % @@ -52565,10 +52882,6 @@ digraph pic { --R tanh : % -> % unit? : % -> Boolean --R unitCanonical : % -> % width : % -> R --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) @@ -52957,11 +53270,13 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coefficient : (%,Expon) -> Coef +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coefficient : (%,Expon) -> Coef coerce : % -> % if Coef has INTDOM --R coerce : Integer -> % coerce : % -> OutputForm --R complete : % -> % degree : % -> Expon --R hash : % -> SingleInteger latex : % -> String @@ -52974,15 +53289,11 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Coef) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if Coef has INTDOM --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coerce : Coef -> % if Coef has COMRING ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R exquo : (%,%) -> Union(%,"failed") if Coef has INTDOM --R monomial : (%,List(Var),List(Expon)) -> % @@ -53299,22 +53610,21 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") --R exquo : (%,%) -> Union(%,"failed") @@ -53550,24 +53860,23 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R one? : % -> Boolean prime? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % one? : % -> Boolean +--R prime? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R exquo : (%,%) -> Union(%,"failed") --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) @@ -53817,23 +54126,23 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,S) -> % ?*? : (Integer,%) -> % --R ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (S,%) -> Integer coerce : S -> % ---R coerce : % -> OutputForm collect : % -> % ---R concat : (%,%) -> % degree : % -> Integer ---R divOfPole : % -> % divOfZero : % -> % ---R effective? : % -> Boolean hash : % -> SingleInteger ---R incr : % -> % latex : % -> String ---R mapGen : ((S -> S),%) -> % nthCoef : (%,Integer) -> Integer ---R nthFactor : (%,Integer) -> S retract : % -> S ---R sample : () -> % size : % -> NonNegativeInteger ---R split : % -> List(%) supp : % -> List(S) ---R suppOfPole : % -> List(S) suppOfZero : % -> List(S) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (S,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (S,%) -> Integer +--R coerce : S -> % coerce : % -> OutputForm +--R collect : % -> % concat : (%,%) -> % +--R degree : % -> Integer divOfPole : % -> % +--R divOfZero : % -> % effective? : % -> Boolean +--R hash : % -> SingleInteger incr : % -> % +--R latex : % -> String mapGen : ((S -> S),%) -> % +--R nthCoef : (%,Integer) -> Integer nthFactor : (%,Integer) -> S +--R retract : % -> S sample : () -> % +--R size : % -> NonNegativeInteger split : % -> List(%) +--R supp : % -> List(S) suppOfPole : % -> List(S) +--R suppOfZero : % -> List(S) zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R highCommonTerms : (%,%) -> % if Integer has OAMON --R mapCoef : ((Integer -> Integer),%) -> % --R retractIfCan : % -> Union(S,"failed") @@ -54074,24 +54383,22 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % one? : % -> Boolean ---R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") ---R ?rem? : (%,%) -> % sample : () -> % ---R sizeLess? : (%,%) -> Boolean unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R one? : % -> Boolean ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) --R euclideanSize : % -> NonNegativeInteger @@ -54486,32 +54793,34 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,Coef) -> % ?*? : (Coef,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R D : (%,List(Var)) -> % D : (%,Var) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % ---R differentiate : (%,Var) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % eval : (%,Var,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R leadingCoefficient : % -> Coef leadingMonomial : % -> % ---R map : ((Coef -> Coef),%) -> % monomial? : % -> Boolean ---R one? : % -> Boolean pole? : % -> Boolean ---R recip : % -> Union(%,"failed") reductum : % -> % ---R sample : () -> % variables : % -> List(Var) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R degree : % -> IndexedExponents(Var) differentiate : (%,List(Var)) -> % +--R differentiate : (%,Var) -> % eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % eval : (%,List(Var),List(%)) -> % +--R eval : (%,Var,%) -> % hash : % -> SingleInteger +--R latex : % -> String leadingCoefficient : % -> Coef +--R leadingMonomial : % -> % map : ((Coef -> Coef),%) -> % +--R monomial? : % -> Boolean one? : % -> Boolean +--R pole? : % -> Boolean recip : % -> Union(%,"failed") +--R reductum : % -> % sample : () -> % +--R variables : % -> List(Var) zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : (%,List(Var),List(NonNegativeInteger)) -> % --R D : (%,Var,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -54531,7 +54840,6 @@ digraph pic { --R coefficient : (%,Var,NonNegativeInteger) -> % --R coefficient : (%,IndexedExponents(Var)) -> Coef --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) ---R coerce : % -> % if Coef has INTDOM --R coerce : Coef -> % if Coef has COMRING --R cos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R cosh : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -54539,13 +54847,8 @@ digraph pic { --R coth : % -> % if Coef has ALGEBRA(FRAC(INT)) --R csc : % -> % if Coef has ALGEBRA(FRAC(INT)) --R csch : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R degree : % -> IndexedExponents(Var) --R differentiate : (%,List(Var),List(NonNegativeInteger)) -> % --R differentiate : (%,Var,NonNegativeInteger) -> % ---R differentiate : (%,List(Var)) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % ---R eval : (%,List(Var),List(%)) -> % --R exp : % -> % if Coef has ALGEBRA(FRAC(INT)) --R exquo : (%,%) -> Union(%,"failed") if Coef has INTDOM --R extend : (%,NonNegativeInteger) -> % @@ -54956,24 +55259,23 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R one? : % -> Boolean prime? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % one? : % -> Boolean +--R prime? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if $ has CHARNZ --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ @@ -55293,12 +55595,14 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef ---R coefficient : (%,Expon) -> Coef coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coefficient : (%,Expon) -> Coef +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % --R degree : % -> Expon ?.? : (%,Expon) -> Coef --R extend : (%,Expon) -> % hash : % -> SingleInteger @@ -55313,8 +55617,6 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Expon,Coef) -> Coef --R D : (%,NonNegativeInteger) -> % if Coef has *: (Expon,Coef) -> Coef @@ -55322,13 +55624,11 @@ digraph pic { --R D : (%,List(Symbol)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Expon,Coef) -> Coef --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Expon,Coef) -> Coef --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Expon,Coef) -> Coef ---R ?^? : (%,NonNegativeInteger) -> % --R approximate : (%,Expon) -> Coef if Coef has **: (Coef,Expon) -> Coef and Coef has coerce: Symbol -> Coef --R associates? : (%,%) -> Boolean if Coef has INTDOM --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coerce : Coef -> % if Coef has COMRING ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R differentiate : % -> % if Coef has *: (Expon,Coef) -> Coef --R differentiate : (%,NonNegativeInteger) -> % if Coef has *: (Expon,Coef) -> Coef @@ -55790,28 +56090,27 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) --R euclideanSize : % -> NonNegativeInteger @@ -56128,32 +56427,34 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R D : % -> % D : (%,NonNegativeInteger) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> % ---R addmod : (%,%,%) -> % associates? : (%,%) -> Boolean ---R base : () -> % binomial : (%,%) -> % ---R bit? : (%,%) -> Boolean coerce : Integer -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm convert : % -> DoubleFloat ---R convert : % -> Float convert : % -> Pattern(Integer) ---R convert : % -> InputForm convert : % -> Integer ---R copy : % -> % dec : % -> % ---R differentiate : % -> % even? : % -> Boolean ---R factor : % -> Factored(%) factorial : % -> % ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> % hash : % -> SingleInteger ---R inc : % -> % init : () -> % ---R invmod : (%,%) -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R length : % -> % mask : % -> % ---R max : (%,%) -> % min : (%,%) -> % ---R mulmod : (%,%,%) -> % negative? : % -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % addmod : (%,%,%) -> % +--R associates? : (%,%) -> Boolean base : () -> % +--R binomial : (%,%) -> % bit? : (%,%) -> Boolean +--R coerce : Integer -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R convert : % -> DoubleFloat convert : % -> Float +--R convert : % -> Pattern(Integer) convert : % -> InputForm +--R convert : % -> Integer copy : % -> % +--R dec : % -> % differentiate : % -> % +--R even? : % -> Boolean factor : % -> Factored(%) +--R factorial : % -> % gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> % +--R hash : % -> SingleInteger inc : % -> % +--R init : () -> % invmod : (%,%) -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % length : % -> % +--R mask : % -> % max : (%,%) -> % +--R min : (%,%) -> % mulmod : (%,%,%) -> % +--R negative? : % -> Boolean nextItem : % -> Union(%,"failed") --R odd? : % -> Boolean one? : % -> Boolean --R permutation : (%,%) -> % positive? : % -> Boolean --R positiveRemainder : (%,%) -> % powmod : (%,%,%) -> % @@ -56168,9 +56469,6 @@ digraph pic { --R symmetricRemainder : (%,%) -> % unit? : % -> Boolean --R unitCanonical : % -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R differentiate : (%,NonNegativeInteger) -> % --R divide : (%,%) -> Record(quotient: %,remainder: %) @@ -56182,7 +56480,6 @@ digraph pic { --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") ---R nextItem : % -> Union(%,"failed") --R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) --R rationalIfCan : % -> Union(Fraction(Integer),"failed") @@ -56744,57 +57041,54 @@ digraph pic { --R ?*? : (%,K) -> % ?*? : (K,%) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?/? : (%,K) -> % if K has FIELD ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R associates? : (%,%) -> Boolean center : % -> K --R coefOfFirstNonZeroTerm : % -> K coefficient : (%,Integer) -> K ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R complete : % -> % degree : % -> Integer ---R delay : (() -> %) -> % ?.? : (%,Integer) -> K ---R extend : (%,Integer) -> % factor : % -> Factored(%) ---R filterUpTo : (%,Integer) -> % findCoef : (%,Integer) -> K ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % leadingCoefficient : % -> K ---R leadingMonomial : % -> % map : ((K -> K),%) -> % ---R monomial : (K,Integer) -> % monomial? : % -> Boolean ---R one? : % -> Boolean order : % -> Integer ---R order : % -> Integer order : (%,Integer) -> Integer ---R pole? : % -> Boolean posExpnPart : % -> % ---R prime? : % -> Boolean printInfo : () -> Boolean ---R printInfo : Boolean -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") reductum : % -> % ---R ?rem? : (%,%) -> % removeFirstZeroes : % -> % ---R removeZeroes : % -> % removeZeroes : (Integer,%) -> % ---R sample : () -> % sbt : (%,%) -> % ---R series : (Integer,K,%) -> % shift : (%,Integer) -> % ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % truncate : (%,Integer) -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R variable : % -> Symbol zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,K) -> % if K has FIELD +--R coerce : K -> % if K has COMRING coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm complete : % -> % +--R degree : % -> Integer delay : (() -> %) -> % +--R ?.? : (%,Integer) -> K extend : (%,Integer) -> % +--R factor : % -> Factored(%) filterUpTo : (%,Integer) -> % +--R findCoef : (%,Integer) -> K gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R leadingCoefficient : % -> K leadingMonomial : % -> % +--R map : ((K -> K),%) -> % monomial : (K,Integer) -> % +--R monomial? : % -> Boolean one? : % -> Boolean +--R order : % -> Integer order : % -> Integer +--R order : (%,Integer) -> Integer pole? : % -> Boolean +--R posExpnPart : % -> % prime? : % -> Boolean +--R printInfo : () -> Boolean printInfo : Boolean -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R reductum : % -> % ?rem? : (%,%) -> % +--R removeFirstZeroes : % -> % removeZeroes : % -> % +--R removeZeroes : (Integer,%) -> % sample : () -> % +--R sbt : (%,%) -> % series : (Integer,K,%) -> % +--R shift : (%,Integer) -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R truncate : (%,Integer) -> % unit? : % -> Boolean +--R unitCanonical : % -> % variable : % -> Symbol +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R D : (%,Symbol,NonNegativeInteger) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R D : (%,List(Symbol)) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R D : (%,Symbol) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R D : (%,NonNegativeInteger) -> % if K has *: (Integer,K) -> K --R D : % -> % if K has *: (Integer,K) -> K ---R ?^? : (%,NonNegativeInteger) -> % --R approximate : (%,Integer) -> K if K has **: (K,Integer) -> K and K has coerce: Symbol -> K --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if K has CHARNZ --R coerce : % -> Stream(Record(k: Integer,c: K)) --R coerce : Stream(Record(k: Integer,c: K)) -> % ---R coerce : K -> % if K has COMRING --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R differentiate : (%,Symbol,NonNegativeInteger) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K --R differentiate : (%,List(Symbol)) -> % if K has PDRING(SYMBOL) and K has *: (Integer,K) -> K @@ -57212,28 +57506,26 @@ digraph pic { --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm complete : % -> % ---R digits : % -> Stream(Integer) extend : (%,Integer) -> % ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R moduloP : % -> Integer modulus : () -> Integer ---R one? : % -> Boolean order : % -> NonNegativeInteger ---R ?quo? : (%,%) -> % quotientByP : % -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R sqrt : (%,Integer) -> % unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complete : % -> % digits : % -> Stream(Integer) +--R extend : (%,Integer) -> % gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % moduloP : % -> Integer +--R modulus : () -> Integer one? : % -> Boolean +--R order : % -> NonNegativeInteger ?quo? : (%,%) -> % +--R quotientByP : % -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean sqrt : (%,Integer) -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R approximate : (%,Integer) -> Integer --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) @@ -57534,41 +57826,44 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,List(VarSet)) -> % D : (%,VarSet) -> % ---R 1 : () -> % 0 : () -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,List(VarSet)) -> % +--R D : (%,VarSet) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R ---R coefficients : % -> List(R) coerce : VarSet -> % ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm degree : % -> E +--R coefficients : % -> List(R) coerce : % -> % if R has INTDOM +--R coerce : VarSet -> % coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R content : % -> R if R has GCDDOM degree : % -> E --R differentiate : (%,VarSet) -> % eval : (%,VarSet,%) -> % ---R eval : (%,VarSet,R) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % ground : % -> R +--R eval : (%,VarSet,R) -> % eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R --R ground? : % -> Boolean hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> R +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leadingCoefficient : % -> R --R leadingMonomial : % -> % map : ((R -> R),%) -> % ---R mapExponents : ((E -> E),%) -> % minimumDegree : % -> E +--R mapExponents : ((E -> E),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET minimumDegree : % -> E --R monomial : (R,E) -> % monomial? : % -> Boolean --R monomials : % -> List(%) one? : % -> Boolean ---R pomopo! : (%,R,E,%) -> % recip : % -> Union(%,"failed") ---R reductum : % -> % retract : % -> VarSet ---R retract : % -> R sample : () -> % ---R variables : % -> List(VarSet) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean +--R pomopo! : (%,R,E,%) -> % primitiveMonomials : % -> List(%) +--R recip : % -> Union(%,"failed") reductum : % -> % +--R retract : % -> VarSet retract : % -> R +--R sample : () -> % variables : % -> List(VarSet) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET --R ?>=? : (%,%) -> Boolean if R has ORDSET --R D : (%,List(VarSet),List(NonNegativeInteger)) -> % --R D : (%,VarSet,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger @@ -57576,10 +57871,8 @@ digraph pic { --R coefficient : (%,List(VarSet),List(NonNegativeInteger)) -> % --R coefficient : (%,VarSet,NonNegativeInteger) -> % --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT)) ---R coerce : % -> % if R has INTDOM --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) --R content : (%,VarSet) -> % if R has GCDDOM ---R content : % -> R if R has GCDDOM --R convert : % -> InputForm if VarSet has KONVERT(INFORM) and R has KONVERT(INFORM) --R convert : % -> Pattern(Integer) if VarSet has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) --R convert : % -> Pattern(Float) if VarSet has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) @@ -57591,25 +57884,17 @@ digraph pic { --R discriminant : (%,VarSet) -> % if R has COMRING --R eval : (%,List(VarSet),List(%)) -> % --R eval : (%,List(VarSet),List(R)) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R factor : % -> Factored(%) if R has PFECAT --R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT --R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT ---R gcd : (%,%) -> % if R has GCDDOM ---R gcd : List(%) -> % if R has GCDDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM --R isExpt : % -> Union(Record(var: VarSet,exponent: NonNegativeInteger),"failed") --R isPlus : % -> Union(List(%),"failed") --R isTimes : % -> Union(List(%),"failed") ---R lcm : (%,%) -> % if R has GCDDOM ---R lcm : List(%) -> % if R has GCDDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM --R mainVariable : % -> Union(VarSet,"failed") ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimumDegree : (%,List(VarSet)) -> List(NonNegativeInteger) --R minimumDegree : (%,VarSet) -> NonNegativeInteger --R monicDivide : (%,%,VarSet) -> Record(quotient: %,remainder: %) @@ -57621,7 +57906,6 @@ digraph pic { --R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if VarSet has PATMAB(INT) and R has PATMAB(INT) --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if VarSet has PATMAB(FLOAT) and R has PATMAB(FLOAT) --R prime? : % -> Boolean if R has PFECAT ---R primitiveMonomials : % -> List(%) --R primitivePart : (%,VarSet) -> % if R has GCDDOM --R primitivePart : % -> % if R has GCDDOM --R reducedSystem : Matrix(%) -> Matrix(R) @@ -58593,12 +58877,14 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef ---R coefficients : % -> Stream(Coef) coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coefficients : % -> Stream(Coef) +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % --R degree : % -> NonNegativeInteger hash : % -> SingleInteger --R latex : % -> String leadingCoefficient : % -> Coef @@ -58611,11 +58897,9 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Coef) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (NonNegativeInteger,Coef) -> Coef --R D : (%,NonNegativeInteger) -> % if Coef has *: (NonNegativeInteger,Coef) -> Coef @@ -58623,7 +58907,6 @@ digraph pic { --R D : (%,List(Symbol)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (NonNegativeInteger,Coef) -> Coef --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has PDRING(SYMBOL) and Coef has *: (NonNegativeInteger,Coef) -> Coef --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (NonNegativeInteger,Coef) -> Coef ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -58642,7 +58925,6 @@ digraph pic { --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coefficient : (%,NonNegativeInteger) -> Coef --R coerce : Coef -> % if Coef has COMRING ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R cos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R cosh : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -59365,7 +59647,7 @@ digraph pic { )set message auto off )clear all ---S 1 of 14 +--S 1 of 15 pi:Polynomial(Integer):=-3*x^3+2*x+13 --R --R @@ -59374,7 +59656,7 @@ pi:Polynomial(Integer):=-3*x^3+2*x+13 --R Type: Polynomial(Integer) --E 1 ---S 2 of 14 +--S 2 of 15 rootOf(pi) --R --R @@ -59382,7 +59664,7 @@ rootOf(pi) --R Type: AlgebraicNumber --E 2 ---S 3 of 14 +--S 3 of 15 rootsOf(pi) --R --R @@ -59390,7 +59672,7 @@ rootsOf(pi) --R Type: List(AlgebraicNumber) --E 3 ---S 4 of 14 +--S 4 of 15 zeroOf(pi) --R --R @@ -59398,7 +59680,7 @@ zeroOf(pi) --R Type: AlgebraicNumber --E 4 ---S 5 of 14 +--S 5 of 15 zerosOf(pi) --R --R @@ -59410,7 +59692,7 @@ zerosOf(pi) --R Type: List(AlgebraicNumber) --E 5 ---S 6 of 14 +--S 6 of 15 sup:SparseUnivariatePolynomial(Integer):=-3*x^3+2*x+13 --R --R @@ -59419,7 +59701,7 @@ sup:SparseUnivariatePolynomial(Integer):=-3*x^3+2*x+13 --R Type: SparseUnivariatePolynomial(Integer) --E 6 ---S 7 of 14 +--S 7 of 15 rootOf(sup) --R --R @@ -59427,7 +59709,7 @@ rootOf(sup) --R Type: AlgebraicNumber --E 7 ---S 8 of 14 +--S 8 of 15 rootOf(sup,x) --R --R @@ -59435,7 +59717,7 @@ rootOf(sup,x) --R Type: AlgebraicNumber --E 8 ---S 9 of 14 +--S 9 of 15 rootsOf(sup) --R --R @@ -59443,7 +59725,7 @@ rootsOf(sup) --R Type: List(AlgebraicNumber) --E 9 ---S 10 of 14 +--S 10 of 15 rootsOf(sup,x) --R --R @@ -59451,7 +59733,7 @@ rootsOf(sup,x) --R Type: List(AlgebraicNumber) --E 10 ---S 11 of 14 +--S 11 of 15 zeroOf(sup) --R --R @@ -59459,7 +59741,7 @@ zeroOf(sup) --R Type: AlgebraicNumber --E 11 ---S 12 of 14 +--S 12 of 15 zeroOf(sup,x) --R --R @@ -59467,7 +59749,7 @@ zeroOf(sup,x) --R Type: AlgebraicNumber --E 12 ---S 13 of 14 +--S 13 of 15 zerosOf(sup) --R --R @@ -59479,7 +59761,7 @@ zerosOf(sup) --R Type: List(AlgebraicNumber) --E 13 ---S 14 of 14 +--S 14 of 15 zerosOf(sup,x) --R --R @@ -59491,6 +59773,65 @@ zerosOf(sup,x) --R Type: List(AlgebraicNumber) --E 14 +--S 15 of 15 +)show AlgebraicallyClosedField +--R +--R AlgebraicallyClosedField is a category constructor +--R Abbreviation for AlgebraicallyClosedField is ACF +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.2.pamphlet to see algebra source code for ACF +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm factor : % -> Factored(%) +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % nthRoot : (%,Integer) -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % rootOf : Polynomial(%) -> % +--R rootsOf : Polynomial(%) -> List(%) sample : () -> % +--R sizeLess? : (%,%) -> Boolean sqrt : % -> % +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean zeroOf : Polynomial(%) -> % +--R zerosOf : Polynomial(%) -> List(%) ?~=? : (%,%) -> Boolean +--R characteristic : () -> NonNegativeInteger +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R euclideanSize : % -> NonNegativeInteger +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R rootOf : (SparseUnivariatePolynomial(%),Symbol) -> % +--R rootOf : SparseUnivariatePolynomial(%) -> % +--R rootsOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) +--R rootsOf : SparseUnivariatePolynomial(%) -> List(%) +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R zeroOf : (SparseUnivariatePolynomial(%),Symbol) -> % +--R zeroOf : SparseUnivariatePolynomial(%) -> % +--R zerosOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) +--R zerosOf : SparseUnivariatePolynomial(%) -> List(%) +--R +--E 15 + )spool )lisp (bye) \end{chunk} @@ -60027,30 +60368,39 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,(R -> R)) -> % D : % -> % if R has DIFRING +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,(R -> R)) -> % +--R D : % -> % if R has DIFRING D : (%,V,NonNegativeInteger) -> % --R D : (%,List(V)) -> % D : (%,V) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R ---R coefficients : % -> List(R) coerce : S -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coefficient : (%,E) -> R coefficients : % -> List(R) +--R coerce : % -> % if R has INTDOM coerce : S -> % --R coerce : V -> % coerce : R -> % --R coerce : Integer -> % coerce : % -> OutputForm ---R degree : % -> E differentiate : (%,List(V)) -> % +--R content : % -> R if R has GCDDOM degree : % -> E +--R differentiate : (%,(R -> R)) -> % differentiate : (%,List(V)) -> % --R differentiate : (%,V) -> % eval : (%,List(V),List(%)) -> % --R eval : (%,V,%) -> % eval : (%,List(V),List(R)) -> % ---R eval : (%,V,R) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % ground : % -> R +--R eval : (%,V,R) -> % eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R --R ground? : % -> Boolean hash : % -> SingleInteger --R initial : % -> % isobaric? : % -> Boolean ---R latex : % -> String leader : % -> V +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leader : % -> V --R leadingCoefficient : % -> R leadingMonomial : % -> % --R map : ((R -> R),%) -> % mapExponents : ((E -> E),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET --R minimumDegree : % -> E monomial : (R,E) -> % --R monomial? : % -> Boolean monomials : % -> List(%) --R one? : % -> Boolean order : % -> NonNegativeInteger ---R pomopo! : (%,R,E,%) -> % recip : % -> Union(%,"failed") +--R order : (%,S) -> NonNegativeInteger pomopo! : (%,R,E,%) -> % +--R primitiveMonomials : % -> List(%) recip : % -> Union(%,"failed") --R reductum : % -> % retract : % -> S --R retract : % -> V retract : % -> R --R sample : () -> % separant : % -> % @@ -60058,9 +60408,6 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET @@ -60072,19 +60419,15 @@ digraph pic { --R D : (%,Symbol) -> % if R has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if R has DIFRING --R D : (%,List(V),List(NonNegativeInteger)) -> % ---R D : (%,V,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) or R has CHARNZ --R coefficient : (%,List(V),List(NonNegativeInteger)) -> % --R coefficient : (%,V,NonNegativeInteger) -> % ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT)) --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) --R content : (%,V) -> % if R has GCDDOM ---R content : % -> R if R has GCDDOM --R convert : % -> InputForm if V has KONVERT(INFORM) and R has KONVERT(INFORM) --R convert : % -> Pattern(Integer) if V has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) --R convert : % -> Pattern(Float) if V has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) @@ -60092,7 +60435,6 @@ digraph pic { --R degree : (%,List(V)) -> List(NonNegativeInteger) --R degree : (%,V) -> NonNegativeInteger --R differentialVariables : % -> List(S) ---R differentiate : (%,(R -> R)) -> % --R differentiate : (%,(R -> R),NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) @@ -60107,27 +60449,19 @@ digraph pic { --R eval : (%,S,R) -> % if R has DIFRING --R eval : (%,List(S),List(%)) -> % if R has DIFRING --R eval : (%,S,%) -> % if R has DIFRING ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R factor : % -> Factored(%) if R has PFECAT --R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT --R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT ---R gcd : (%,%) -> % if R has GCDDOM ---R gcd : List(%) -> % if R has GCDDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM --R isExpt : % -> Union(Record(var: V,exponent: NonNegativeInteger),"failed") --R isPlus : % -> Union(List(%),"failed") --R isTimes : % -> Union(List(%),"failed") ---R lcm : (%,%) -> % if R has GCDDOM ---R lcm : List(%) -> % if R has GCDDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM --R mainVariable : % -> Union(V,"failed") --R makeVariable : % -> (NonNegativeInteger -> %) if R has DIFRING --R makeVariable : S -> (NonNegativeInteger -> %) ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimumDegree : (%,List(V)) -> List(NonNegativeInteger) --R minimumDegree : (%,V) -> NonNegativeInteger --R monicDivide : (%,%,V) -> Record(quotient: %,remainder: %) @@ -60136,11 +60470,9 @@ digraph pic { --R multivariate : (SparseUnivariatePolynomial(%),V) -> % --R multivariate : (SparseUnivariatePolynomial(R),V) -> % --R numberOfMonomials : % -> NonNegativeInteger ---R order : (%,S) -> NonNegativeInteger --R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if V has PATMAB(INT) and R has PATMAB(INT) --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if V has PATMAB(FLOAT) and R has PATMAB(FLOAT) --R prime? : % -> Boolean if R has PFECAT ---R primitiveMonomials : % -> List(%) --R primitivePart : (%,V) -> % if R has GCDDOM --R primitivePart : % -> % if R has GCDDOM --R reducedSystem : Matrix(%) -> Matrix(R) @@ -60932,29 +61264,27 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % one? : % -> Boolean ---R prime? : % -> Boolean primeFrobenius : % -> % ---R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") ---R ?rem? : (%,%) -> % sample : () -> % ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R primeFrobenius : % -> % ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") --R discreteLog : (%,%) -> Union(NonNegativeInteger,"failed") @@ -61251,11 +61581,13 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : R -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R characteristicPolynomial : % -> UP coerce : R -> % --R coerce : Integer -> % coerce : % -> OutputForm --R discriminant : Vector(%) -> R hash : % -> SingleInteger --R latex : % -> String norm : % -> R @@ -61263,11 +61595,7 @@ digraph pic { --R recip : % -> Union(%,"failed") sample : () -> % --R trace : % -> R zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger ---R characteristicPolynomial : % -> UP --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coordinates : (Vector(%),Vector(%)) -> Matrix(R) --R coordinates : (%,Vector(%)) -> Vector(R) @@ -61565,61 +61893,63 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for FS --R --R------------------------------- Operations -------------------------------- ---R -? : % -> % if R has ABELGRP ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R ?*? : (%,%) -> % if R has SGROUP ?*? : (%,R) -> % if R has COMRING +--R ?*? : (R,%) -> % if R has COMRING ?+? : (%,%) -> % if R has ABELSG +--R -? : % -> % if R has ABELGRP ?-? : (%,%) -> % if R has ABELGRP +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean D : (%,Symbol) -> % if R has RING --R 1 : () -> % if R has SGROUP 0 : () -> % if R has ABELSG +--R applyQuote : (Symbol,List(%)) -> % applyQuote : (Symbol,%,%,%) -> % --R applyQuote : (Symbol,%,%) -> % applyQuote : (Symbol,%) -> % --R belong? : BasicOperator -> Boolean box : List(%) -> % ---R box : % -> % coerce : R -> % ---R coerce : Symbol -> % coerce : Kernel(%) -> % ---R coerce : % -> OutputForm distribute : (%,%) -> % ---R distribute : % -> % elt : (BasicOperator,%,%) -> % ---R elt : (BasicOperator,%) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % eval : (%,Kernel(%),%) -> % +--R box : % -> % coerce : % -> % if R has INTDOM +--R coerce : R -> % coerce : Symbol -> % +--R coerce : Kernel(%) -> % coerce : % -> OutputForm +--R distribute : (%,%) -> % distribute : % -> % +--R elt : (BasicOperator,List(%)) -> % elt : (BasicOperator,%,%,%) -> % +--R elt : (BasicOperator,%,%) -> % elt : (BasicOperator,%) -> % +--R eval : (%,Symbol,(% -> %)) -> % eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % eval : (%,Kernel(%),%) -> % --R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean +--R gcd : (%,%) -> % if R has INTDOM gcd : List(%) -> % if R has INTDOM --R ground : % -> R ground? : % -> Boolean --R hash : % -> SingleInteger height : % -> NonNegativeInteger ---R is? : (%,Symbol) -> Boolean kernel : (BasicOperator,%) -> % ---R kernels : % -> List(Kernel(%)) latex : % -> String +--R is? : (%,Symbol) -> Boolean is? : (%,BasicOperator) -> Boolean +--R kernel : (BasicOperator,%) -> % kernels : % -> List(Kernel(%)) +--R latex : % -> String lcm : (%,%) -> % if R has INTDOM +--R lcm : List(%) -> % if R has INTDOM map : ((% -> %),Kernel(%)) -> % --R max : (%,%) -> % min : (%,%) -> % +--R numerator : % -> % if R has RING one? : % -> Boolean if R has SGROUP --R paren : List(%) -> % paren : % -> % --R retract : % -> R retract : % -> Symbol --R retract : % -> Kernel(%) subst : (%,Equation(%)) -> % --R tower : % -> List(Kernel(%)) variables : % -> List(Symbol) --R ?~=? : (%,%) -> Boolean ---R ?*? : (%,%) -> % if R has SGROUP --R ?*? : (PositiveInteger,%) -> % if R has ABELSG --R ?*? : (NonNegativeInteger,%) -> % if R has ABELSG --R ?*? : (Integer,%) -> % if R has ABELGRP ---R ?*? : (%,R) -> % if R has COMRING ---R ?*? : (R,%) -> % if R has COMRING --R ?*? : (%,Fraction(Integer)) -> % if R has INTDOM --R ?*? : (Fraction(Integer),%) -> % if R has INTDOM --R ?**? : (%,PositiveInteger) -> % if R has SGROUP --R ?**? : (%,NonNegativeInteger) -> % if R has SGROUP --R ?**? : (%,Integer) -> % if R has GROUP or R has INTDOM ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has ABELGRP --R ?/? : (%,%) -> % if R has GROUP or R has INTDOM --R ?/? : (SparseMultivariatePolynomial(R,Kernel(%)),SparseMultivariatePolynomial(R,Kernel(%))) -> % if R has INTDOM ---R D : (%,Symbol) -> % if R has RING --R D : (%,List(Symbol)) -> % if R has RING --R D : (%,Symbol,NonNegativeInteger) -> % if R has RING --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has RING --R ?^? : (%,PositiveInteger) -> % if R has SGROUP --R ?^? : (%,NonNegativeInteger) -> % if R has SGROUP --R ?^? : (%,Integer) -> % if R has GROUP or R has INTDOM ---R applyQuote : (Symbol,List(%)) -> % --R applyQuote : (Symbol,%,%,%,%) -> % ---R applyQuote : (Symbol,%,%,%) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R characteristic : () -> NonNegativeInteger if R has RING --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coerce : Integer -> % if R has RING or R has RETRACT(INT) --R coerce : Fraction(Integer) -> % if R has INTDOM or R has RETRACT(INT) and R has INTDOM or R has RETRACT(FRAC(INT)) --R coerce : Polynomial(R) -> % if R has RING ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Polynomial(R)) -> % if R has INTDOM --R coerce : Fraction(Polynomial(Fraction(R))) -> % if R has INTDOM --R coerce : Polynomial(Fraction(R)) -> % if R has INTDOM @@ -61639,9 +61969,7 @@ digraph pic { --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has RING --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has RING --R divide : (%,%) -> Record(quotient: %,remainder: %) if R has INTDOM ---R elt : (BasicOperator,List(%)) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,%,%,%) -> % --R euclideanSize : % -> NonNegativeInteger if R has INTDOM --R eval : (%,Symbol,NonNegativeInteger,(% -> %)) -> % if R has RING --R eval : (%,Symbol,NonNegativeInteger,(List(%) -> %)) -> % if R has RING @@ -61656,12 +61984,9 @@ digraph pic { --R eval : (%,BasicOperator,(List(%) -> %)) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R eval : (%,List(Kernel(%)),List(%)) -> % --R even? : % -> Boolean if $ has RETRACT(INT) --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if R has INTDOM @@ -61669,11 +61994,8 @@ digraph pic { --R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) if R has INTDOM --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") if R has INTDOM --R factor : % -> Factored(%) if R has INTDOM ---R gcd : (%,%) -> % if R has INTDOM ---R gcd : List(%) -> % if R has INTDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has INTDOM --R inv : % -> % if R has GROUP or R has INTDOM ---R is? : (%,BasicOperator) -> Boolean --R isExpt : (%,Symbol) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING --R isExpt : (%,BasicOperator) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING --R isExpt : % -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has SGROUP @@ -61682,17 +62004,12 @@ digraph pic { --R isPower : % -> Union(Record(val: %,exponent: Integer),"failed") if R has RING --R isTimes : % -> Union(List(%),"failed") if R has SGROUP --R kernel : (BasicOperator,List(%)) -> % ---R lcm : (%,%) -> % if R has INTDOM ---R lcm : List(%) -> % if R has INTDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has INTDOM --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") if R has INTDOM --R numer : % -> SparseMultivariatePolynomial(R,Kernel(%)) if R has RING ---R numerator : % -> % if R has RING --R odd? : % -> Boolean if $ has RETRACT(INT) ---R one? : % -> Boolean if R has SGROUP --R operator : BasicOperator -> BasicOperator --R operators : % -> List(BasicOperator) --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if R has PATMAB(FLOAT) @@ -63018,6 +63335,7 @@ digraph pic { --S 1 of 1 )show InfinitlyClosePointCategory +--R --I InfinitlyClosePointCategory(K: Field, --I symb: List Symbol, --I PolyRing: PolynomialCategory(t#1,t#4,OrderedVariableList t#2), @@ -63285,32 +63603,31 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm conjugate : % -> % ---R extDegree : % -> PositiveInteger factor : % -> Factored(%) ---R fullOutput : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % ground? : % -> Boolean ---R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % maxTower : List(%) -> % ---R one? : % -> Boolean previousTower : % -> % ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % setTower! : % -> Void ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % unit? : % -> Boolean ---R unitCanonical : % -> % vectorise : (%,%) -> Vector(%) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R conjugate : % -> % extDegree : % -> PositiveInteger +--R factor : % -> Factored(%) fullOutput : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R ground? : % -> Boolean hash : % -> SingleInteger +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R maxTower : List(%) -> % one? : % -> Boolean +--R previousTower : % -> % prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R setTower! : % -> Void sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R vectorise : (%,%) -> Vector(%) zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R definingPolynomial : % -> SparseUnivariatePolynomial(%) --R definingPolynomial : () -> SparseUnivariatePolynomial(%) @@ -63664,24 +63981,28 @@ digraph pic { --R ?*? : (%,S) -> % ?*? : (S,%) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (S,S) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : (%,(S -> S)) -> % --R D : % -> % if S has DIFRING 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % if S has OINTDOM associates? : (%,%) -> Boolean --R ceiling : % -> S if S has INS coerce : S -> % --R coerce : Fraction(Integer) -> % coerce : % -> % --R coerce : Integer -> % coerce : % -> OutputForm ---R denom : % -> S denominator : % -> % +--R convert : % -> Float if S has REAL denom : % -> S +--R denominator : % -> % differentiate : (%,(S -> S)) -> % --R factor : % -> Factored(%) floor : % -> S if S has INS --R gcd : List(%) -> % gcd : (%,%) -> % --R hash : % -> SingleInteger init : () -> % if S has STEP --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % ---R map : ((S -> S),%) -> % numer : % -> S +--R map : ((S -> S),%) -> % max : (%,%) -> % if S has ORDSET +--R min : (%,%) -> % if S has ORDSET numer : % -> S --R numerator : % -> % one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % --R random : () -> % if S has INS recip : % -> Union(%,"failed") @@ -63689,9 +64010,8 @@ digraph pic { --R sample : () -> % sizeLess? : (%,%) -> Boolean --R squareFree : % -> Factored(%) squareFreePart : % -> % --R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R wholePart : % -> S if S has EUCDOM zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET --R ?>? : (%,%) -> Boolean if S has ORDSET @@ -63702,18 +64022,14 @@ digraph pic { --R D : (%,List(Symbol)) -> % if S has PDRING(SYMBOL) --R D : (%,Symbol) -> % if S has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if S has DIFRING ---R ?^? : (%,NonNegativeInteger) -> % ---R abs : % -> % if S has OINTDOM --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if S has CHARNZ or and(has($,CharacteristicNonZero),has(S,PolynomialFactorizationExplicit)) --R coerce : Symbol -> % if S has RETRACT(SYMBOL) --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(S,PolynomialFactorizationExplicit)) --R convert : % -> DoubleFloat if S has REAL ---R convert : % -> Float if S has REAL --R convert : % -> InputForm if S has KONVERT(INFORM) --R convert : % -> Pattern(Float) if S has KONVERT(PATTERN(FLOAT)) --R convert : % -> Pattern(Integer) if S has KONVERT(PATTERN(INT)) ---R differentiate : (%,(S -> S)) -> % --R differentiate : (%,(S -> S),NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if S has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if S has PDRING(SYMBOL) @@ -63739,8 +64055,6 @@ digraph pic { --R fractionPart : % -> % if S has EUCDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) ---R max : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") --R negative? : % -> Boolean if S has OINTDOM --R nextItem : % -> Union(%,"failed") if S has STEP @@ -63764,7 +64078,6 @@ digraph pic { --R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if S has PFECAT --R subtractIfCan : (%,%) -> Union(%,"failed") --R unitNormal : % -> Record(unit: %,canonical: %,associate: %) ---R wholePart : % -> S if S has EUCDOM --R --E 1 @@ -64366,14 +64679,16 @@ digraph pic { --R ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % --R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R abs : % -> % associates? : (%,%) -> Boolean --R coerce : Fraction(Integer) -> % coerce : Integer -> % --R coerce : Fraction(Integer) -> % coerce : % -> % @@ -64391,14 +64706,11 @@ digraph pic { --R rename! : (%,OutputForm) -> % retract : % -> Fraction(Integer) --R sample : () -> % sign : % -> Integer --R sizeLess? : (%,%) -> Boolean sqrt : Integer -> % ---R sqrt : Fraction(Integer) -> % sqrt : % -> % ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R sqrt : Fraction(Integer) -> % sqrt : (%,NonNegativeInteger) -> % +--R sqrt : % -> % squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R allRootsOf : Polynomial(Integer) -> List(%) --R allRootsOf : Polynomial(Fraction(Integer)) -> List(%) --R allRootsOf : Polynomial(%) -> List(%) @@ -64427,7 +64739,6 @@ digraph pic { --R retractIfCan : % -> Union(Integer,"failed") if Fraction(Integer) has RETRACT(INT) --R rootOf : (SparseUnivariatePolynomial(%),PositiveInteger) -> Union(%,"failed") --R rootOf : (SparseUnivariatePolynomial(%),PositiveInteger,OutputForm) -> Union(%,"failed") ---R sqrt : (%,NonNegativeInteger) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") --R unitNormal : % -> Record(unit: %,canonical: %,associate: %) --R @@ -64915,14 +65226,16 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R abs : % -> % associates? : (%,%) -> Boolean --R ceiling : % -> % coerce : Fraction(Integer) -> % --R coerce : Integer -> % coerce : Fraction(Integer) -> % @@ -64947,10 +65260,6 @@ digraph pic { --R truncate : % -> % unit? : % -> Boolean --R unitCanonical : % -> % wholePart : % -> Integer --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) --R euclideanSize : % -> NonNegativeInteger @@ -65351,20 +65660,26 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,V,NonNegativeInteger) -> % --R D : (%,List(V)) -> % D : (%,V) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R ---R coefficients : % -> List(R) coerce : V -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coefficient : (%,E) -> R coefficients : % -> List(R) +--R coerce : % -> % if R has INTDOM coerce : V -> % --R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm deepestInitial : % -> % ---R deepestTail : % -> % degree : % -> E ---R differentiate : (%,List(V)) -> % differentiate : (%,V) -> % ---R eval : (%,List(V),List(%)) -> % eval : (%,V,%) -> % ---R eval : (%,List(V),List(R)) -> % eval : (%,V,R) -> % +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM +--R deepestInitial : % -> % deepestTail : % -> % +--R degree : % -> E differentiate : (%,List(V)) -> % +--R differentiate : (%,V) -> % eval : (%,List(V),List(%)) -> % +--R eval : (%,V,%) -> % eval : (%,List(V),List(R)) -> % +--R eval : (%,V,R) -> % eval : (%,List(%),List(%)) -> % --R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM gcd : (R,%) -> R if R has GCDDOM --R ground : % -> R ground? : % -> Boolean --R hash : % -> SingleInteger head : % -> % --R headReduce : (%,%) -> % headReduced? : (%,%) -> Boolean @@ -65372,20 +65687,23 @@ digraph pic { --R initiallyReduce : (%,%) -> % iteratedInitials : % -> List(%) --R latex : % -> String lazyPquo : (%,%,V) -> % --R lazyPquo : (%,%) -> % lazyPrem : (%,%,V) -> % ---R lazyPrem : (%,%) -> % leadingCoefficient : (%,V) -> % +--R lazyPrem : (%,%) -> % lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leadingCoefficient : (%,V) -> % --R leadingCoefficient : % -> R leadingMonomial : % -> % --R leastMonomial : % -> % mainCoefficients : % -> List(%) --R mainMonomial : % -> % mainMonomials : % -> List(%) --R map : ((R -> R),%) -> % mapExponents : ((E -> E),%) -> % ---R mdeg : % -> NonNegativeInteger minimumDegree : % -> E +--R max : (%,%) -> % if R has ORDSET mdeg : % -> NonNegativeInteger +--R min : (%,%) -> % if R has ORDSET minimumDegree : % -> E --R monic? : % -> Boolean monicModulo : (%,%) -> % --R monomial : (R,E) -> % monomial? : % -> Boolean --R monomials : % -> List(%) mvar : % -> V --R normalized? : (%,%) -> Boolean one? : % -> Boolean --R pomopo! : (%,R,E,%) -> % pquo : (%,%,V) -> % --R pquo : (%,%) -> % prem : (%,%,V) -> % ---R prem : (%,%) -> % quasiMonic? : % -> Boolean ---R recip : % -> Union(%,"failed") reduced? : (%,%) -> Boolean +--R prem : (%,%) -> % primitiveMonomials : % -> List(%) +--R quasiMonic? : % -> Boolean recip : % -> Union(%,"failed") +--R reduced? : (%,List(%)) -> Boolean reduced? : (%,%) -> Boolean --R reductum : (%,V) -> % reductum : % -> % --R retract : % -> V retract : % -> R --R sample : () -> % supRittWu? : (%,%) -> Boolean @@ -65393,31 +65711,24 @@ digraph pic { --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET --R ?>=? : (%,%) -> Boolean if R has ORDSET --R D : (%,List(V),List(NonNegativeInteger)) -> % ---R D : (%,V,NonNegativeInteger) -> % --R LazardQuotient : (%,%,NonNegativeInteger) -> % if R has INTDOM --R LazardQuotient2 : (%,%,%,NonNegativeInteger) -> % if R has INTDOM --R RittWuCompare : (%,%) -> Union(Boolean,"failed") ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) or R has CHARNZ --R coefficient : (%,List(V),List(NonNegativeInteger)) -> % --R coefficient : (%,V,NonNegativeInteger) -> % ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT)) --R coerce : % -> Polynomial(R) if V has KONVERT(SYMBOL) --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) --R content : (%,V) -> % if R has GCDDOM ---R content : % -> R if R has GCDDOM --R convert : % -> Polynomial(R) if V has KONVERT(SYMBOL) --R convert : % -> String if R has RETRACT(INT) and V has KONVERT(SYMBOL) --R convert : Polynomial(R) -> % if V has KONVERT(SYMBOL) @@ -65431,8 +65742,6 @@ digraph pic { --R differentiate : (%,List(V),List(NonNegativeInteger)) -> % --R differentiate : (%,V,NonNegativeInteger) -> % --R discriminant : (%,V) -> % if R has COMRING ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R exactQuotient : (%,%) -> % if R has INTDOM --R exactQuotient : (%,R) -> % if R has INTDOM --R exactQuotient! : (%,%) -> % if R has INTDOM @@ -65443,9 +65752,6 @@ digraph pic { --R factor : % -> Factored(%) if R has PFECAT --R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT --R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT ---R gcd : (%,%) -> % if R has GCDDOM ---R gcd : List(%) -> % if R has GCDDOM ---R gcd : (R,%) -> R if R has GCDDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM --R halfExtendedSubResultantGcd1 : (%,%) -> Record(gcd: %,coef1: %) if R has INTDOM --R halfExtendedSubResultantGcd2 : (%,%) -> Record(gcd: %,coef2: %) if R has INTDOM @@ -65461,15 +65767,11 @@ digraph pic { --R lazyPseudoDivide : (%,%,V) -> Record(coef: %,gap: NonNegativeInteger,quotient: %,remainder: %) --R lazyPseudoDivide : (%,%) -> Record(coef: %,gap: NonNegativeInteger,quotient: %,remainder: %) --R lazyResidueClass : (%,%) -> Record(polnum: %,polden: %,power: NonNegativeInteger) ---R lcm : (%,%) -> % if R has GCDDOM ---R lcm : List(%) -> % if R has GCDDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM --R mainContent : % -> % if R has GCDDOM --R mainPrimitivePart : % -> % if R has GCDDOM --R mainSquareFreePart : % -> % if R has GCDDOM --R mainVariable : % -> Union(V,"failed") ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimumDegree : (%,List(V)) -> List(NonNegativeInteger) --R minimumDegree : (%,V) -> NonNegativeInteger --R monicDivide : (%,%,V) -> Record(quotient: %,remainder: %) @@ -65485,12 +65787,10 @@ digraph pic { --R primPartElseUnitCanonical : % -> % if R has INTDOM --R primPartElseUnitCanonical! : % -> % if R has INTDOM --R prime? : % -> Boolean if R has PFECAT ---R primitiveMonomials : % -> List(%) --R primitivePart : (%,V) -> % if R has GCDDOM --R primitivePart : % -> % if R has GCDDOM --R primitivePart! : % -> % if R has GCDDOM --R pseudoDivide : (%,%) -> Record(quotient: %,remainder: %) ---R reduced? : (%,List(%)) -> Boolean --R reducedSystem : Matrix(%) -> Matrix(R) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) @@ -67333,31 +67633,31 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef ---R coefficient : (%,Integer) -> Coef coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coefficient : (%,Integer) -> Coef +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % --R degree : % -> Integer ?.? : (%,Integer) -> Coef --R extend : (%,Integer) -> % hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> Coef ---R leadingMonomial : % -> % map : ((Coef -> Coef),%) -> % ---R monomial : (Coef,Integer) -> % monomial? : % -> Boolean ---R one? : % -> Boolean order : (%,Integer) -> Integer ---R order : % -> Integer pole? : % -> Boolean ---R recip : % -> Union(%,"failed") reductum : % -> % ---R sample : () -> % truncate : (%,Integer) -> % ---R variable : % -> Symbol zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean +--R inv : % -> % if Coef has FIELD latex : % -> String +--R leadingCoefficient : % -> Coef leadingMonomial : % -> % +--R map : ((Coef -> Coef),%) -> % monomial : (Coef,Integer) -> % +--R monomial? : % -> Boolean one? : % -> Boolean +--R order : (%,Integer) -> Integer order : % -> Integer +--R pole? : % -> Boolean recip : % -> Union(%,"failed") +--R reductum : % -> % sample : () -> % +--R truncate : (%,Integer) -> % variable : % -> Symbol +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Coef has FIELD --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Integer,Coef) -> Coef @@ -67367,7 +67667,6 @@ digraph pic { --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Integer,Coef) -> Coef --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Integer,Coef) -> Coef --R ?^? : (%,Integer) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -67384,7 +67683,6 @@ digraph pic { --R atanh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R coerce : Coef -> % if Coef has COMRING --R cos : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -67414,7 +67712,6 @@ digraph pic { --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if Coef has FIELD --R integrate : (%,Symbol) -> % if Coef has ACFS(INT) and Coef has PRIMCAT and Coef has TRANFUN and Coef has ALGEBRA(FRAC(INT)) or Coef has variables: Coef -> List(Symbol) and Coef has integrate: (Coef,Symbol) -> Coef and Coef has ALGEBRA(FRAC(INT)) --R integrate : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R inv : % -> % if Coef has FIELD --R lcm : (%,%) -> % if Coef has FIELD --R lcm : List(%) -> % if Coef has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if Coef has FIELD @@ -67988,14 +68285,17 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coerce : % -> % if Coef has INTDOM --R coerce : Integer -> % coerce : % -> OutputForm --R complete : % -> % degree : % -> Fraction(Integer) ---R hash : % -> SingleInteger latex : % -> String +--R ?.? : (%,Fraction(Integer)) -> Coef hash : % -> SingleInteger +--R inv : % -> % if Coef has FIELD latex : % -> String --R leadingCoefficient : % -> Coef leadingMonomial : % -> % --R map : ((Coef -> Coef),%) -> % monomial? : % -> Boolean --R one? : % -> Boolean order : % -> Fraction(Integer) @@ -68005,11 +68305,9 @@ digraph pic { --R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Coef has FIELD --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Fraction(Integer),Coef) -> Coef @@ -68019,7 +68317,6 @@ digraph pic { --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R ?^? : (%,Integer) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -68037,7 +68334,6 @@ digraph pic { --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coefficient : (%,Fraction(Integer)) -> Coef ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R coerce : Coef -> % if Coef has COMRING --R cos : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -68054,7 +68350,6 @@ digraph pic { --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R divide : (%,%) -> Record(quotient: %,remainder: %) if Coef has FIELD --R ?.? : (%,%) -> % if Fraction(Integer) has SGROUP ---R ?.? : (%,Fraction(Integer)) -> Coef --R euclideanSize : % -> NonNegativeInteger if Coef has FIELD --R eval : (%,Coef) -> Stream(Coef) if Coef has **: (Coef,Fraction(Integer)) -> Coef --R exp : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -68069,7 +68364,6 @@ digraph pic { --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if Coef has FIELD --R integrate : (%,Symbol) -> % if Coef has ACFS(INT) and Coef has PRIMCAT and Coef has TRANFUN and Coef has ALGEBRA(FRAC(INT)) or Coef has variables: Coef -> List(Symbol) and Coef has integrate: (Coef,Symbol) -> Coef and Coef has ALGEBRA(FRAC(INT)) --R integrate : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R inv : % -> % if Coef has FIELD --R lcm : (%,%) -> % if Coef has FIELD --R lcm : List(%) -> % if Coef has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if Coef has FIELD @@ -68602,32 +68896,39 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,(R -> R)) -> % D : % -> % ---R D : (%,NonNegativeInteger) -> % 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coefficients : % -> List(R) coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R degree : % -> NonNegativeInteger differentiate : % -> % ---R ?.? : (%,%) -> % ?.? : (%,R) -> R +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,(R -> R)) -> % +--R D : % -> % D : (%,NonNegativeInteger) -> % +--R D : (%,SingletonAsOrderedSet) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficients : % -> List(R) +--R coerce : % -> % if R has INTDOM coerce : SingletonAsOrderedSet -> % +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM +--R degree : % -> NonNegativeInteger differentiate : (%,(R -> R)) -> % +--R differentiate : % -> % ?.? : (%,%) -> % +--R ?.? : (%,R) -> R eval : (%,List(%),List(%)) -> % --R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % ---R ground : % -> R ground? : % -> Boolean ---R hash : % -> SingleInteger init : () -> % if R has STEP ---R latex : % -> String leadingCoefficient : % -> R ---R leadingMonomial : % -> % map : ((R -> R),%) -> % ---R monomial? : % -> Boolean monomials : % -> List(%) ---R one? : % -> Boolean pseudoRemainder : (%,%) -> % ---R recip : % -> Union(%,"failed") reductum : % -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R +--R ground? : % -> Boolean hash : % -> SingleInteger +--R init : () -> % if R has STEP latex : % -> String +--R lcm : (%,%) -> % if R has GCDDOM lcm : List(%) -> % if R has GCDDOM +--R leadingCoefficient : % -> R leadingMonomial : % -> % +--R map : ((R -> R),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET monomial? : % -> Boolean +--R monomials : % -> List(%) one? : % -> Boolean +--R primitiveMonomials : % -> List(%) pseudoRemainder : (%,%) -> % +--R ?quo? : (%,%) -> % if R has FIELD recip : % -> Union(%,"failed") +--R reductum : % -> % ?rem? : (%,%) -> % if R has FIELD --R retract : % -> R sample : () -> % --R unvectorise : Vector(R) -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET @@ -68640,8 +68941,6 @@ digraph pic { --R D : (%,List(SingletonAsOrderedSet),List(NonNegativeInteger)) -> % --R D : (%,SingletonAsOrderedSet,NonNegativeInteger) -> % --R D : (%,List(SingletonAsOrderedSet)) -> % ---R D : (%,SingletonAsOrderedSet) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger @@ -68649,21 +68948,17 @@ digraph pic { --R coefficient : (%,List(SingletonAsOrderedSet),List(NonNegativeInteger)) -> % --R coefficient : (%,SingletonAsOrderedSet,NonNegativeInteger) -> % --R coefficient : (%,NonNegativeInteger) -> R ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) or R has ALGEBRA(FRAC(INT)) ---R coerce : SingletonAsOrderedSet -> % --R composite : (Fraction(%),%) -> Union(Fraction(%),"failed") if R has INTDOM --R composite : (%,%) -> Union(%,"failed") if R has INTDOM --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if and(has($,CharacteristicNonZero),has(R,PolynomialFactorizationExplicit)) --R content : (%,SingletonAsOrderedSet) -> % if R has GCDDOM ---R content : % -> R if R has GCDDOM --R convert : % -> InputForm if SingletonAsOrderedSet has KONVERT(INFORM) and R has KONVERT(INFORM) --R convert : % -> Pattern(Integer) if SingletonAsOrderedSet has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) --R convert : % -> Pattern(Float) if SingletonAsOrderedSet has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) --R degree : (%,List(SingletonAsOrderedSet)) -> List(NonNegativeInteger) --R degree : (%,SingletonAsOrderedSet) -> NonNegativeInteger --R differentiate : (%,(R -> R),%) -> % ---R differentiate : (%,(R -> R)) -> % --R differentiate : (%,(R -> R),NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) @@ -68686,8 +68981,6 @@ digraph pic { --R eval : (%,SingletonAsOrderedSet,%) -> % --R eval : (%,List(SingletonAsOrderedSet),List(R)) -> % --R eval : (%,SingletonAsOrderedSet,R) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if R has FIELD --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM @@ -68696,22 +68989,16 @@ digraph pic { --R factor : % -> Factored(%) if R has PFECAT --R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT --R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT ---R gcd : (%,%) -> % if R has GCDDOM ---R gcd : List(%) -> % if R has GCDDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM --R integrate : % -> % if R has ALGEBRA(FRAC(INT)) --R isExpt : % -> Union(Record(var: SingletonAsOrderedSet,exponent: NonNegativeInteger),"failed") --R isPlus : % -> Union(List(%),"failed") --R isTimes : % -> Union(List(%),"failed") --R karatsubaDivide : (%,NonNegativeInteger) -> Record(quotient: %,remainder: %) ---R lcm : (%,%) -> % if R has GCDDOM ---R lcm : List(%) -> % if R has GCDDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM --R mainVariable : % -> Union(SingletonAsOrderedSet,"failed") --R makeSUP : % -> SparseUnivariatePolynomial(R) --R mapExponents : ((NonNegativeInteger -> NonNegativeInteger),%) -> % ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimumDegree : (%,List(SingletonAsOrderedSet)) -> List(NonNegativeInteger) --R minimumDegree : (%,SingletonAsOrderedSet) -> NonNegativeInteger --R minimumDegree : % -> NonNegativeInteger @@ -68731,18 +69018,15 @@ digraph pic { --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if SingletonAsOrderedSet has PATMAB(FLOAT) and R has PATMAB(FLOAT) --R pomopo! : (%,R,NonNegativeInteger,%) -> % --R prime? : % -> Boolean if R has PFECAT ---R primitiveMonomials : % -> List(%) --R primitivePart : (%,SingletonAsOrderedSet) -> % if R has GCDDOM --R primitivePart : % -> % if R has GCDDOM --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if R has FIELD --R pseudoDivide : (%,%) -> Record(coef: R,quotient: %,remainder: %) if R has INTDOM --R pseudoQuotient : (%,%) -> % if R has INTDOM ---R ?quo? : (%,%) -> % if R has FIELD --R reducedSystem : Matrix(%) -> Matrix(R) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) --R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) ---R ?rem? : (%,%) -> % if R has FIELD --R resultant : (%,%) -> R if R has COMRING --R resultant : (%,%,SingletonAsOrderedSet) -> % if R has COMRING --R retract : % -> SingletonAsOrderedSet @@ -69823,43 +70107,52 @@ digraph pic { --R Issue )edit bookvol10.2.pamphlet to see algebra source code for ACFS --R --R------------------------------- Operations -------------------------------- +--R ?*? : (R,%) -> % if R has COMRING ?*? : (%,R) -> % if R has COMRING --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean D : (%,Symbol) -> % if R has RING --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R applyQuote : (Symbol,%) -> % applyQuote : (Symbol,%,%) -> % ---R associates? : (%,%) -> Boolean belong? : BasicOperator -> Boolean ---R box : % -> % box : List(%) -> % ---R coerce : Kernel(%) -> % coerce : Symbol -> % ---R coerce : R -> % coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm distribute : % -> % ---R distribute : (%,%) -> % elt : (BasicOperator,%) -> % ---R elt : (BasicOperator,%,%) -> % eval : (%,Kernel(%),%) -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % applyQuote : (Symbol,%) -> % +--R applyQuote : (Symbol,%,%) -> % applyQuote : (Symbol,%,%,%) -> % +--R applyQuote : (Symbol,List(%)) -> % associates? : (%,%) -> Boolean +--R belong? : BasicOperator -> Boolean box : % -> % +--R box : List(%) -> % coerce : Kernel(%) -> % +--R coerce : Symbol -> % coerce : R -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R distribute : % -> % distribute : (%,%) -> % +--R elt : (BasicOperator,%) -> % elt : (BasicOperator,%,%) -> % +--R elt : (BasicOperator,%,%,%) -> % elt : (BasicOperator,List(%)) -> % +--R eval : (%,Kernel(%),%) -> % eval : (%,List(Equation(%))) -> % --R eval : (%,Equation(%)) -> % eval : (%,%,%) -> % +--R eval : (%,List(%),List(%)) -> % eval : (%,Symbol,(% -> %)) -> % --R factor : % -> Factored(%) freeOf? : (%,%) -> Boolean --R freeOf? : (%,Symbol) -> Boolean gcd : List(%) -> % --R gcd : (%,%) -> % ground : % -> R --R ground? : % -> Boolean hash : % -> SingleInteger --R height : % -> NonNegativeInteger inv : % -> % ---R is? : (%,Symbol) -> Boolean kernel : (BasicOperator,%) -> % ---R kernels : % -> List(Kernel(%)) latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % +--R is? : (%,BasicOperator) -> Boolean is? : (%,Symbol) -> Boolean +--R kernel : (BasicOperator,%) -> % kernels : % -> List(Kernel(%)) +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % map : ((% -> %),Kernel(%)) -> % --R max : (%,%) -> % min : (%,%) -> % ---R nthRoot : (%,Integer) -> % one? : % -> Boolean ---R paren : % -> % paren : List(%) -> % ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R retract : % -> Kernel(%) retract : % -> Symbol ---R retract : % -> R rootOf : (%,Symbol) -> % ---R rootOf : % -> % rootOf : Polynomial(%) -> % ---R rootsOf : (%,Symbol) -> List(%) rootsOf : % -> List(%) +--R nthRoot : (%,Integer) -> % numerator : % -> % if R has RING +--R one? : % -> Boolean paren : % -> % +--R paren : List(%) -> % prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % retract : % -> Kernel(%) +--R retract : % -> Symbol retract : % -> R +--R rootOf : (%,Symbol) -> % rootOf : % -> % +--R rootOf : Polynomial(%) -> % rootsOf : (%,Symbol) -> List(%) +--R rootsOf : % -> List(%) rootsOf : Polynomial(%) -> List(%) --R sample : () -> % sizeLess? : (%,%) -> Boolean --R sqrt : % -> % squareFree : % -> Factored(%) --R squareFreePart : % -> % subst : (%,Equation(%)) -> % @@ -69868,21 +70161,12 @@ digraph pic { --R zero? : % -> Boolean zeroOf : (%,Symbol) -> % --R zeroOf : % -> % zeroOf : Polynomial(%) -> % --R zerosOf : (%,Symbol) -> List(%) zerosOf : % -> List(%) ---R ?~=? : (%,%) -> Boolean ---R ?*? : (R,%) -> % if R has COMRING ---R ?*? : (%,R) -> % if R has COMRING ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R zerosOf : Polynomial(%) -> List(%) ?~=? : (%,%) -> Boolean --R ?/? : (SparseMultivariatePolynomial(R,Kernel(%)),SparseMultivariatePolynomial(R,Kernel(%))) -> % if R has INTDOM --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has RING --R D : (%,Symbol,NonNegativeInteger) -> % if R has RING --R D : (%,List(Symbol)) -> % if R has RING ---R D : (%,Symbol) -> % if R has RING ---R ?^? : (%,NonNegativeInteger) -> % ---R applyQuote : (Symbol,%,%,%) -> % --R applyQuote : (Symbol,%,%,%,%) -> % ---R applyQuote : (Symbol,List(%)) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coerce : SparseMultivariatePolynomial(R,Kernel(%)) -> % if R has RING @@ -69905,17 +70189,12 @@ digraph pic { --R differentiate : (%,List(Symbol)) -> % if R has RING --R differentiate : (%,Symbol) -> % if R has RING --R divide : (%,%) -> Record(quotient: %,remainder: %) ---R elt : (BasicOperator,%,%,%) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,List(%)) -> % --R euclideanSize : % -> NonNegativeInteger --R eval : (%,List(Kernel(%)),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % ---R eval : (%,List(%),List(%)) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,BasicOperator,(List(%) -> %)) -> % @@ -69935,7 +70214,6 @@ digraph pic { --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") --R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) ---R is? : (%,BasicOperator) -> Boolean --R isExpt : % -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has SGROUP --R isExpt : (%,BasicOperator) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING --R isExpt : (%,Symbol) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING @@ -69946,11 +70224,9 @@ digraph pic { --R kernel : (BasicOperator,List(%)) -> % --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") --R numer : % -> SparseMultivariatePolynomial(R,Kernel(%)) if R has RING ---R numerator : % -> % if R has RING --R odd? : % -> Boolean if $ has RETRACT(INT) --R operator : BasicOperator -> BasicOperator --R operators : % -> List(BasicOperator) @@ -69976,7 +70252,6 @@ digraph pic { --R rootOf : SparseUnivariatePolynomial(%) -> % --R rootsOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R rootsOf : SparseUnivariatePolynomial(%) -> List(%) ---R rootsOf : Polynomial(%) -> List(%) --R subst : (%,List(Equation(%))) -> % --R subst : (%,List(Kernel(%)),List(%)) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -69986,7 +70261,6 @@ digraph pic { --R zeroOf : SparseUnivariatePolynomial(%) -> % --R zerosOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R zerosOf : SparseUnivariatePolynomial(%) -> List(%) ---R zerosOf : Polynomial(%) -> List(%) --R --E 1 @@ -70601,12 +70875,14 @@ digraph pic { --R ?*? : (F,%) -> % ?*? : (%,F) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,F) -> % ?/? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,Integer) -> % +--R ?=? : (%,%) -> Boolean Frobenius : % -> % if F has FINITE +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean --R associates? : (%,%) -> Boolean coerce : F -> % --R coerce : Fraction(Integer) -> % coerce : % -> % @@ -70624,11 +70900,7 @@ digraph pic { --R transcendent? : % -> Boolean unit? : % -> Boolean --R unitCanonical : % -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R Frobenius : (%,NonNegativeInteger) -> % if F has FINITE ---R Frobenius : % -> % if F has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if F has CHARNZ or F has FINITE --R degree : % -> OnePointCompletion(PositiveInteger) @@ -71009,23 +71281,25 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R D : % -> % D : (%,NonNegativeInteger) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean charthRoot : % -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R createPrimitiveElement : () -> % differentiate : % -> % ---R enumerate : () -> List(%) factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger index : PositiveInteger -> % ---R init : () -> % inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % lookup : % -> PositiveInteger +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R charthRoot : % -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm createPrimitiveElement : () -> % +--R differentiate : % -> % enumerate : () -> List(%) +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R index : PositiveInteger -> % init : () -> % +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R lookup : % -> PositiveInteger nextItem : % -> Union(%,"failed") --R one? : % -> Boolean order : % -> PositiveInteger --R prime? : % -> Boolean primeFrobenius : % -> % --R primitive? : % -> Boolean primitiveElement : () -> % @@ -71036,9 +71310,6 @@ digraph pic { --R squareFreePart : % -> % unit? : % -> Boolean --R unitCanonical : % -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") --R conditionP : Matrix(%) -> Union(Vector(%),"failed") @@ -71055,7 +71326,6 @@ digraph pic { --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") ---R nextItem : % -> Union(%,"failed") --R order : % -> OnePointCompletion(PositiveInteger) --R primeFrobenius : (%,NonNegativeInteger) -> % --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) @@ -71595,14 +71865,16 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Fraction(Integer)) -> % ?**? : (%,Integer) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R abs : % -> % associates? : (%,%) -> Boolean --R base : () -> PositiveInteger bits : () -> PositiveInteger --R ceiling : % -> % coerce : Fraction(Integer) -> % @@ -71631,10 +71903,6 @@ digraph pic { --R truncate : % -> % unit? : % -> Boolean --R unitCanonical : % -> % wholePart : % -> Integer --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R bits : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision --R characteristic : () -> NonNegativeInteger --R decreasePrecision : Integer -> PositiveInteger if $ has arbitraryPrecision @@ -72086,11 +72354,13 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % basis : () -> Vector(%) +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R basis : () -> Vector(%) characteristicPolynomial : % -> UP --R coerce : R -> % coerce : Integer -> % --R coerce : % -> OutputForm convert : Vector(R) -> % --R convert : % -> Vector(R) coordinates : % -> Vector(R) @@ -72101,11 +72371,7 @@ digraph pic { --R represents : Vector(R) -> % sample : () -> % --R trace : % -> R traceMatrix : () -> Matrix(R) --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger ---R characteristicPolynomial : % -> UP --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R coordinates : Vector(%) -> Matrix(R) --R coordinates : (Vector(%),Vector(%)) -> Matrix(R) @@ -72411,26 +72677,28 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R D : % -> % D : (%,NonNegativeInteger) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean charthRoot : % -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R conjugate : % -> % createPrimitiveElement : () -> % ---R differentiate : % -> % enumerate : () -> List(%) ---R extDegree : % -> PositiveInteger factor : % -> Factored(%) ---R fullOutput : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % ground? : % -> Boolean ---R hash : % -> SingleInteger index : PositiveInteger -> % ---R init : () -> % inv : % -> % ---R latex : % -> String lcm : List(%) -> % ---R lcm : (%,%) -> % lookup : % -> PositiveInteger ---R maxTower : List(%) -> % one? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R charthRoot : % -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm conjugate : % -> % +--R createPrimitiveElement : () -> % differentiate : % -> % +--R enumerate : () -> List(%) extDegree : % -> PositiveInteger +--R factor : % -> Factored(%) fullOutput : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R ground? : % -> Boolean hash : % -> SingleInteger +--R index : PositiveInteger -> % init : () -> % +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R lookup : % -> PositiveInteger maxTower : List(%) -> % +--R nextItem : % -> Union(%,"failed") one? : % -> Boolean --R order : % -> PositiveInteger previousTower : % -> % --R prime? : % -> Boolean primeFrobenius : % -> % --R primitive? : % -> Boolean primitiveElement : () -> % @@ -72442,9 +72710,6 @@ digraph pic { --R unit? : % -> Boolean unitCanonical : % -> % --R vectorise : (%,%) -> Vector(%) zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") --R conditionP : Matrix(%) -> Union(Vector(%),"failed") @@ -72468,7 +72733,6 @@ digraph pic { --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") --R newElement : (SparseUnivariatePolynomial(%),%,Symbol) -> % --R newElement : (SparseUnivariatePolynomial(%),Symbol) -> % ---R nextItem : % -> Union(%,"failed") --R order : % -> OnePointCompletion(PositiveInteger) --R primeFrobenius : (%,NonNegativeInteger) -> % --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) @@ -72806,36 +73070,38 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef ---R coefficient : (%,Integer) -> Coef coerce : UTS -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coefficient : (%,Integer) -> Coef +--R coerce : % -> % if Coef has INTDOM coerce : UTS -> % --R coerce : Integer -> % coerce : % -> OutputForm --R complete : % -> % degree : % -> Integer ---R ?.? : (%,Integer) -> Coef extend : (%,Integer) -> % ---R hash : % -> SingleInteger latex : % -> String +--R denom : % -> UTS if Coef has FIELD ?.? : (%,Integer) -> Coef +--R extend : (%,Integer) -> % hash : % -> SingleInteger +--R inv : % -> % if Coef has FIELD latex : % -> String --R laurent : (Integer,UTS) -> % leadingCoefficient : % -> Coef --R leadingMonomial : % -> % map : ((Coef -> Coef),%) -> % --R monomial : (Coef,Integer) -> % monomial? : % -> Boolean ---R one? : % -> Boolean order : (%,Integer) -> Integer ---R order : % -> Integer pole? : % -> Boolean ---R recip : % -> Union(%,"failed") reductum : % -> % ---R removeZeroes : (Integer,%) -> % removeZeroes : % -> % ---R retract : % -> UTS sample : () -> % ---R taylor : % -> UTS taylorRep : % -> UTS ---R truncate : (%,Integer) -> % variable : % -> Symbol ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R numer : % -> UTS if Coef has FIELD one? : % -> Boolean +--R order : (%,Integer) -> Integer order : % -> Integer +--R pole? : % -> Boolean recip : % -> Union(%,"failed") +--R reductum : % -> % removeZeroes : (Integer,%) -> % +--R removeZeroes : % -> % retract : % -> UTS +--R sample : () -> % taylor : % -> UTS +--R taylorRep : % -> UTS truncate : (%,Integer) -> % +--R variable : % -> Symbol zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (UTS,%) -> % if Coef has FIELD --R ?*? : (%,UTS) -> % if Coef has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (UTS,UTS) -> % if Coef has FIELD --R ?/? : (%,%) -> % if Coef has FIELD --R ?/? : (%,Coef) -> % if Coef has FIELD @@ -72852,7 +73118,6 @@ digraph pic { --R D : (%,(UTS -> UTS),NonNegativeInteger) -> % if Coef has FIELD --R D : (%,(UTS -> UTS)) -> % if Coef has FIELD --R ?^? : (%,Integer) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R abs : % -> % if and(has(UTS,OrderedIntegralDomain),has(Coef,Field)) --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -72871,7 +73136,6 @@ digraph pic { --R ceiling : % -> UTS if and(has(UTS,IntegerNumberSystem),has(Coef,Field)) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if and(OR(has(UTS,CharacteristicNonZero),and(has($,CharacteristicNonZero),has(UTS,PolynomialFactorizationExplicit))),has(Coef,Field)) or Coef has CHARNZ ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R coerce : Symbol -> % if and(has(UTS,RetractableTo(Symbol)),has(Coef,Field)) --R coerce : Coef -> % if Coef has COMRING @@ -72887,7 +73151,6 @@ digraph pic { --R coth : % -> % if Coef has ALGEBRA(FRAC(INT)) --R csc : % -> % if Coef has ALGEBRA(FRAC(INT)) --R csch : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R denom : % -> UTS if Coef has FIELD --R denominator : % -> % if Coef has FIELD --R differentiate : (%,Symbol) -> % if and(has(UTS,PartialDifferentialRing(Symbol)),has(Coef,Field)) or Coef has PDRING(SYMBOL) and Coef has *: (Integer,Coef) -> Coef --R differentiate : (%,List(Symbol)) -> % if and(has(UTS,PartialDifferentialRing(Symbol)),has(Coef,Field)) or Coef has PDRING(SYMBOL) and Coef has *: (Integer,Coef) -> Coef @@ -72924,7 +73187,6 @@ digraph pic { --R init : () -> % if and(has(UTS,StepThrough),has(Coef,Field)) --R integrate : (%,Symbol) -> % if Coef has ACFS(INT) and Coef has PRIMCAT and Coef has TRANFUN and Coef has ALGEBRA(FRAC(INT)) or Coef has variables: Coef -> List(Symbol) and Coef has integrate: (Coef,Symbol) -> Coef and Coef has ALGEBRA(FRAC(INT)) --R integrate : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R inv : % -> % if Coef has FIELD --R lcm : (%,%) -> % if Coef has FIELD --R lcm : List(%) -> % if Coef has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if Coef has FIELD @@ -72940,7 +73202,6 @@ digraph pic { --R negative? : % -> Boolean if and(has(UTS,OrderedIntegralDomain),has(Coef,Field)) --R nextItem : % -> Union(%,"failed") if and(has(UTS,StepThrough),has(Coef,Field)) --R nthRoot : (%,Integer) -> % if Coef has ALGEBRA(FRAC(INT)) ---R numer : % -> UTS if Coef has FIELD --R numerator : % -> % if Coef has FIELD --R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if and(has(UTS,PatternMatchable(Integer)),has(Coef,Field)) --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if and(has(UTS,PatternMatchable(Float)),has(Coef,Field)) @@ -73673,14 +73934,17 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coerce : % -> % if Coef has INTDOM --R coerce : ULS -> % coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % ---R degree : % -> Fraction(Integer) hash : % -> SingleInteger +--R degree : % -> Fraction(Integer) ?.? : (%,Fraction(Integer)) -> Coef +--R hash : % -> SingleInteger inv : % -> % if Coef has FIELD --R latex : % -> String laurent : % -> ULS --R laurentRep : % -> ULS leadingCoefficient : % -> Coef --R leadingMonomial : % -> % map : ((Coef -> Coef),%) -> % @@ -73692,11 +73956,9 @@ digraph pic { --R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Coef has FIELD --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Fraction(Integer),Coef) -> Coef @@ -73706,7 +73968,6 @@ digraph pic { --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R ?^? : (%,Integer) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -73724,7 +73985,6 @@ digraph pic { --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coefficient : (%,Fraction(Integer)) -> Coef ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R coerce : Coef -> % if Coef has COMRING --R cos : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -73741,7 +74001,6 @@ digraph pic { --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has PDRING(SYMBOL) and Coef has *: (Fraction(Integer),Coef) -> Coef --R divide : (%,%) -> Record(quotient: %,remainder: %) if Coef has FIELD --R ?.? : (%,%) -> % if Fraction(Integer) has SGROUP ---R ?.? : (%,Fraction(Integer)) -> Coef --R euclideanSize : % -> NonNegativeInteger if Coef has FIELD --R eval : (%,Coef) -> Stream(Coef) if Coef has **: (Coef,Fraction(Integer)) -> Coef --R exp : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -73756,7 +74015,6 @@ digraph pic { --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if Coef has FIELD --R integrate : (%,Symbol) -> % if Coef has ACFS(INT) and Coef has PRIMCAT and Coef has TRANFUN and Coef has ALGEBRA(FRAC(INT)) or Coef has variables: Coef -> List(Symbol) and Coef has integrate: (Coef,Symbol) -> Coef and Coef has ALGEBRA(FRAC(INT)) --R integrate : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R inv : % -> % if Coef has FIELD --R laurentIfCan : % -> Union(ULS,"failed") --R lcm : (%,%) -> % if Coef has FIELD --R lcm : List(%) -> % if Coef has FIELD @@ -74295,13 +74553,15 @@ digraph pic { --R ?*? : (F,%) -> % ?*? : (%,F) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,F) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if F has FINITE ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R Frobenius : % -> % if F has FINITE 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R algebraic? : % -> Boolean associates? : (%,%) -> Boolean --R basis : () -> Vector(%) coerce : F -> % --R coerce : Fraction(Integer) -> % coerce : % -> % @@ -74309,24 +74569,22 @@ digraph pic { --R coordinates : % -> Vector(F) degree : % -> PositiveInteger --R dimension : () -> CardinalNumber factor : % -> Factored(%) --R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R generator : () -> % if F has FINITE hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if F has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> F one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(F) -> % retract : % -> F ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> F transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if F has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(F) -> % +--R retract : % -> F sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> F +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if F has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if F has FINITE ---R Frobenius : % -> % if F has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if F has CHARNZ or F has FINITE @@ -74352,9 +74610,7 @@ digraph pic { --R extensionDegree : () -> OnePointCompletion(PositiveInteger) --R factorsOfCyclicGroupSize : () -> List(Record(factor: Integer,exponent: Integer)) if F has FINITE --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) ---R generator : () -> % if F has FINITE --R index : PositiveInteger -> % if F has FINITE ---R init : () -> % if F has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(F)) -> % if F has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(F),"failed") if F has FINITE @@ -74375,7 +74631,6 @@ digraph pic { --R primitive? : % -> Boolean if F has FINITE --R primitiveElement : () -> % if F has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if F has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if F has FINITE --R retractIfCan : % -> Union(F,"failed") --R size : () -> NonNegativeInteger if F has FINITE @@ -75110,31 +75365,36 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % +--R -? : % -> % ?/? : (%,%) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % basis : () -> Vector(%) +--R characteristicPolynomial : % -> UP coerce : % -> % if R has FIELD --R coerce : R -> % coerce : Integer -> % --R coerce : % -> OutputForm convert : UP -> % --R convert : % -> UP convert : Vector(R) -> % --R convert : % -> Vector(R) coordinates : % -> Vector(R) --R definingPolynomial : () -> UP discriminant : () -> R ---R discriminant : Vector(%) -> R generator : () -> % ---R hash : % -> SingleInteger inv : % -> % if R has FIELD ---R latex : % -> String lift : % -> UP ---R norm : % -> R one? : % -> Boolean ---R rank : () -> PositiveInteger recip : % -> Union(%,"failed") ---R reduce : UP -> % represents : Vector(R) -> % +--R discriminant : Vector(%) -> R gcd : (%,%) -> % if R has FIELD +--R gcd : List(%) -> % if R has FIELD generator : () -> % +--R hash : % -> SingleInteger init : () -> % if R has FFIELDC +--R inv : % -> % if R has FIELD latex : % -> String +--R lcm : (%,%) -> % if R has FIELD lcm : List(%) -> % if R has FIELD +--R lift : % -> UP norm : % -> R +--R one? : % -> Boolean ?quo? : (%,%) -> % if R has FIELD +--R random : () -> % if R has FINITE rank : () -> PositiveInteger +--R recip : % -> Union(%,"failed") reduce : UP -> % +--R ?rem? : (%,%) -> % if R has FIELD represents : Vector(R) -> % --R retract : % -> R sample : () -> % --R trace : % -> R traceMatrix : () -> Matrix(R) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R unit? : % -> Boolean if R has FIELD zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if R has FIELD --R ?*? : (Fraction(Integer),%) -> % if R has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if R has FIELD ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,%) -> % if R has FIELD --R D : (%,(R -> R)) -> % if R has FIELD --R D : (%,(R -> R),NonNegativeInteger) -> % if R has FIELD --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if and(has(R,PartialDifferentialRing(Symbol)),has(R,Field)) @@ -75144,14 +75404,11 @@ digraph pic { --R D : (%,NonNegativeInteger) -> % if and(has(R,DifferentialRing),has(R,Field)) or R has FFIELDC --R D : % -> % if and(has(R,DifferentialRing),has(R,Field)) or R has FFIELDC --R ?^? : (%,Integer) -> % if R has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has FIELD --R characteristic : () -> NonNegativeInteger ---R characteristicPolynomial : % -> UP --R charthRoot : % -> Union(%,"failed") if R has CHARNZ --R charthRoot : % -> % if R has FFIELDC --R coerce : Fraction(Integer) -> % if R has FIELD or R has RETRACT(FRAC(INT)) ---R coerce : % -> % if R has FIELD --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if R has FFIELDC --R coordinates : Vector(%) -> Matrix(R) --R coordinates : (Vector(%),Vector(%)) -> Matrix(R) @@ -75177,13 +75434,8 @@ digraph pic { --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") if R has FIELD --R factor : % -> Factored(%) if R has FIELD --R factorsOfCyclicGroupSize : () -> List(Record(factor: Integer,exponent: Integer)) if R has FFIELDC ---R gcd : (%,%) -> % if R has FIELD ---R gcd : List(%) -> % if R has FIELD --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has FIELD --R index : PositiveInteger -> % if R has FINITE ---R init : () -> % if R has FFIELDC ---R lcm : (%,%) -> % if R has FIELD ---R lcm : List(%) -> % if R has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has FIELD --R lookup : % -> PositiveInteger if R has FINITE --R minimalPolynomial : % -> UP if R has FIELD @@ -75197,8 +75449,6 @@ digraph pic { --R primitive? : % -> Boolean if R has FFIELDC --R primitiveElement : () -> % if R has FFIELDC --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if R has FIELD ---R ?quo? : (%,%) -> % if R has FIELD ---R random : () -> % if R has FINITE --R reduce : Fraction(UP) -> Union(%,"failed") if R has FIELD --R reducedSystem : Matrix(%) -> Matrix(R) --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) @@ -75206,7 +75456,6 @@ digraph pic { --R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) --R regularRepresentation : % -> Matrix(R) --R regularRepresentation : (%,Vector(%)) -> Matrix(R) ---R ?rem? : (%,%) -> % if R has FIELD --R representationType : () -> Union("prime",polynomial,normal,cyclic) if R has FFIELDC --R represents : (Vector(R),Vector(%)) -> % --R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) @@ -75221,7 +75470,6 @@ digraph pic { --R subtractIfCan : (%,%) -> Union(%,"failed") --R tableForDiscreteLogarithm : Integer -> Table(PositiveInteger,NonNegativeInteger) if R has FFIELDC --R traceMatrix : Vector(%) -> Matrix(R) ---R unit? : % -> Boolean if R has FIELD --R unitCanonical : % -> % if R has FIELD --R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has FIELD --R @@ -75750,40 +75998,38 @@ digraph pic { --R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,Fraction(Integer)) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : Fraction(Integer) -> % coerce : Integer -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R conjugate : % -> % dimension : () -> CardinalNumber ---R extDegree : % -> PositiveInteger factor : % -> Factored(%) ---R fullOutput : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % ground? : % -> Boolean ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R maxTower : List(%) -> % one? : % -> Boolean ---R previousTower : % -> % prime? : % -> Boolean ---R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") ---R ?rem? : (%,%) -> % retract : % -> Fraction(Integer) ---R retract : % -> Fraction(Integer) retract : % -> Integer ---R sample : () -> % setTower! : % -> Void ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R vectorise : (%,%) -> Vector(%) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R algebraic? : % -> Boolean associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : Fraction(Integer) -> % +--R coerce : Integer -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm conjugate : % -> % +--R dimension : () -> CardinalNumber extDegree : % -> PositiveInteger +--R factor : % -> Factored(%) fullOutput : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R ground? : % -> Boolean hash : % -> SingleInteger +--R inGroundField? : % -> Boolean inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % maxTower : List(%) -> % +--R one? : % -> Boolean previousTower : % -> % +--R prime? : % -> Boolean ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Fraction(Integer) +--R retract : % -> Integer sample : () -> % +--R setTower! : % -> Void sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % vectorise : (%,%) -> Vector(%) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R Frobenius : % -> % if Fraction(Integer) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if Fraction(Integer) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Fraction(Integer) has CHARNZ or Fraction(Integer) has FINITE --R definingPolynomial : % -> SparseUnivariatePolynomial(%) @@ -76150,35 +76396,50 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,(R -> R)) -> % 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R abs : % -> % if R has RNS basis : () -> Vector(%) ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm complex : (R,R) -> % ---R conjugate : % -> % convert : Vector(R) -> % ---R convert : % -> Vector(R) coordinates : % -> Vector(R) +--R -? : % -> % ?/? : (%,%) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,(R -> R)) -> % +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % if R has RNS acos : % -> % if R has TRANFUN +--R acosh : % -> % if R has TRANFUN acot : % -> % if R has TRANFUN +--R acoth : % -> % if R has TRANFUN acsc : % -> % if R has TRANFUN +--R acsch : % -> % if R has TRANFUN argument : % -> R if R has TRANFUN +--R asec : % -> % if R has TRANFUN asech : % -> % if R has TRANFUN +--R asin : % -> % if R has TRANFUN asinh : % -> % if R has TRANFUN +--R atan : % -> % if R has TRANFUN atanh : % -> % if R has TRANFUN +--R basis : () -> Vector(%) coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complex : (R,R) -> % conjugate : % -> % +--R convert : Vector(R) -> % convert : % -> Vector(R) +--R coordinates : % -> Vector(R) cos : % -> % if R has TRANFUN +--R cosh : % -> % if R has TRANFUN cot : % -> % if R has TRANFUN +--R coth : % -> % if R has TRANFUN csc : % -> % if R has TRANFUN +--R csch : % -> % if R has TRANFUN differentiate : (%,(R -> R)) -> % --R discriminant : () -> R discriminant : Vector(%) -> R ---R generator : () -> % hash : % -> SingleInteger ---R imag : % -> R imaginary : () -> % +--R exp : % -> % if R has TRANFUN generator : () -> % +--R hash : % -> SingleInteger imag : % -> R +--R imaginary : () -> % init : () -> % if R has FFIELDC --R inv : % -> % if R has FIELD latex : % -> String ---R map : ((R -> R),%) -> % norm : % -> R ---R one? : % -> Boolean pi : () -> % if R has TRANFUN +--R log : % -> % if R has TRANFUN map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET +--R norm : % -> R one? : % -> Boolean +--R pi : () -> % if R has TRANFUN random : () -> % if R has FINITE --R rank : () -> PositiveInteger real : % -> R --R recip : % -> Union(%,"failed") represents : Vector(R) -> % --R retract : % -> R sample : () -> % +--R sec : % -> % if R has TRANFUN sech : % -> % if R has TRANFUN +--R sin : % -> % if R has TRANFUN sinh : % -> % if R has TRANFUN +--R tan : % -> % if R has TRANFUN tanh : % -> % if R has TRANFUN --R trace : % -> R traceMatrix : () -> Matrix(R) --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if R has FIELD --R ?*? : (Fraction(Integer),%) -> % if R has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if R has FIELD --R ?**? : (%,%) -> % if R has TRANFUN --R ?**? : (%,Fraction(Integer)) -> % if R has RADCAT and R has TRANFUN ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,%) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET @@ -76191,21 +76452,7 @@ digraph pic { --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R D : (%,(R -> R),NonNegativeInteger) -> % --R ?^? : (%,Integer) -> % if R has FIELD ---R ?^? : (%,NonNegativeInteger) -> % ---R acos : % -> % if R has TRANFUN ---R acosh : % -> % if R has TRANFUN ---R acot : % -> % if R has TRANFUN ---R acoth : % -> % if R has TRANFUN ---R acsc : % -> % if R has TRANFUN ---R acsch : % -> % if R has TRANFUN ---R argument : % -> R if R has TRANFUN ---R asec : % -> % if R has TRANFUN ---R asech : % -> % if R has TRANFUN ---R asin : % -> % if R has TRANFUN ---R asinh : % -> % if R has TRANFUN --R associates? : (%,%) -> Boolean if R has INTDOM or R has EUCDOM and R has PFECAT ---R atan : % -> % if R has TRANFUN ---R atanh : % -> % if R has TRANFUN --R characteristic : () -> NonNegativeInteger --R characteristicPolynomial : % -> SparseUnivariatePolynomial(R) --R charthRoot : % -> Union(%,"failed") if and(has($,CharacteristicNonZero),AND(has(R,EuclideanDomain),has(R,PolynomialFactorizationExplicit))) or R has CHARNZ @@ -76223,13 +76470,7 @@ digraph pic { --R coordinates : Vector(%) -> Matrix(R) --R coordinates : (Vector(%),Vector(%)) -> Matrix(R) --R coordinates : (%,Vector(%)) -> Vector(R) ---R cos : % -> % if R has TRANFUN ---R cosh : % -> % if R has TRANFUN ---R cot : % -> % if R has TRANFUN ---R coth : % -> % if R has TRANFUN --R createPrimitiveElement : () -> % if R has FFIELDC ---R csc : % -> % if R has TRANFUN ---R csch : % -> % if R has TRANFUN --R definingPolynomial : () -> SparseUnivariatePolynomial(R) --R derivationCoordinates : (Vector(%),(R -> R)) -> Matrix(R) if R has FIELD --R differentiate : % -> % if and(has(R,Field),has(R,DifferentialRing)) or R has DIFRING or and(has(R,DifferentialRing),has(R,Field)) @@ -76238,7 +76479,6 @@ digraph pic { --R differentiate : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) ---R differentiate : (%,(R -> R)) -> % --R differentiate : (%,(R -> R),NonNegativeInteger) -> % --R discreteLog : (%,%) -> Union(NonNegativeInteger,"failed") if R has FFIELDC --R discreteLog : % -> NonNegativeInteger if R has FFIELDC @@ -76252,7 +76492,6 @@ digraph pic { --R eval : (%,Equation(R)) -> % if R has EVALAB(R) --R eval : (%,R,R) -> % if R has EVALAB(R) --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) ---R exp : % -> % if R has TRANFUN --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if R has EUCDOM --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM or R has EUCDOM and R has PFECAT --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM @@ -76266,15 +76505,11 @@ digraph pic { --R gcd : List(%) -> % if R has EUCDOM or R has EUCDOM and R has PFECAT --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has EUCDOM or R has EUCDOM and R has PFECAT --R index : PositiveInteger -> % if R has FINITE ---R init : () -> % if R has FFIELDC --R lcm : (%,%) -> % if R has EUCDOM or R has EUCDOM and R has PFECAT --R lcm : List(%) -> % if R has EUCDOM or R has EUCDOM and R has PFECAT --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has EUCDOM or R has EUCDOM and R has PFECAT --R lift : % -> SparseUnivariatePolynomial(R) ---R log : % -> % if R has TRANFUN --R lookup : % -> PositiveInteger if R has FINITE ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has FIELD --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") if R has EUCDOM --R nextItem : % -> Union(%,"failed") if R has FFIELDC @@ -76291,7 +76526,6 @@ digraph pic { --R primitiveElement : () -> % if R has FFIELDC --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if R has EUCDOM --R ?quo? : (%,%) -> % if R has EUCDOM ---R random : () -> % if R has FINITE --R rational : % -> Fraction(Integer) if R has INS --R rational? : % -> Boolean if R has INS --R rationalIfCan : % -> Union(Fraction(Integer),"failed") if R has INS @@ -76311,10 +76545,6 @@ digraph pic { --R retractIfCan : % -> Union(R,"failed") --R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) --R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) ---R sec : % -> % if R has TRANFUN ---R sech : % -> % if R has TRANFUN ---R sin : % -> % if R has TRANFUN ---R sinh : % -> % if R has TRANFUN --R size : () -> NonNegativeInteger if R has FINITE --R sizeLess? : (%,%) -> Boolean if R has EUCDOM --R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has EUCDOM and R has PFECAT @@ -76324,8 +76554,6 @@ digraph pic { --R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has EUCDOM and R has PFECAT --R subtractIfCan : (%,%) -> Union(%,"failed") --R tableForDiscreteLogarithm : Integer -> Table(PositiveInteger,NonNegativeInteger) if R has FFIELDC ---R tan : % -> % if R has TRANFUN ---R tanh : % -> % if R has TRANFUN --R traceMatrix : Vector(%) -> Matrix(R) --R unit? : % -> Boolean if R has INTDOM or R has EUCDOM and R has PFECAT --R unitCanonical : % -> % if R has INTDOM or R has EUCDOM and R has PFECAT @@ -77295,34 +77523,37 @@ digraph pic { --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(UP),%) -> % ?*? : (%,Fraction(UP)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % basis : () -> Vector(%) ---R branchPoint? : UP -> Boolean branchPoint? : F -> Boolean ---R coerce : Fraction(UP) -> % coerce : Integer -> % ---R coerce : % -> OutputForm convert : UPUP -> % ---R convert : % -> UPUP definingPolynomial : () -> UPUP ---R discriminant : () -> Fraction(UP) elt : (%,F,F) -> F +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R basis : () -> Vector(%) branchPoint? : UP -> Boolean +--R branchPoint? : F -> Boolean coerce : Fraction(UP) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R convert : UPUP -> % convert : % -> UPUP +--R convert : Vector(Fraction(UP)) -> % convert : % -> Vector(Fraction(UP)) +--R definingPolynomial : () -> UPUP discriminant : () -> Fraction(UP) +--R elliptic : () -> Union(UP,"failed") elt : (%,F,F) -> F --R generator : () -> % genus : () -> NonNegativeInteger --R hash : % -> SingleInteger integral? : (%,UP) -> Boolean --R integral? : (%,F) -> Boolean integral? : % -> Boolean ---R integralBasis : () -> Vector(%) latex : % -> String ---R lift : % -> UPUP norm : % -> Fraction(UP) ---R one? : % -> Boolean primitivePart : % -> % ---R ramified? : UP -> Boolean ramified? : F -> Boolean +--R integralAtInfinity? : % -> Boolean integralBasis : () -> Vector(%) +--R latex : % -> String lift : % -> UPUP +--R norm : % -> Fraction(UP) one? : % -> Boolean +--R primitivePart : % -> % ramified? : UP -> Boolean +--R ramified? : F -> Boolean ramifiedAtInfinity? : () -> Boolean --R rank : () -> PositiveInteger rationalPoint? : (F,F) -> Boolean --R recip : % -> Union(%,"failed") reduce : UPUP -> % --R represents : (Vector(UP),UP) -> % retract : % -> Fraction(UP) --R sample : () -> % singular? : UP -> Boolean ---R singular? : F -> Boolean trace : % -> Fraction(UP) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R singular? : F -> Boolean singularAtInfinity? : () -> Boolean +--R trace : % -> Fraction(UP) zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Fraction(UP) has FIELD --R ?*? : (Fraction(Integer),%) -> % if Fraction(UP) has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if Fraction(UP) has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Fraction(UP) has FIELD --R D : % -> % if and(has(Fraction(UP),Field),has(Fraction(UP),DifferentialRing)) or and(has(Fraction(UP),DifferentialRing),has(Fraction(UP),Field)) or Fraction(UP) has FFIELDC --R D : (%,NonNegativeInteger) -> % if and(has(Fraction(UP),Field),has(Fraction(UP),DifferentialRing)) or and(has(Fraction(UP),DifferentialRing),has(Fraction(UP),Field)) or Fraction(UP) has FFIELDC @@ -77333,7 +77564,6 @@ digraph pic { --R D : (%,(Fraction(UP) -> Fraction(UP))) -> % if Fraction(UP) has FIELD --R D : (%,(Fraction(UP) -> Fraction(UP)),NonNegativeInteger) -> % if Fraction(UP) has FIELD --R ?^? : (%,Integer) -> % if Fraction(UP) has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R absolutelyIrreducible? : () -> Boolean --R algSplitSimple : (%,(UP -> UP)) -> Record(num: %,den: UP,derivden: UP,gd: UP) --R associates? : (%,%) -> Boolean if Fraction(UP) has FIELD @@ -77346,8 +77576,6 @@ digraph pic { --R coerce : Fraction(Integer) -> % if Fraction(UP) has FIELD or Fraction(UP) has RETRACT(FRAC(INT)) --R complementaryBasis : Vector(%) -> Vector(%) --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if Fraction(UP) has FFIELDC ---R convert : Vector(Fraction(UP)) -> % ---R convert : % -> Vector(Fraction(UP)) --R coordinates : Vector(%) -> Matrix(Fraction(UP)) --R coordinates : % -> Vector(Fraction(UP)) --R coordinates : (Vector(%),Vector(%)) -> Matrix(Fraction(UP)) @@ -77367,7 +77595,6 @@ digraph pic { --R discreteLog : % -> NonNegativeInteger if Fraction(UP) has FFIELDC --R discriminant : Vector(%) -> Fraction(UP) --R divide : (%,%) -> Record(quotient: %,remainder: %) if Fraction(UP) has FIELD ---R elliptic : () -> Union(UP,"failed") --R enumerate : () -> List(%) if Fraction(UP) has FINITE --R euclideanSize : % -> NonNegativeInteger if Fraction(UP) has FIELD --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if Fraction(UP) has FIELD @@ -77382,7 +77609,6 @@ digraph pic { --R hyperelliptic : () -> Union(UP,"failed") --R index : PositiveInteger -> % if Fraction(UP) has FINITE --R init : () -> % if Fraction(UP) has FFIELDC ---R integralAtInfinity? : % -> Boolean --R integralBasisAtInfinity : () -> Vector(%) --R integralCoordinates : % -> Record(num: Vector(UP),den: UP) --R integralDerivationMatrix : (UP -> UP) -> Record(num: Matrix(UP),den: UP) @@ -77411,7 +77637,6 @@ digraph pic { --R primitiveElement : () -> % if Fraction(UP) has FFIELDC --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if Fraction(UP) has FIELD --R ?quo? : (%,%) -> % if Fraction(UP) has FIELD ---R ramifiedAtInfinity? : () -> Boolean --R random : () -> % if Fraction(UP) has FINITE --R rationalPoints : () -> List(List(F)) if F has FINITE --R reduce : Fraction(UPUP) -> Union(%,"failed") if Fraction(UP) has FIELD @@ -77431,7 +77656,6 @@ digraph pic { --R retractIfCan : % -> Union(Fraction(UP),"failed") --R retractIfCan : % -> Union(Fraction(Integer),"failed") if Fraction(UP) has RETRACT(FRAC(INT)) --R retractIfCan : % -> Union(Integer,"failed") if Fraction(UP) has RETRACT(INT) ---R singularAtInfinity? : () -> Boolean --R size : () -> NonNegativeInteger if Fraction(UP) has FINITE --R sizeLess? : (%,%) -> Boolean if Fraction(UP) has FIELD --R squareFree : % -> Factored(%) if Fraction(UP) has FIELD @@ -78507,43 +78731,41 @@ digraph pic { --R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,Fraction(Integer)) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : Fraction(Integer) -> % coerce : Integer -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R conjugate : % -> % dimension : () -> CardinalNumber ---R extDegree : % -> PositiveInteger factor : % -> Factored(%) ---R fullOutput : % -> OutputForm gcd : List(%) -> % ---R gcd : (%,%) -> % ground? : % -> Boolean ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R maxTower : List(%) -> % one? : % -> Boolean ---R previousTower : % -> % prime? : % -> Boolean ---R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") ---R ?rem? : (%,%) -> % retract : % -> Fraction(Integer) ---R retract : % -> Fraction(Integer) retract : % -> Integer ---R sample : () -> % setTower! : % -> Void ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R vectorise : (%,%) -> Vector(%) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R algebraic? : % -> Boolean associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : Fraction(Integer) -> % +--R coerce : Integer -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm conjugate : % -> % +--R dimension : () -> CardinalNumber extDegree : % -> PositiveInteger +--R factor : % -> Factored(%) fullOutput : % -> OutputForm +--R gcd : List(%) -> % gcd : (%,%) -> % +--R ground? : % -> Boolean hash : % -> SingleInteger +--R inGroundField? : % -> Boolean inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % maxTower : List(%) -> % +--R one? : % -> Boolean previousTower : % -> % +--R prime? : % -> Boolean ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Fraction(Integer) +--R retract : % -> Integer sample : () -> % +--R setTower! : % -> Void sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % vectorise : (%,%) -> Vector(%) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,PseudoAlgebraicClosureOfRationalNumber) -> % --R ?*? : (PseudoAlgebraicClosureOfRationalNumber,%) -> % ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,PseudoAlgebraicClosureOfRationalNumber) -> % --R Frobenius : % -> % if PseudoAlgebraicClosureOfRationalNumber has FINITE or Fraction(Integer) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if PseudoAlgebraicClosureOfRationalNumber has FINITE or Fraction(Integer) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if PseudoAlgebraicClosureOfRationalNumber has CHARNZ or PseudoAlgebraicClosureOfRationalNumber has FINITE or Fraction(Integer) has CHARNZ or Fraction(Integer) has FINITE --R coerce : PseudoAlgebraicClosureOfRationalNumber -> % diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index 0585c7f..5329aa3 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -445,7 +445,7 @@ A Gonshor genetic algebra ([WB], p. 41-42) of dimension 4: The coefficient ring: \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 1 of 98 +--S 1 of 99 R := FRAC POLY INT --R --R @@ -457,7 +457,7 @@ R := FRAC POLY INT The following multiplication constants may be chosen arbitrarily (notice that we write ckij for $c_(i,j)^k$): \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 2 of 98 +--S 2 of 99 (c100, c101, _ c200, c201, c202, c211, _ c300, c301, c302, c303, c311, c312, c322) : R @@ -465,14 +465,14 @@ c300, c301, c302, c303, c311, c312, c322) : R --R Type: Void --E 2 ---S 3 of 98 +--S 3 of 99 c100 := 1 ; c101 := -1 ; --R --R --R Type: Fraction(Polynomial(Integer)) --E 3 ---S 4 of 98 +--S 4 of 99 c200 := 0 ; c201 := 1 ; c202 := -1 ; c211 := 2 ; --R @@ -480,7 +480,7 @@ c200 := 0 ; c201 := 1 ; c202 := -1 ; --R Type: Fraction(Polynomial(Integer)) --E 4 ---S 5 of 98 +--S 5 of 99 c300 := 1 ; c301 := 0 ; c302 := -1 ; c303 := 1 ; c311 := 1 ; c312 := 0 ; c322 := 2 ; @@ -492,7 +492,7 @@ c300 := 1 ; c301 := 0 ; c302 := -1 ; c303 := 1 ; \end{chunk} The matrices of the multiplication constants: \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 6 of 98 +--S 6 of 99 gonshor : List SquareMatrix(4,R) := [matrix [ [1, 0, 0, 0], [0, 0, 0, 0],_ [0, 0, 0, 0], [0, 0, 0, 0] ],_ @@ -507,7 +507,7 @@ gonshor : List SquareMatrix(4,R) := --R Type: List(SquareMatrix(4,Fraction(Polynomial(Integer)))) --E 6 ---S 7 of 98 +--S 7 of 99 basisSymbols : List Symbol := [subscript(e,[i]) for i in 0..3] --R --R @@ -516,7 +516,7 @@ basisSymbols : List Symbol := [subscript(e,[i]) for i in 0..3] --R Type: List(Symbol) --E 7 ---S 8 of 98 +--S 8 of 99 GonshorGenetic := ALGSC(R, 4, basisSymbols, gonshor) --R --R @@ -526,7 +526,7 @@ GonshorGenetic := ALGSC(R, 4, basisSymbols, gonshor) --R Type: Domain --E 8 ---S 9 of 98 +--S 9 of 99 commutative?()$GonshorGenetic --R --R algebra is commutative @@ -535,7 +535,7 @@ commutative?()$GonshorGenetic --R Type: Boolean --E 9 ---S 10 of 98 +--S 10 of 99 associative?()$GonshorGenetic --R --R algebra is not associative @@ -547,28 +547,28 @@ associative?()$GonshorGenetic \end{chunk} The canonical basis: \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 11 of 98 +--S 11 of 99 e0 : GonshorGenetic := [1, 0, 0, 0] :: Vector R ; --R --R --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 11 ---S 12 of 98 +--S 12 of 99 e1 : GonshorGenetic := [0, 1, 0, 0] :: Vector R ; --R --R --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 12 ---S 13 of 98 +--S 13 of 99 e2 : GonshorGenetic := [0, 0, 1, 0] :: Vector R ; --R --R --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 13 ---S 14 of 98 +--S 14 of 99 e3 : GonshorGenetic := [0, 0, 0, 1] :: Vector R ; --R --R @@ -578,7 +578,7 @@ e3 : GonshorGenetic := [0, 0, 0, 1] :: Vector R ; \end{chunk} A generic element of the algebra: \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 15 of 98 +--S 15 of 99 x : GonshorGenetic := x0*e0 + x1*e1 + x2*e2 + x3*e3 --R --R @@ -590,7 +590,7 @@ x : GonshorGenetic := x0*e0 + x1*e1 + x2*e2 + x3*e3 \end{chunk} The matrix of the left multiplication with x : \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 16 of 98 +--S 16 of 99 Lx := leftRegularRepresentation x --R --R @@ -611,7 +611,7 @@ defined by 8(e0) := 1 and 8(ei) := 0 for i = 1,2,3 . The coefficients of the characteristic polynomial of Lx depend only on 8(x) = x0 : \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 17 of 98 +--S 17 of 99 p := characteristicPolynomial(Lx,Y) --R --R @@ -623,7 +623,7 @@ p := characteristicPolynomial(Lx,Y) \end{chunk} The left minimal polynomial of x divides Y * p(Y) : \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 18 of 98 +--S 18 of 99 leftMinimalPolynomial x --R --R @@ -634,7 +634,7 @@ leftMinimalPolynomial x )clear prop A a b c r s ---S 19 of 98 +--S 19 of 99 A := GonshorGenetic --R --R @@ -644,7 +644,7 @@ A := GonshorGenetic --R Type: Domain --E 19 ---S 20 of 98 +--S 20 of 99 a := x --R --R @@ -653,7 +653,7 @@ a := x --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 20 ---S 21 of 98 +--S 21 of 99 b := (1/4)*e1 + (1/5)*e2 + (3/20)*e3 + (2/5)*e0 --R --R @@ -663,7 +663,7 @@ b := (1/4)*e1 + (1/5)*e2 + (3/20)*e3 + (2/5)*e0 --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 21 ---S 22 of 98 +--S 22 of 99 c := (1/3)*e1 + (1/7)*e2 + (8/21)*e3 + (1/7)*e0 --R --R @@ -673,7 +673,7 @@ c := (1/3)*e1 + (1/7)*e2 + (8/21)*e3 + (1/7)*e0 --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 22 ---S 23 of 98 +--S 23 of 99 r : R := r --R --R @@ -681,7 +681,7 @@ r : R := r --R Type: Fraction(Polynomial(Integer)) --E 23 ---S 24 of 98 +--S 24 of 99 s : R := s --R --R @@ -689,7 +689,7 @@ s : R := s --R Type: Fraction(Polynomial(Integer)) --E 24 ---S 25 of 98 +--S 25 of 99 b*c --R --R @@ -699,7 +699,7 @@ b*c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 25 ---S 26 of 98 +--S 26 of 99 (b*c)*b --R --R @@ -709,7 +709,7 @@ b*c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 26 ---S 27 of 98 +--S 27 of 99 b*(c*b) --R --R @@ -726,7 +726,7 @@ r,s : R \begin{chunk}{AlgebraGivenByStructuralConstants.input} )clear prop AP ---S 28 of 98 +--S 28 of 99 AP := ALGPKG(R,A) --R --R @@ -737,7 +737,7 @@ AP := ALGPKG(R,A) --R Type: Domain --E 28 ---S 29 of 98 +--S 29 of 99 r*a --R --R @@ -746,7 +746,7 @@ r*a --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 29 ---S 30 of 98 +--S 30 of 99 a*r --R --R @@ -755,7 +755,7 @@ a*r --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 30 ---S 31 of 98 +--S 31 of 99 a*b --R --R @@ -765,7 +765,7 @@ a*b --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 31 ---S 32 of 98 +--S 32 of 99 b*c --R --R @@ -775,7 +775,7 @@ b*c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 32 ---S 33 of 98 +--S 33 of 99 12 * c --R --R @@ -785,7 +785,7 @@ b*c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 32 ---S 34 of 98 +--S 34 of 99 (-3) * a --R --R @@ -794,7 +794,7 @@ b*c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 34 ---S 35 of 98 +--S 35 of 99 d := a ** 12 --R --R @@ -814,7 +814,7 @@ d := a ** 12 --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 35 ---S 36 of 98 +--S 36 of 99 -d --R --R @@ -834,7 +834,7 @@ d := a ** 12 --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 36 ---S 37 of 98 +--S 37 of 99 a + b --R --R @@ -844,7 +844,7 @@ a + b --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 37 ---S 38 of 98 +--S 38 of 99 d-c --R --R @@ -872,7 +872,7 @@ d-c --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 38 ---S 39 of 98 +--S 39 of 99 (a*(a*a) = leftPower(a,3)) :: Boolean --R --R @@ -880,7 +880,7 @@ d-c --R Type: Boolean --E 39 ---S 40 of 98 +--S 40 of 99 (a ** 11 = (a**8 * a**2) * a) :: Boolean --R --R @@ -888,7 +888,7 @@ d-c --R Type: Boolean --E 40 ---S 41 of 98 +--S 41 of 99 (a ** 11 = a**8 * (a**2 * a)) :: Boolean --R --R @@ -896,7 +896,7 @@ d-c --R Type: Boolean --E 41 ---S 42 of 98 +--S 42 of 99 zero := 0$A --R --R @@ -904,7 +904,7 @@ zero := 0$A --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 42 ---S 43 of 98 +--S 43 of 99 zero : A := 0 --R --R @@ -912,7 +912,7 @@ zero : A := 0 --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 43 ---S 44 of 98 +--S 44 of 99 alternative?()$A --R --R algebra is not left alternative @@ -921,7 +921,7 @@ alternative?()$A --R Type: Boolean --E 44 ---S 45 of 98 +--S 45 of 99 antiCommutative?()$A --R --R algebra is not anti-commutative @@ -930,7 +930,7 @@ antiCommutative?()$A --R Type: Boolean --E 45 ---S 46 of 98 +--S 46 of 99 associative?()$A --R --R algebra is not associative @@ -939,7 +939,7 @@ associative?()$A --R Type: Boolean --E 46 ---S 47 of 98 +--S 47 of 99 commutative?()$A --R --R algebra is commutative @@ -948,7 +948,7 @@ commutative?()$A --R Type: Boolean --E 47 ---S 48 of 98 +--S 48 of 99 commutator(a,b) --R --R @@ -956,7 +956,7 @@ commutator(a,b) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 48 ---S 49 of 98 +--S 49 of 99 antiCommutator(a,b) --R --R @@ -966,7 +966,7 @@ antiCommutator(a,b) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 49 ---S 50 of 98 +--S 50 of 99 associator(a,b,c) --R --R @@ -976,7 +976,7 @@ associator(a,b,c) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 50 ---S 51 of 98 +--S 51 of 99 basis()$A --R --R @@ -985,7 +985,7 @@ basis()$A --RType: Vector(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 51 ---S 52 of 98 +--S 52 of 99 n := rank()$A --R --R @@ -993,7 +993,7 @@ n := rank()$A --R Type: PositiveInteger --E 52 ---S 53 of 98 +--S 53 of 99 v : Vector R := [i for i in 1..n] --R --R @@ -1001,7 +1001,7 @@ v : Vector R := [i for i in 1..n] --R Type: Vector(Fraction(Polynomial(Integer))) --E 53 ---S 54 of 98 +--S 54 of 99 g : A := represents v --R --R @@ -1010,7 +1010,7 @@ g : A := represents v --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 54 ---S 55 of 98 +--S 55 of 99 coordinates a --R --R @@ -1018,7 +1018,7 @@ coordinates a --R Type: Vector(Fraction(Polynomial(Integer))) --E 55 ---S 56 of 98 +--S 56 of 99 coordinates [a,b] --R --R @@ -1030,7 +1030,7 @@ coordinates [a,b] --R Type: Matrix(Fraction(Polynomial(Integer))) --E 56 ---S 57 of 98 +--S 57 of 99 a.3 --R --R @@ -1038,7 +1038,7 @@ a.3 --R Type: Fraction(Polynomial(Integer)) --E 57 ---S 58 of 98 +--S 58 of 99 flexible?()$A --R --R algebra is flexible @@ -1047,7 +1047,7 @@ flexible?()$A --R Type: Boolean --E 58 ---S 59 of 98 +--S 59 of 99 leftAlternative?()$A --R --R algebra is not left alternative @@ -1056,7 +1056,7 @@ leftAlternative?()$A --R Type: Boolean --E 59 ---S 60 of 98 +--S 60 of 99 rightAlternative?()$A --R --R algebra is not right alternative @@ -1065,7 +1065,7 @@ rightAlternative?()$A --R Type: Boolean --E 60 ---S 61 of 98 +--S 61 of 99 sB := someBasis()$A --R --R @@ -1074,7 +1074,7 @@ sB := someBasis()$A --RType: Vector(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 61 ---S 62 of 98 +--S 62 of 99 zero? a --R --R @@ -1082,7 +1082,7 @@ zero? a --R Type: Boolean --E 62 ---S 63 of 98 +--S 63 of 99 associatorDependence()$A --R --R @@ -1093,7 +1093,7 @@ associatorDependence()$A \end{chunk} ConditionsForIdempotents()\$A \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 64 of 98 +--S 64 of 99 jacobiIdentity?()$A --R --R Jacobi identity does not hold @@ -1102,7 +1102,7 @@ jacobiIdentity?()$A --R Type: Boolean --E 64 ---S 65 of 98 +--S 65 of 99 jordanAlgebra?()$A --R --R algebra is commutative @@ -1112,7 +1112,7 @@ jordanAlgebra?()$A --R Type: Boolean --E 65 ---S 66 of 98 +--S 66 of 99 jordanAdmissible?()$A --R --R algebra is not Jordan admissible @@ -1121,7 +1121,7 @@ jordanAdmissible?()$A --R Type: Boolean --E 66 ---S 67 of 98 +--S 67 of 99 lieAdmissible?()$A --R --R algebra is Lie admissible @@ -1133,7 +1133,7 @@ lieAdmissible?()$A \end{chunk} ConditionsForIdempotents \begin{chunk}{AlgebraGivenByStructuralConstants.input} ---S 68 of 98 +--S 68 of 99 b2 := [reduce(+,[sB.i for i in 1..k]) for k in 1..n] --R --R @@ -1142,7 +1142,7 @@ b2 := [reduce(+,[sB.i for i in 1..k]) for k in 1..n] --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 68 ---S 69 of 98 +--S 69 of 99 coordinates (a ,b2 :: Vector A) --R --R @@ -1150,7 +1150,7 @@ coordinates (a ,b2 :: Vector A) --R Type: Vector(Fraction(Polynomial(Integer))) --E 69 ---S 70 of 98 +--S 70 of 99 coordinates ([a,b] ,bb := (b2 :: Vector A)) --R --R @@ -1162,7 +1162,7 @@ coordinates ([a,b] ,bb := (b2 :: Vector A)) --R Type: Matrix(Fraction(Polynomial(Integer))) --E 70 ---S 71 of 98 +--S 71 of 99 leftMinimalPolynomial a --R --R @@ -1171,7 +1171,7 @@ leftMinimalPolynomial a --R Type: SparseUnivariatePolynomial(Fraction(Polynomial(Integer))) --E 71 ---S 72 of 98 +--S 72 of 99 leftPower (a,10) --R --R @@ -1186,7 +1186,7 @@ leftPower (a,10) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 72 ---S 73 of 98 +--S 73 of 99 rightPower(a,10) --R --R @@ -1201,7 +1201,7 @@ rightPower(a,10) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 73 ---S 74 of 98 +--S 74 of 99 leftRegularRepresentation a --R --R @@ -1215,7 +1215,7 @@ leftRegularRepresentation a --R Type: Matrix(Fraction(Polynomial(Integer))) --E 74 ---S 75 of 98 +--S 75 of 99 leftRegularRepresentation (a,bb) --R --R @@ -1229,7 +1229,7 @@ leftRegularRepresentation (a,bb) --R Type: Matrix(Fraction(Polynomial(Integer))) --E 75 ---S 76 of 98 +--S 76 of 99 leftUnit()$A --R --R this algebra has no left unit @@ -1238,7 +1238,7 @@ leftUnit()$A --R Type: Union("failed",...) --E 76 ---S 77 of 98 +--S 77 of 99 represents (v,bb) --R --R @@ -1247,7 +1247,7 @@ represents (v,bb) --RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) --E 77 ---S 78 of 98 +--S 78 of 99 rightMinimalPolynomial a --R --R @@ -1256,7 +1256,7 @@ rightMinimalPolynomial a --R Type: SparseUnivariatePolynomial(Fraction(Polynomial(Integer))) --E 78 ---S 79 of 98 +--S 79 of 99 rightRegularRepresentation a --R --R @@ -1270,7 +1270,7 @@ rightRegularRepresentation a --R Type: Matrix(Fraction(Polynomial(Integer))) --E 79 ---S 80 of 98 +--S 80 of 99 rightRegularRepresentation (a,bb) --R --R @@ -1284,7 +1284,7 @@ rightRegularRepresentation (a,bb) --R Type: Matrix(Fraction(Polynomial(Integer))) --E 80 ---S 81 of 98 +--S 81 of 99 rightUnit()$A --R --R this algebra has no right unit @@ -1293,7 +1293,7 @@ rightUnit()$A --R Type: Union("failed",...) --E 81 ---S 82 of 98 +--S 82 of 99 structuralConstants()$A --R --R @@ -1307,7 +1307,7 @@ structuralConstants()$A --R Type: Vector(Matrix(Fraction(Polynomial(Integer)))) --E 82 ---S 83 of 98 +--S 83 of 99 structuralConstants(bb) --R --R @@ -1321,7 +1321,7 @@ structuralConstants(bb) --R Type: Vector(Matrix(Fraction(Polynomial(Integer)))) --E 83 ---S 84 of 98 +--S 84 of 99 unit()$A --R --R this algebra has no unit @@ -1330,7 +1330,7 @@ unit()$A --R Type: Union("failed",...) --E 84 ---S 85 of 98 +--S 85 of 99 biRank a --R --R @@ -1338,7 +1338,7 @@ biRank a --R Type: PositiveInteger --E 85 ---S 86 of 98 +--S 86 of 99 leftRank a --R --R @@ -1346,7 +1346,7 @@ leftRank a --R Type: PositiveInteger --E 86 ---S 87 of 98 +--S 87 of 99 doubleRank a --R --R @@ -1354,7 +1354,7 @@ doubleRank a --R Type: PositiveInteger --E 87 ---S 88 of 98 +--S 88 of 99 rightRank a --R --R @@ -1362,7 +1362,7 @@ rightRank a --R Type: PositiveInteger --E 88 ---S 89 of 98 +--S 89 of 99 weakBiRank a --R --R @@ -1370,7 +1370,7 @@ weakBiRank a --R Type: PositiveInteger --E 89 ---S 90 of 98 +--S 90 of 99 basisOfCenter()$AP --R --R @@ -1379,7 +1379,7 @@ basisOfCenter()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 90 ---S 91 of 98 +--S 91 of 99 basisOfLeftNucleus()$AP --R --R @@ -1388,7 +1388,7 @@ basisOfLeftNucleus()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 91 ---S 92 of 98 +--S 92 of 99 basisOfNucleus()$AP --R --R @@ -1397,7 +1397,7 @@ basisOfNucleus()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 92 ---S 93 of 98 +--S 93 of 99 basisOfRightNucleus()$AP --R --R @@ -1406,7 +1406,7 @@ basisOfRightNucleus()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 93 ---S 94 of 98 +--S 94 of 99 basisOfCentroid()$AP --R --R @@ -1420,7 +1420,7 @@ basisOfCentroid()$AP --R Type: List(Matrix(Fraction(Polynomial(Integer)))) --E 94 ---S 95 of 98 +--S 95 of 99 basisOfCommutingElements()$AP --R --R @@ -1429,7 +1429,7 @@ basisOfCommutingElements()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 95 ---S 96 of 98 +--S 96 of 99 basisOfLeftNucloid()$AP --R --R @@ -1443,7 +1443,7 @@ basisOfLeftNucloid()$AP --R Type: List(Matrix(Fraction(Polynomial(Integer)))) --E 96 ---S 97 of 98 +--S 97 of 99 basisOfMiddleNucleus()$AP --R --R @@ -1452,7 +1452,7 @@ basisOfMiddleNucleus()$AP --RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) --E 97 ---S 98 of 98 +--S 98 of 99 basisOfRightNucloid()$AP --R --R @@ -1466,6 +1466,80 @@ basisOfRightNucloid()$AP --R Type: List(Matrix(Fraction(Polynomial(Integer)))) --E 98 +--S 99 of 99 +)show AlgebraGivenByStructuralConstants +--R +--R AlgebraGivenByStructuralConstants(R: Field,n: PositiveInteger,ls: List(Symbol),gamma: Vector(Matrix(R))) is a domain constructor +--R Abbreviation for AlgebraGivenByStructuralConstants is ALGSC +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for ALGSC +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (SquareMatrix(n,R),%) -> % ?*? : (R,%) -> % +--R ?*? : (%,R) -> % ?*? : (%,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % alternative? : () -> Boolean +--R antiAssociative? : () -> Boolean antiCommutative? : () -> Boolean +--R antiCommutator : (%,%) -> % apply : (Matrix(R),%) -> % +--R associative? : () -> Boolean associator : (%,%,%) -> % +--R basis : () -> Vector(%) coerce : Vector(R) -> % +--R coerce : % -> OutputForm commutative? : () -> Boolean +--R commutator : (%,%) -> % convert : Vector(R) -> % +--R convert : % -> Vector(R) coordinates : % -> Vector(R) +--R ?.? : (%,Integer) -> R flexible? : () -> Boolean +--R hash : % -> SingleInteger jacobiIdentity? : () -> Boolean +--R jordanAdmissible? : () -> Boolean jordanAlgebra? : () -> Boolean +--R latex : % -> String leftAlternative? : () -> Boolean +--R leftDiscriminant : () -> R leftDiscriminant : Vector(%) -> R +--R leftNorm : % -> R leftTrace : % -> R +--R leftTraceMatrix : () -> Matrix(R) lieAdmissible? : () -> Boolean +--R lieAlgebra? : () -> Boolean powerAssociative? : () -> Boolean +--R rank : () -> PositiveInteger represents : Vector(R) -> % +--R rightAlternative? : () -> Boolean rightDiscriminant : () -> R +--R rightDiscriminant : Vector(%) -> R rightNorm : % -> R +--R rightTrace : % -> R rightTraceMatrix : () -> Matrix(R) +--R sample : () -> % someBasis : () -> Vector(%) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R associatorDependence : () -> List(Vector(R)) if R has INTDOM +--R conditionsForIdempotents : () -> List(Polynomial(R)) +--R conditionsForIdempotents : Vector(%) -> List(Polynomial(R)) +--R coordinates : Vector(%) -> Matrix(R) +--R coordinates : (Vector(%),Vector(%)) -> Matrix(R) +--R coordinates : (%,Vector(%)) -> Vector(R) +--R leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial(R) +--R leftMinimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has INTDOM +--R leftPower : (%,PositiveInteger) -> % +--R leftRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(R)) if R has FIELD +--R leftRecip : % -> Union(%,"failed") if R has INTDOM +--R leftRegularRepresentation : % -> Matrix(R) +--R leftRegularRepresentation : (%,Vector(%)) -> Matrix(R) +--R leftTraceMatrix : Vector(%) -> Matrix(R) +--R leftUnit : () -> Union(%,"failed") if R has INTDOM +--R leftUnits : () -> Union(Record(particular: %,basis: List(%)),"failed") if R has INTDOM +--R noncommutativeJordanAlgebra? : () -> Boolean +--R plenaryPower : (%,PositiveInteger) -> % +--R recip : % -> Union(%,"failed") if R has INTDOM +--R represents : (Vector(R),Vector(%)) -> % +--R rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(R) +--R rightMinimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has INTDOM +--R rightPower : (%,PositiveInteger) -> % +--R rightRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(R)) if R has FIELD +--R rightRecip : % -> Union(%,"failed") if R has INTDOM +--R rightRegularRepresentation : % -> Matrix(R) +--R rightRegularRepresentation : (%,Vector(%)) -> Matrix(R) +--R rightTraceMatrix : Vector(%) -> Matrix(R) +--R rightUnit : () -> Union(%,"failed") if R has INTDOM +--R rightUnits : () -> Union(Record(particular: %,basis: List(%)),"failed") if R has INTDOM +--R structuralConstants : () -> Vector(Matrix(R)) +--R structuralConstants : Vector(%) -> Vector(Matrix(R)) +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unit : () -> Union(%,"failed") if R has INTDOM +--R +--E 99 + )spool )lisp (bye) \end{chunk} @@ -2616,34 +2690,37 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(UP),%) -> % ?*? : (%,Fraction(UP)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % basis : () -> Vector(%) ---R branchPoint? : UP -> Boolean branchPoint? : F -> Boolean ---R coerce : Fraction(UP) -> % coerce : Integer -> % ---R coerce : % -> OutputForm convert : UPUP -> % ---R convert : % -> UPUP definingPolynomial : () -> UPUP ---R discriminant : () -> Fraction(UP) elt : (%,F,F) -> F +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R basis : () -> Vector(%) branchPoint? : UP -> Boolean +--R branchPoint? : F -> Boolean coerce : Fraction(UP) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R convert : UPUP -> % convert : % -> UPUP +--R convert : Vector(Fraction(UP)) -> % convert : % -> Vector(Fraction(UP)) +--R definingPolynomial : () -> UPUP discriminant : () -> Fraction(UP) +--R elliptic : () -> Union(UP,"failed") elt : (%,F,F) -> F --R generator : () -> % genus : () -> NonNegativeInteger --R hash : % -> SingleInteger integral? : (%,UP) -> Boolean --R integral? : (%,F) -> Boolean integral? : % -> Boolean ---R integralBasis : () -> Vector(%) latex : % -> String ---R lift : % -> UPUP norm : % -> Fraction(UP) ---R one? : % -> Boolean primitivePart : % -> % ---R ramified? : UP -> Boolean ramified? : F -> Boolean +--R integralAtInfinity? : % -> Boolean integralBasis : () -> Vector(%) +--R latex : % -> String lift : % -> UPUP +--R norm : % -> Fraction(UP) one? : % -> Boolean +--R primitivePart : % -> % ramified? : UP -> Boolean +--R ramified? : F -> Boolean ramifiedAtInfinity? : () -> Boolean --R rank : () -> PositiveInteger rationalPoint? : (F,F) -> Boolean --R recip : % -> Union(%,"failed") reduce : UPUP -> % --R represents : (Vector(UP),UP) -> % retract : % -> Fraction(UP) --R sample : () -> % singular? : UP -> Boolean ---R singular? : F -> Boolean trace : % -> Fraction(UP) ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R singular? : F -> Boolean singularAtInfinity? : () -> Boolean +--R trace : % -> Fraction(UP) zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Fraction(UP) has FIELD --R ?*? : (Fraction(Integer),%) -> % if Fraction(UP) has FIELD ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Integer) -> % if Fraction(UP) has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Fraction(UP) has FIELD --R D : % -> % if Fraction(UP) has DIFRING and Fraction(UP) has FIELD or Fraction(UP) has FFIELDC --R D : (%,NonNegativeInteger) -> % if Fraction(UP) has DIFRING and Fraction(UP) has FIELD or Fraction(UP) has FFIELDC @@ -2654,7 +2731,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R D : (%,(Fraction(UP) -> Fraction(UP))) -> % if Fraction(UP) has FIELD --R D : (%,(Fraction(UP) -> Fraction(UP)),NonNegativeInteger) -> % if Fraction(UP) has FIELD --R ?^? : (%,Integer) -> % if Fraction(UP) has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R absolutelyIrreducible? : () -> Boolean --R algSplitSimple : (%,(UP -> UP)) -> Record(num: %,den: UP,derivden: UP,gd: UP) --R associates? : (%,%) -> Boolean if Fraction(UP) has FIELD @@ -2667,8 +2743,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R coerce : Fraction(Integer) -> % if Fraction(UP) has FIELD or Fraction(UP) has RETRACT(FRAC(INT)) --R complementaryBasis : Vector(%) -> Vector(%) --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if Fraction(UP) has FFIELDC ---R convert : Vector(Fraction(UP)) -> % ---R convert : % -> Vector(Fraction(UP)) --R coordinates : Vector(%) -> Matrix(Fraction(UP)) --R coordinates : % -> Vector(Fraction(UP)) --R coordinates : (Vector(%),Vector(%)) -> Matrix(Fraction(UP)) @@ -2688,7 +2762,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R discreteLog : % -> NonNegativeInteger if Fraction(UP) has FFIELDC --R discriminant : Vector(%) -> Fraction(UP) --R divide : (%,%) -> Record(quotient: %,remainder: %) if Fraction(UP) has FIELD ---R elliptic : () -> Union(UP,"failed") --R enumerate : () -> List(%) if Fraction(UP) has FINITE --R euclideanSize : % -> NonNegativeInteger if Fraction(UP) has FIELD --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if Fraction(UP) has FIELD @@ -2703,7 +2776,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R hyperelliptic : () -> Union(UP,"failed") --R index : PositiveInteger -> % if Fraction(UP) has FINITE --R init : () -> % if Fraction(UP) has FFIELDC ---R integralAtInfinity? : % -> Boolean --R integralBasisAtInfinity : () -> Vector(%) --R integralCoordinates : % -> Record(num: Vector(UP),den: UP) --R integralDerivationMatrix : (UP -> UP) -> Record(num: Matrix(UP),den: UP) @@ -2733,7 +2805,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R primitiveElement : () -> % if Fraction(UP) has FFIELDC --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if Fraction(UP) has FIELD --R ?quo? : (%,%) -> % if Fraction(UP) has FIELD ---R ramifiedAtInfinity? : () -> Boolean --R random : () -> % if Fraction(UP) has FINITE --R rationalPoints : () -> List(List(F)) if F has FINITE --R reduce : Fraction(UPUP) -> Union(%,"failed") if Fraction(UP) has FIELD @@ -2753,7 +2824,6 @@ AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_ --R retractIfCan : % -> Union(Fraction(UP),"failed") --R retractIfCan : % -> Union(Fraction(Integer),"failed") if Fraction(UP) has RETRACT(FRAC(INT)) --R retractIfCan : % -> Union(Integer,"failed") if Fraction(UP) has RETRACT(INT) ---R singularAtInfinity? : () -> Boolean --R size : () -> NonNegativeInteger if Fraction(UP) has FINITE --R sizeLess? : (%,%) -> Boolean if Fraction(UP) has FIELD --R squareFree : % -> Factored(%) if Fraction(UP) has FIELD @@ -3151,17 +3221,19 @@ AlgebraicFunctionField(F, UP, UPUP, modulus): Exports == Implementation where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for AN --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (%,%) -> % ?*? : (%,Fraction(Integer)) -> % ---R ?*? : (Fraction(Integer),%) -> % ?**? : (%,PositiveInteger) -> % ---R ?**? : (%,Integer) -> % ?+? : (%,%) -> % ---R -? : % -> % ?-? : (%,%) -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R D : % -> % D : (%,NonNegativeInteger) -> % ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % ?^? : (%,Integer) -> % +--R ?*? : (PositiveInteger,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (%,%) -> % +--R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % +--R ?**? : (%,PositiveInteger) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,Fraction(Integer)) -> % +--R ?+? : (%,%) -> % -? : % -> % +--R ?-? : (%,%) -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean D : % -> % +--R D : (%,NonNegativeInteger) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,PositiveInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,Integer) -> % --R associates? : (%,%) -> Boolean belong? : BasicOperator -> Boolean --R box : List(%) -> % box : % -> % --R coerce : Integer -> % coerce : % -> % @@ -3169,55 +3241,50 @@ AlgebraicFunctionField(F, UP, UPUP, modulus): Exports == Implementation where --R coerce : % -> OutputForm convert : % -> Complex(Float) --R convert : % -> DoubleFloat convert : % -> Float --R differentiate : % -> % distribute : (%,%) -> % ---R distribute : % -> % elt : (BasicOperator,%,%) -> % ---R elt : (BasicOperator,%) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % eval : (%,Kernel(%),%) -> % ---R factor : % -> Factored(%) freeOf? : (%,Symbol) -> Boolean ---R freeOf? : (%,%) -> Boolean gcd : (%,%) -> % ---R gcd : List(%) -> % hash : % -> SingleInteger ---R height : % -> NonNegativeInteger inv : % -> % ---R is? : (%,Symbol) -> Boolean kernel : (BasicOperator,%) -> % +--R distribute : % -> % elt : (BasicOperator,List(%)) -> % +--R elt : (BasicOperator,%,%,%) -> % elt : (BasicOperator,%,%) -> % +--R elt : (BasicOperator,%) -> % eval : (%,Symbol,(% -> %)) -> % +--R eval : (%,List(%),List(%)) -> % eval : (%,%,%) -> % +--R eval : (%,Equation(%)) -> % eval : (%,List(Equation(%))) -> % +--R eval : (%,Kernel(%),%) -> % factor : % -> Factored(%) +--R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean +--R gcd : (%,%) -> % gcd : List(%) -> % +--R hash : % -> SingleInteger height : % -> NonNegativeInteger +--R inv : % -> % is? : (%,Symbol) -> Boolean +--R is? : (%,BasicOperator) -> Boolean kernel : (BasicOperator,%) -> % --R kernels : % -> List(Kernel(%)) latex : % -> String --R lcm : (%,%) -> % lcm : List(%) -> % ---R max : (%,%) -> % min : (%,%) -> % ---R norm : (%,List(Kernel(%))) -> % norm : (%,Kernel(%)) -> % ---R nthRoot : (%,Integer) -> % one? : % -> Boolean ---R paren : List(%) -> % paren : % -> % ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") reduce : % -> % ---R ?rem? : (%,%) -> % retract : % -> Fraction(Integer) ---R retract : % -> Integer retract : % -> Kernel(%) ---R rootOf : Polynomial(%) -> % sample : () -> % +--R map : ((% -> %),Kernel(%)) -> % max : (%,%) -> % +--R min : (%,%) -> % norm : (%,List(Kernel(%))) -> % +--R norm : (%,Kernel(%)) -> % nthRoot : (%,Integer) -> % +--R one? : % -> Boolean paren : List(%) -> % +--R paren : % -> % prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R reduce : % -> % ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Integer +--R retract : % -> Kernel(%) rootOf : Polynomial(%) -> % +--R rootsOf : Polynomial(%) -> List(%) sample : () -> % --R sizeLess? : (%,%) -> Boolean sqrt : % -> % --R squareFree : % -> Factored(%) squareFreePart : % -> % --R subst : (%,Equation(%)) -> % tower : % -> List(Kernel(%)) --R unit? : % -> Boolean unitCanonical : % -> % --R zero? : % -> Boolean zeroOf : Polynomial(%) -> % ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R zerosOf : Polynomial(%) -> List(%) ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R coerce : SparseMultivariatePolynomial(Integer,Kernel(%)) -> % --R definingPolynomial : % -> % if $ has RING --R denom : % -> SparseMultivariatePolynomial(Integer,Kernel(%)) --R differentiate : (%,NonNegativeInteger) -> % --R divide : (%,%) -> Record(quotient: %,remainder: %) ---R elt : (BasicOperator,List(%)) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,%,%,%) -> % --R euclideanSize : % -> NonNegativeInteger --R eval : (%,BasicOperator,(% -> %)) -> % --R eval : (%,BasicOperator,(List(%) -> %)) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R eval : (%,List(Kernel(%)),List(%)) -> % --R even? : % -> Boolean if $ has RETRACT(INT) --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") @@ -3225,11 +3292,9 @@ AlgebraicFunctionField(F, UP, UPUP, modulus): Exports == Implementation where --R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) ---R is? : (%,BasicOperator) -> Boolean --R kernel : (BasicOperator,List(%)) -> % --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") --R norm : (SparseUnivariatePolynomial(%),List(Kernel(%))) -> SparseUnivariatePolynomial(%) @@ -3248,7 +3313,6 @@ AlgebraicFunctionField(F, UP, UPUP, modulus): Exports == Implementation where --R retractIfCan : % -> Union(Kernel(%),"failed") --R rootOf : SparseUnivariatePolynomial(%) -> % --R rootOf : (SparseUnivariatePolynomial(%),Symbol) -> % ---R rootsOf : Polynomial(%) -> List(%) --R rootsOf : SparseUnivariatePolynomial(%) -> List(%) --R rootsOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R subst : (%,List(Kernel(%)),List(%)) -> % @@ -3257,7 +3321,6 @@ AlgebraicFunctionField(F, UP, UPUP, modulus): Exports == Implementation where --R unitNormal : % -> Record(unit: %,canonical: %,associate: %) --R zeroOf : SparseUnivariatePolynomial(%) -> % --R zeroOf : (SparseUnivariatePolynomial(%),Symbol) -> % ---R zerosOf : Polynomial(%) -> List(%) --R zerosOf : SparseUnivariatePolynomial(%) -> List(%) --R zerosOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R @@ -3898,6 +3961,7 @@ AlgebraicNumber(): Exports == Implementation where --S 1 of 1 )show AnonymousFunction +--R --R AnonymousFunction is a domain constructor --R Abbreviation for AnonymousFunction is ANON --R This constructor is not exposed in this frame. @@ -3978,14 +4042,16 @@ AnonymousFunction():SetCategory == add --R --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coefficient : (%,%) -> R coerce : R -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R degree : % -> NonNegativeInteger exp : List(Integer) -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficient : (%,%) -> R +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm degree : % -> NonNegativeInteger +--R exp : List(Integer) -> % generator : NonNegativeInteger -> % --R hash : % -> SingleInteger homogeneous? : % -> Boolean --R latex : % -> String leadingBasisTerm : % -> % --R leadingCoefficient : % -> R map : ((R -> R),%) -> % @@ -3993,11 +4059,7 @@ AnonymousFunction():SetCategory == add --R reductum : % -> % retract : % -> R --R retractable? : % -> Boolean sample : () -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger ---R generator : NonNegativeInteger -> % --R retractIfCan : % -> Union(R,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -4965,8 +5027,8 @@ b := generator()$Sae --R coerce : % -> OutputForm dom : % -> SExpression --R domainOf : % -> OutputForm hash : % -> SingleInteger --R latex : % -> String obj : % -> None ---R objectOf : % -> OutputForm ?~=? : (%,%) -> Boolean ---R showTypeInOutput : Boolean -> String +--R objectOf : % -> OutputForm showTypeInOutput : Boolean -> String +--R ?~=? : (%,%) -> Boolean --R --E 18 @@ -5539,9 +5601,9 @@ latex a --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean extract! : % -> S --R insert! : (S,%) -> % inspect : % -> S ---R map : ((S -> S),%) -> % pop! : % -> S ---R push! : (S,%) -> S sample : () -> % ---R top : % -> S +--R latex : % -> String if S has SETCAT map : ((S -> S),%) -> % +--R pop! : % -> S push! : (S,%) -> S +--R sample : () -> % top : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -5554,7 +5616,6 @@ latex a --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -6578,13 +6639,12 @@ ArrayStack(S:SetCategory): StackAggregate(S) with --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : FortranExpression([construct,QUOTEX],[construct],MachineFloat) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -7792,13 +7852,12 @@ Asp20(name): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : FortranExpression([construct],[construct,QUOTEXC],MachineFloat) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -7986,9 +8045,9 @@ Asp24(name): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Matrix(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Matrix(MachineFloat) -> % --R --E 1 @@ -8153,9 +8212,9 @@ Asp27(name): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Matrix(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Matrix(MachineFloat) -> % --R --E 1 @@ -8678,9 +8737,9 @@ Asp29(name): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Matrix(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Matrix(MachineFloat) -> % --R --E 1 @@ -9251,9 +9310,9 @@ Asp33(name): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Matrix(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Matrix(MachineFloat) -> % --R --E 1 @@ -9683,13 +9742,12 @@ Asp35(name): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : FortranExpression([construct],[construct,QUOTEX],MachineFloat) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -10534,13 +10592,12 @@ Asp42(nameOne,nameTwo,nameThree): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : FortranExpression([construct],[construct,QUOTEX],MachineFloat) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -12709,9 +12766,9 @@ Asp78(name): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % ---R coerce : % -> OutputForm outputAsFortran : % -> Void +--R coerce : Vector(MachineFloat) -> % coerce : % -> OutputForm +--R outputAsFortran : % -> Void --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % ---R coerce : Vector(MachineFloat) -> % --R --E 1 @@ -13147,13 +13204,12 @@ Asp80(name): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R coerce : FortranCode -> % coerce : List(FortranCode) -> % --R coerce : % -> OutputForm outputAsFortran : % -> Void ---R retract : Polynomial(Float) -> % retract : Expression(Float) -> % +--R retract : Polynomial(Integer) -> % retract : Polynomial(Float) -> % +--R retract : Expression(Integer) -> % retract : Expression(Float) -> % --R coerce : FortranExpression([construct,QUOTEX],[construct,QUOTEY],MachineFloat) -> % --R coerce : Record(localSymbols: SymbolTable,code: List(FortranCode)) -> % --R retract : Fraction(Polynomial(Integer)) -> % --R retract : Fraction(Polynomial(Float)) -> % ---R retract : Polynomial(Integer) -> % ---R retract : Expression(Integer) -> % --R retractIfCan : Fraction(Polynomial(Integer)) -> Union(%,"failed") --R retractIfCan : Fraction(Polynomial(Float)) -> Union(%,"failed") --R retractIfCan : Polynomial(Integer) -> Union(%,"failed") @@ -13351,16 +13407,16 @@ Asp9(name): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % antiCommutator : (%,%) -> % ---R associator : (%,%,%) -> % coerce : A -> % ---R coerce : % -> A coerce : % -> OutputForm ---R commutator : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R antiCommutator : (%,%) -> % associator : (%,%,%) -> % +--R coerce : A -> % coerce : % -> A +--R coerce : % -> OutputForm commutator : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R alternative? : () -> Boolean if A has FINAALG(R) --R antiAssociative? : () -> Boolean if A has FINAALG(R) --R antiCommutative? : () -> Boolean if A has FINAALG(R) @@ -13625,16 +13681,16 @@ AssociatedJordanAlgebra(R:CommutativeRing,A:NonAssociativeAlgebra R): --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % antiCommutator : (%,%) -> % ---R associator : (%,%,%) -> % coerce : A -> % ---R coerce : % -> A coerce : % -> OutputForm ---R commutator : (%,%) -> % hash : % -> SingleInteger ---R latex : % -> String sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R antiCommutator : (%,%) -> % associator : (%,%,%) -> % +--R coerce : A -> % coerce : % -> A +--R coerce : % -> OutputForm commutator : (%,%) -> % +--R hash : % -> SingleInteger latex : % -> String +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R alternative? : () -> Boolean if A has FINAALG(R) --R antiAssociative? : () -> Boolean if A has FINAALG(R) --R antiCommutative? : () -> Boolean if A has FINAALG(R) @@ -13883,7 +13939,8 @@ AssociatedLieAlgebra(R:CommutativeRing,A:NonAssociativeAlgebra R): )set message test on )set message auto off )clear all ---S 1 of 10 + +--S 1 of 11 Data := Record(monthsOld : Integer, gender : String) --R --R @@ -13891,13 +13948,13 @@ Data := Record(monthsOld : Integer, gender : String) --R Type: Domain --E 1 ---S 2 of 10 +--S 2 of 11 al : AssociationList(String,Data) --R --R Type: Void --E 2 ---S 3 of 10 +--S 3 of 11 al := table() --R --R @@ -13905,7 +13962,7 @@ al := table() --R Type: AssociationList(String,Record(monthsOld: Integer,gender: String)) --E 3 ---S 4 of 10 +--S 4 of 11 al."bob" := [407,"male"]$Data --R --R @@ -13913,7 +13970,7 @@ al."bob" := [407,"male"]$Data --R Type: Record(monthsOld: Integer,gender: String) --E 4 ---S 5 of 10 +--S 5 of 11 al."judith" := [366,"female"]$Data --R --R @@ -13921,7 +13978,7 @@ al."judith" := [366,"female"]$Data --R Type: Record(monthsOld: Integer,gender: String) --E 5 ---S 6 of 10 +--S 6 of 11 al."katie" := [24,"female"]$Data --R --R @@ -13929,7 +13986,7 @@ al."katie" := [24,"female"]$Data --R Type: Record(monthsOld: Integer,gender: String) --E 6 ---S 7 of 10 +--S 7 of 11 al."smokie" := [200,"female"]$Data --R --R @@ -13937,7 +13994,7 @@ al."smokie" := [200,"female"]$Data --R Type: Record(monthsOld: Integer,gender: String) --E 7 ---S 8 of 10 +--S 8 of 11 al --R --R @@ -13953,7 +14010,7 @@ al --R Type: AssociationList(String,Record(monthsOld: Integer,gender: String)) --E 8 ---S 9 of 10 +--S 9 of 11 al."katie" := [23,"female"]$Data --R --R @@ -13961,7 +14018,7 @@ al."katie" := [23,"female"]$Data --R Type: Record(monthsOld: Integer,gender: String) --E 9 ---S 10 of 10 +--S 10 of 11 delete!(al,1) --R --R @@ -13974,6 +14031,197 @@ delete!(al,1) --R "bob"= [monthsOld= 407,gender= "male"] --R Type: AssociationList(String,Record(monthsOld: Integer,gender: String)) --E 10 + +--S 11 of 11 +)show AssociationList +--R +--R AssociationList(Key: SetCategory,Entry: SetCategory) is a domain constructor +--R Abbreviation for AssociationList is ALIST +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for ALIST +--R +--R------------------------------- Operations -------------------------------- +--R children : % -> List(%) concat : (%,%) -> % +--R concat : List(%) -> % concat! : (%,%) -> % +--R copy : % -> % cycleEntry : % -> % +--R cycleTail : % -> % cyclic? : % -> Boolean +--R delete : (%,Integer) -> % delete! : (%,Integer) -> % +--R dictionary : () -> % distance : (%,%) -> Integer +--R ?.rest : (%,rest) -> % elt : (%,Key,Entry) -> Entry +--R ?.? : (%,Key) -> Entry empty : () -> % +--R empty? : % -> Boolean entries : % -> List(Entry) +--R eq? : (%,%) -> Boolean explicitlyFinite? : % -> Boolean +--R index? : (Integer,%) -> Boolean index? : (Key,%) -> Boolean +--R indices : % -> List(Integer) indices : % -> List(Key) +--R insert : (%,%,Integer) -> % insert! : (%,%,Integer) -> % +--R key? : (Key,%) -> Boolean keys : % -> List(Key) +--R last : (%,NonNegativeInteger) -> % leaf? : % -> Boolean +--R map : ((Entry -> Entry),%) -> % nodes : % -> List(%) +--R possiblyInfinite? : % -> Boolean qelt : (%,Key) -> Entry +--R rest : % -> % rest : (%,NonNegativeInteger) -> % +--R reverse : % -> % sample : () -> % +--R setelt : (%,Key,Entry) -> Entry table : () -> % +--R tail : % -> % +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ? Boolean if Record(key: Key,entry: Entry) has ORDSET +--R ?<=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has ORDSET +--R ?=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R ?>? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has ORDSET +--R ?>=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has ORDSET +--R any? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R any? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R any? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R assoc : (Key,%) -> Union(Record(key: Key,entry: Entry),"failed") +--R bag : List(Record(key: Key,entry: Entry)) -> % +--R child? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT +--R coerce : % -> OutputForm if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R concat : (Record(key: Key,entry: Entry),%) -> % +--R concat : (%,Record(key: Key,entry: Entry)) -> % +--R concat! : (%,Record(key: Key,entry: Entry)) -> % +--R construct : List(Record(key: Key,entry: Entry)) -> % +--R construct : List(Record(key: Key,entry: Entry)) -> % +--R convert : % -> InputForm if Record(key: Key,entry: Entry) has KONVERT(INFORM) +--R copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable +--R count : ((Record(key: Key,entry: Entry) -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Record(key: Key,entry: Entry),%) -> NonNegativeInteger if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R count : ((Entry -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Entry,%) -> NonNegativeInteger if $ has finiteAggregate and Entry has SETCAT +--R count : (Record(key: Key,entry: Entry),%) -> NonNegativeInteger if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R count : ((Record(key: Key,entry: Entry) -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R cycleLength : % -> NonNegativeInteger +--R cycleSplit! : % -> % if $ has shallowlyMutable +--R delete : (%,UniversalSegment(Integer)) -> % +--R delete! : (%,UniversalSegment(Integer)) -> % +--R dictionary : List(Record(key: Key,entry: Entry)) -> % +--R ?.value : (%,value) -> Record(key: Key,entry: Entry) +--R ?.first : (%,first) -> Record(key: Key,entry: Entry) +--R ?.last : (%,last) -> Record(key: Key,entry: Entry) +--R ?.? : (%,UniversalSegment(Integer)) -> % +--R ?.? : (%,Integer) -> Record(key: Key,entry: Entry) +--R elt : (%,Integer,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) +--R entries : % -> List(Record(key: Key,entry: Entry)) +--R entry? : (Record(key: Key,entry: Entry),%) -> Boolean if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R entry? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R eval : (%,List(Equation(Record(key: Key,entry: Entry)))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Equation(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,List(Record(key: Key,entry: Entry)),List(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,List(Equation(Entry))) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Equation(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Entry,Entry) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Entry),List(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Record(key: Key,entry: Entry)),List(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Equation(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,List(Equation(Record(key: Key,entry: Entry)))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R every? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R every? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R every? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R extract! : % -> Record(key: Key,entry: Entry) +--R fill! : (%,Record(key: Key,entry: Entry)) -> % if $ has shallowlyMutable +--R fill! : (%,Entry) -> % if $ has shallowlyMutable +--R find : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Union(Record(key: Key,entry: Entry),"failed") +--R find : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Union(Record(key: Key,entry: Entry),"failed") +--R first : % -> Record(key: Key,entry: Entry) +--R first : (%,NonNegativeInteger) -> % +--R first : % -> Entry if Key has ORDSET +--R hash : % -> SingleInteger if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R insert : (Record(key: Key,entry: Entry),%,Integer) -> % +--R insert! : (Record(key: Key,entry: Entry),%,Integer) -> % +--R insert! : (Record(key: Key,entry: Entry),%) -> % +--R inspect : % -> Record(key: Key,entry: Entry) +--R last : % -> Record(key: Key,entry: Entry) +--R latex : % -> String if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R leaves : % -> List(Record(key: Key,entry: Entry)) +--R less? : (%,NonNegativeInteger) -> Boolean +--R list : Record(key: Key,entry: Entry) -> % +--R map : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % +--R map : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,%) -> % +--R map : (((Entry,Entry) -> Entry),%,%) -> % +--R map : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % +--R map! : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % if $ has shallowlyMutable +--R map! : ((Entry -> Entry),%) -> % if $ has shallowlyMutable +--R map! : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % if $ has shallowlyMutable +--R max : (%,%) -> % if Record(key: Key,entry: Entry) has ORDSET +--R maxIndex : % -> Integer if Integer has ORDSET +--R maxIndex : % -> Key if Key has ORDSET +--R member? : (Record(key: Key,entry: Entry),%) -> Boolean if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R member? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R member? : (Record(key: Key,entry: Entry),%) -> Boolean if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R members : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R members : % -> List(Entry) if $ has finiteAggregate +--R members : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R merge : (%,%) -> % if Record(key: Key,entry: Entry) has ORDSET +--R merge : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Boolean),%,%) -> % +--R merge! : (%,%) -> % if Record(key: Key,entry: Entry) has ORDSET +--R merge! : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Boolean),%,%) -> % +--R min : (%,%) -> % if Record(key: Key,entry: Entry) has ORDSET +--R minIndex : % -> Integer if Integer has ORDSET +--R minIndex : % -> Key if Key has ORDSET +--R more? : (%,NonNegativeInteger) -> Boolean +--R new : (NonNegativeInteger,Record(key: Key,entry: Entry)) -> % +--R node? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT +--R parts : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R parts : % -> List(Entry) if $ has finiteAggregate +--R parts : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R position : (Record(key: Key,entry: Entry),%,Integer) -> Integer if Record(key: Key,entry: Entry) has SETCAT +--R position : (Record(key: Key,entry: Entry),%) -> Integer if Record(key: Key,entry: Entry) has SETCAT +--R position : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Integer +--R qelt : (%,Integer) -> Record(key: Key,entry: Entry) +--R qsetelt! : (%,Integer,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R qsetelt! : (%,Key,Entry) -> Entry if $ has shallowlyMutable +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove! : (Record(key: Key,entry: Entry),%) -> % if Record(key: Key,entry: Entry) has SETCAT +--R remove! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % +--R remove! : (Key,%) -> Union(Entry,"failed") +--R remove! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove! : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate +--R removeDuplicates : % -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R removeDuplicates! : % -> % if Record(key: Key,entry: Entry) has SETCAT +--R reverse! : % -> % if $ has shallowlyMutable +--R search : (Key,%) -> Union(Entry,"failed") +--R second : % -> Record(key: Key,entry: Entry) +--R select : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R select : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R select! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % +--R select! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable +--R setelt : (%,value,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setelt : (%,first,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setelt : (%,rest,%) -> % if $ has shallowlyMutable +--R setelt : (%,last,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setelt : (%,UniversalSegment(Integer),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setelt : (%,Integer,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setfirst! : (%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setlast! : (%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R setrest! : (%,%) -> % if $ has shallowlyMutable +--R setvalue! : (%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has shallowlyMutable +--R size? : (%,NonNegativeInteger) -> Boolean +--R sort : % -> % if Record(key: Key,entry: Entry) has ORDSET +--R sort : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Boolean),%) -> % +--R sort! : % -> % if $ has shallowlyMutable and Record(key: Key,entry: Entry) has ORDSET +--R sort! : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Boolean),%) -> % if $ has shallowlyMutable +--R sorted? : % -> Boolean if Record(key: Key,entry: Entry) has ORDSET +--R sorted? : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Boolean),%) -> Boolean +--R split! : (%,Integer) -> % if $ has shallowlyMutable +--R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable +--R swap! : (%,Key,Key) -> Void if $ has shallowlyMutable +--R table : List(Record(key: Key,entry: Entry)) -> % +--R third : % -> Record(key: Key,entry: Entry) +--R value : % -> Record(key: Key,entry: Entry) +--R ?~=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R +--E 11 + )spool )lisp (bye) \end{chunk} @@ -14282,6 +14530,7 @@ AssociationList(Key:SetCategory, Entry:SetCategory): --S 1 of 1 )show AttributeButtons +--R --R AttributeButtons is a domain constructor --R Abbreviation for AttributeButtons is ATTRBUT --R This constructor is exposed in this frame. @@ -14289,12 +14538,11 @@ AssociationList(Key:SetCategory, Entry:SetCategory): --R --R------------------------------- Operations -------------------------------- --R ?=? : (%,%) -> Boolean coerce : % -> OutputForm ---R decrease : String -> Float hash : % -> SingleInteger ---R increase : String -> Float latex : % -> String +--R decrease : String -> Float decrease : (String,String) -> Float +--R hash : % -> SingleInteger increase : String -> Float +--R increase : (String,String) -> Float latex : % -> String --R resetAttributeButtons : () -> Void ?~=? : (%,%) -> Boolean ---R decrease : (String,String) -> Float --R getButtonValue : (String,String) -> Float ---R increase : (String,String) -> Float --R setAttributeButtonStep : Float -> Float --R setButtonValue : (String,String,Float) -> Float --R setButtonValue : (String,Float) -> Float @@ -14588,19 +14836,17 @@ AttributeButtons(): E == I where --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?/? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R coerce : % -> OutputForm commutator : (%,%) -> % --R conjugate : (%,%) -> % ?.? : (%,R) -> R --R hash : % -> SingleInteger inv : % -> % ---R latex : % -> String morphism : (R -> R) -> % +--R latex : % -> String morphism : ((R,Integer) -> R) -> % +--R morphism : ((R -> R),(R -> R)) -> % morphism : (R -> R) -> % --R one? : % -> Boolean recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % ---R morphism : ((R,Integer) -> R) -> % ---R morphism : ((R -> R),(R -> R)) -> % --R --E 1 @@ -14725,34 +14971,119 @@ Automorphism(R:Ring): Join(Group, Eltable(R, R)) with )set message test on )set message auto off )clear all ---S 1 of 7 + +--S 1 of 8 lm := [3,5,7,11] +--R +--R +--R (1) [3,5,7,11] +--R Type: List(PositiveInteger) --E 1 ---S 2 of 7 +--S 2 of 8 t := balancedBinaryTree(#lm, 0) +--R +--R +--R (2) [[0,0,0],0,[0,0,0]] +--R Type: BalancedBinaryTree(NonNegativeInteger) --E 2 ---S 3 of 7 +--S 3 of 8 setleaves!(t,lm) +--R +--R +--R (3) [[3,0,5],0,[7,0,11]] +--R Type: BalancedBinaryTree(NonNegativeInteger) --E 3 ---S 4 of 7 +--S 4 of 8 mapUp!(t,_*) +--R +--R +--R (4) 1155 +--R Type: PositiveInteger --E 4 ---S 5 of 7 +--S 5 of 8 t +--R +--R +--R (5) [[3,15,5],1155,[7,77,11]] +--R Type: BalancedBinaryTree(NonNegativeInteger) --E 5 ---S 6 of 7 +--S 6 of 8 mapDown!(t,12,_rem) +--R +--R +--R (6) [[0,12,2],12,[5,12,1]] +--R Type: BalancedBinaryTree(NonNegativeInteger) --E 6 ---S 7 of 7 +--S 7 of 8 leaves % +--R +--R +--R (7) [0,2,5,1] +--R Type: List(NonNegativeInteger) --E 7 +--S 8 of 8 +)show BalancedBinaryTree +--R +--R BalancedBinaryTree(S: SetCategory) is a domain constructor +--R Abbreviation for BalancedBinaryTree is BBTREE +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for BBTREE +--R +--R------------------------------- Operations -------------------------------- +--R children : % -> List(%) copy : % -> % +--R cyclic? : % -> Boolean distance : (%,%) -> Integer +--R ?.right : (%,right) -> % ?.left : (%,left) -> % +--R ?.value : (%,value) -> S empty : () -> % +--R empty? : % -> Boolean eq? : (%,%) -> Boolean +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) left : % -> % +--R map : ((S -> S),%) -> % mapDown! : (%,S,((S,S) -> S)) -> % +--R mapUp! : (%,((S,S) -> S)) -> S node : (%,S,%) -> % +--R nodes : % -> List(%) right : % -> % +--R sample : () -> % setleaves! : (%,List(S)) -> % +--R value : % -> S +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ?=? : (%,%) -> Boolean if S has SETCAT +--R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate +--R balancedBinaryTree : (NonNegativeInteger,S) -> % +--R child? : (%,%) -> Boolean if S has SETCAT +--R coerce : % -> OutputForm if S has SETCAT +--R count : (S,%) -> NonNegativeInteger if $ has finiteAggregate and S has SETCAT +--R count : ((S -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,S,S) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,Equation(S)) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT +--R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate +--R hash : % -> SingleInteger if S has SETCAT +--R less? : (%,NonNegativeInteger) -> Boolean +--R map! : ((S -> S),%) -> % if $ has shallowlyMutable +--R mapDown! : (%,S,((S,S,S) -> List(S))) -> % +--R mapUp! : (%,%,((S,S,S,S) -> S)) -> % +--R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT +--R members : % -> List(S) if $ has finiteAggregate +--R more? : (%,NonNegativeInteger) -> Boolean +--R node? : (%,%) -> Boolean if S has SETCAT +--R parts : % -> List(S) if $ has finiteAggregate +--R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable +--R setelt : (%,right,%) -> % if $ has shallowlyMutable +--R setelt : (%,left,%) -> % if $ has shallowlyMutable +--R setelt : (%,value,S) -> S if $ has shallowlyMutable +--R setleft! : (%,%) -> % if $ has shallowlyMutable +--R setright! : (%,%) -> % if $ has shallowlyMutable +--R setvalue! : (%,S) -> S if $ has shallowlyMutable +--R size? : (%,NonNegativeInteger) -> Boolean +--R ?~=? : (%,%) -> Boolean if S has SETCAT +--R +--E 8 + )spool )lisp (bye) \end{chunk} @@ -15107,28 +15438,26 @@ BalancedBinaryTree(S: SetCategory): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm complete : % -> % ---R digits : % -> Stream(Integer) extend : (%,Integer) -> % ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R moduloP : % -> Integer modulus : () -> Integer ---R one? : % -> Boolean order : % -> NonNegativeInteger ---R ?quo? : (%,%) -> % quotientByP : % -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R sqrt : (%,Integer) -> % unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complete : % -> % digits : % -> Stream(Integer) +--R extend : (%,Integer) -> % gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % moduloP : % -> Integer +--R modulus : () -> Integer one? : % -> Boolean +--R order : % -> NonNegativeInteger ?quo? : (%,%) -> % +--R quotientByP : % -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean sqrt : (%,Integer) -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R approximate : (%,Integer) -> Integer --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) @@ -15262,31 +15591,31 @@ BalancedPAdicInteger(p:Integer) == InnerPAdicInteger(p,false$Boolean) --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm denominator : % -> % ---R factor : % -> Factored(%) gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R numerator : % -> % one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R removeZeroes : (Integer,%) -> % removeZeroes : % -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R denominator : % -> % factor : % -> Factored(%) +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % numerator : % -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % removeZeroes : (Integer,%) -> % +--R removeZeroes : % -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,BalancedPAdicInteger(p)) -> % --R ?*? : (BalancedPAdicInteger(p),%) -> % ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (BalancedPAdicInteger(p),BalancedPAdicInteger(p)) -> % --R ? Boolean if BalancedPAdicInteger(p) has ORDSET --R ?<=? : (%,%) -> Boolean if BalancedPAdicInteger(p) has ORDSET @@ -15300,7 +15629,6 @@ BalancedPAdicInteger(p:Integer) == InnerPAdicInteger(p,false$Boolean) --R D : (%,Symbol) -> % if BalancedPAdicInteger(p) has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if BalancedPAdicInteger(p) has DIFRING --R D : % -> % if BalancedPAdicInteger(p) has DIFRING ---R ?^? : (%,NonNegativeInteger) -> % --R abs : % -> % if BalancedPAdicInteger(p) has OINTDOM --R approximate : (%,Integer) -> Fraction(Integer) --R ceiling : % -> BalancedPAdicInteger(p) if BalancedPAdicInteger(p) has INS @@ -15646,7 +15974,7 @@ BasicFunctions(): E == I where )set message test on )set message auto off )clear all ---S 1 of 18 +--S 1 of 19 y := operator 'y --R --R @@ -15654,7 +15982,7 @@ y := operator 'y --R Type: BasicOperator --E 1 ---S 2 of 18 +--S 2 of 19 deq := D(y x, x, 2) + D(y x, x) + y x = 0 --R --R @@ -15664,7 +15992,7 @@ deq := D(y x, x, 2) + D(y x, x) + y x = 0 --R Type: Equation(Expression(Integer)) --E 2 ---S 3 of 18 +--S 3 of 19 solve(deq, y, x) --R --R @@ -15676,7 +16004,7 @@ solve(deq, y, x) --IType: Union(Record(particular: Expression(Integer),basis: ... --E 3 ---S 4 of 18 +--S 4 of 19 nary? y --R --R @@ -15684,7 +16012,7 @@ nary? y --R Type: Boolean --E 4 ---S 5 of 18 +--S 5 of 19 unary? y --R --R @@ -15692,7 +16020,7 @@ unary? y --R Type: Boolean --E 5 ---S 6 of 18 +--S 6 of 19 opOne := operator('opOne, 1) --R --R @@ -15700,7 +16028,7 @@ opOne := operator('opOne, 1) --R Type: BasicOperator --E 6 ---S 7 of 18 +--S 7 of 19 nary? opOne --R --R @@ -15708,7 +16036,7 @@ nary? opOne --R Type: Boolean --E 7 ---S 8 of 18 +--S 8 of 19 unary? opOne --R --R @@ -15716,7 +16044,7 @@ unary? opOne --R Type: Boolean --E 8 ---S 9 of 18 +--S 9 of 19 arity opOne --R --R @@ -15724,7 +16052,7 @@ arity opOne --R Type: Union(NonNegativeInteger,...) --E 9 ---S 10 of 18 +--S 10 of 19 name opOne --R --R @@ -15732,7 +16060,7 @@ name opOne --R Type: Symbol --E 10 ---S 11 of 18 +--S 11 of 19 is?(opOne, 'z2) --R --R @@ -15740,7 +16068,7 @@ is?(opOne, 'z2) --R Type: Boolean --E 11 ---S 12 of 18 +--S 12 of 19 is?(opOne, "opOne") --R --R @@ -15748,7 +16076,7 @@ is?(opOne, "opOne") --R Type: Boolean --E 12 ---S 13 of 18 +--S 13 of 19 properties y --R --R @@ -15756,7 +16084,7 @@ properties y --R Type: AssociationList(String,None) --E 13 ---S 14 of 18 +--S 14 of 19 setProperty(y, "use", "unknown function" :: None ) --R --R @@ -15764,7 +16092,7 @@ setProperty(y, "use", "unknown function" :: None ) --R Type: BasicOperator --E 14 ---S 15 of 18 +--S 15 of 19 properties y --R --R @@ -15772,7 +16100,7 @@ properties y --R Type: AssociationList(String,None) --E 15 ---S 16 of 18 +--S 16 of 19 property(y, "use") :: None pretend String --R --R @@ -15780,7 +16108,7 @@ property(y, "use") :: None pretend String --R Type: String --E 16 ---S 17 of 18 +--S 17 of 19 deleteProperty!(y, "use") --R --R @@ -15788,13 +16116,51 @@ deleteProperty!(y, "use") --R Type: BasicOperator --E 17 ---S 18 of 18 +--S 18 of 19 properties y --R --R --R (18) table() --R Type: AssociationList(String,None) --E 18 + +--S 19 of 19 +)show BasicOperator +--R +--R BasicOperator is a domain constructor +--R Abbreviation for BasicOperator is BOP +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for BOP +--R +--R------------------------------- Operations -------------------------------- +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean assert : (%,String) -> % +--R coerce : % -> OutputForm copy : % -> % +--R deleteProperty! : (%,String) -> % has? : (%,String) -> Boolean +--R hash : % -> SingleInteger is? : (%,Symbol) -> Boolean +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % name : % -> Symbol +--R nary? : % -> Boolean nullary? : % -> Boolean +--R operator : Symbol -> % setProperty : (%,String,None) -> % +--R unary? : % -> Boolean weight : % -> NonNegativeInteger +--R ?~=? : (%,%) -> Boolean +--R arity : % -> Union(NonNegativeInteger,"failed") +--R comparison : (%,((%,%) -> Boolean)) -> % +--R display : (%,(OutputForm -> OutputForm)) -> % +--R display : (%,(List(OutputForm) -> OutputForm)) -> % +--R display : % -> Union((List(OutputForm) -> OutputForm),"failed") +--R equality : (%,((%,%) -> Boolean)) -> % +--R input : % -> Union((List(InputForm) -> InputForm),"failed") +--R input : (%,(List(InputForm) -> InputForm)) -> % +--R operator : (Symbol,NonNegativeInteger) -> % +--R properties : % -> AssociationList(String,None) +--R property : (%,String) -> Union(None,"failed") +--R setProperties : (%,AssociationList(String,None)) -> % +--R weight : (%,NonNegativeInteger) -> % +--R +--E 19 + )spool )lisp (bye) \end{chunk} @@ -16183,21 +16549,21 @@ BasicOperator(): Exports == Implementation where )set message auto off )clear all ---S 1 of 30 +--S 1 of 31 print copyBSD() --R --R [] --R Type: Void --E 1 ---S 2 of 30 +--S 2 of 31 print copyIto() --R --R table() --R Type: Void --E 2 ---S 3 of 30 +--S 3 of 31 dX:=introduce!(X,dX) -- dX is a new stochastic differential --R --R @@ -16205,21 +16571,21 @@ dX:=introduce!(X,dX) -- dX is a new stochastic differential --R Type: Union(BasicStochasticDifferential,...) --E 3 ---S 4 of 30 +--S 4 of 31 print copyBSD() --R --R [dX] --R Type: Void --E 4 ---S 5 of 30 +--S 5 of 31 print copyIto() --R --R table(X= dX) --R Type: Void --E 5 ---S 6 of 30 +--S 6 of 31 getSmgl(dX) --R --R @@ -16227,42 +16593,42 @@ getSmgl(dX) --R Type: Union(Symbol,...) --E 6 ---S 7 of 30 +--S 7 of 31 print dX --R --R dX --R Type: Void --E 7 ---S 8 of 30 +--S 8 of 31 print domainOf(dX) --R --R BasicStochasticDifferential() --R Type: Void --E 8 ---S 9 of 30 +--S 9 of 31 print d X --R --R dX --R Type: Void --E 9 ---S 10 of 30 +--S 10 of 31 print ((d$BSD) X) --R --R dX --R Type: Void --E 10 ---S 11 of 30 +--S 11 of 31 print domainOf((d$BSD) X) --R --R BasicStochasticDifferential() --R Type: Void --E 11 ---S 12 of 30 +--S 12 of 31 introduce!(t,dt) -- dt is a new stochastic differential --R --R @@ -16270,21 +16636,21 @@ introduce!(t,dt) -- dt is a new stochastic differential --R Type: Union(BasicStochasticDifferential,...) --E 12 ---S 13 of 30 +--S 13 of 31 print copyBSD() --R --R [dX,dt] --R Type: Void --E 13 ---S 14 of 30 +--S 14 of 31 print copyIto() --R --R table(t= dt,X= dX) --R Type: Void --E 14 ---S 15 of 30 +--S 15 of 31 introduce!(t,dt) -- fails (can't repeat an introduction!) --R --R @@ -16292,21 +16658,21 @@ introduce!(t,dt) -- fails (can't repeat an introduction!) --R Type: Union("failed",...) --E 15 ---S 16 of 30 +--S 16 of 31 print copyBSD() --R --R [dX,dt] --R Type: Void --E 16 ---S 17 of 30 +--S 17 of 31 print copyIto() --R --R table(t= dt,X= dX) --R Type: Void --E 17 ---S 18 of 30 +--S 18 of 31 'dZ::BSD -- can fail --R --R dZ is not a stochastic differential @@ -16319,7 +16685,7 @@ print copyIto() --R --E 18 ---S 19 of 30 +--S 19 of 31 dZ::BSD -- fails --R --R dZ is not a stochastic differential @@ -16332,7 +16698,7 @@ dZ::BSD -- fails --R --E 19 ---S 20 of 30 +--S 20 of 31 string(dY)::BSD -- fails --R --R @@ -16343,14 +16709,14 @@ string(dY)::BSD -- fails --R --E 20 ---S 21 of 30 +--S 21 of 31 print domainOf(dt) --R --R Variable(dt) --R Type: Void --E 21 ---S 22 of 30 +--S 22 of 31 dt::BSD -- succeeds! --R --R @@ -16358,7 +16724,7 @@ dt::BSD -- succeeds! --R Type: BasicStochasticDifferential --E 22 ---S 23 of 30 +--S 23 of 31 getSmgl(dt::BSD) --R --R @@ -16366,7 +16732,7 @@ getSmgl(dt::BSD) --R Type: Union(Symbol,...) --E 23 ---S 24 of 30 +--S 24 of 31 dX::Symbol -- succeeds --R --R @@ -16374,7 +16740,7 @@ dX::Symbol -- succeeds --R Type: Symbol --E 24 ---S 25 of 30 +--S 25 of 31 string(dX) -- fails --R --R There are 3 exposed and 1 unexposed library operations named string @@ -16393,7 +16759,7 @@ string(dX) -- fails --R $ to specify which version of the function you need. --E 25 ---S 26 of 30 +--S 26 of 31 string(dX::Symbol) -- succeeds --R --R @@ -16401,7 +16767,7 @@ string(dX::Symbol) -- succeeds --R Type: String --E 26 ---S 27 of 30 +--S 27 of 31 [introduce!(A[i],dA[i]) for i in 1..2] --R --R @@ -16410,7 +16776,7 @@ string(dX::Symbol) -- succeeds --R Type: List(Union(BasicStochasticDifferential,"failed")) --E 27 ---S 28 of 30 +--S 28 of 31 print copyBSD() --R --R [dA ,dA ,dX,dt] @@ -16418,7 +16784,7 @@ print copyBSD() --R Type: Void --E 28 ---S 29 of 30 +--S 29 of 31 print copyIto() --R --R table(A = dA ,A = dA ,t= dt,X= dX) @@ -16426,7 +16792,7 @@ print copyIto() --R Type: Void --E 29 ---S 30 of 30 +--S 30 of 31 [d A[i] for i in 1..2] --R --R @@ -16435,6 +16801,28 @@ print copyIto() --R Type: List(Union(BasicStochasticDifferential,Integer)) --E 30 +--S 31 of 31 +)show BasicStochasticDifferential +--R +--R BasicStochasticDifferential is a domain constructor +--R Abbreviation for BasicStochasticDifferential is BSD +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for BSD +--R +--R------------------------------- Operations -------------------------------- +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean coerce : % -> OutputForm +--R convert : % -> Symbol convert : Symbol -> % +--R copyBSD : () -> List(%) copyIto : () -> Table(Symbol,%) +--R d : Symbol -> Union(%,Integer) hash : % -> SingleInteger +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % ?~=? : (%,%) -> Boolean +--R convertIfCan : Symbol -> Union(%,"failed") +--R getSmgl : % -> Union(Symbol,"failed") +--R introduce! : (Symbol,Symbol) -> Union(%,"failed") +--R +--E 31 )spool )lisp (bye) @@ -16678,7 +17066,8 @@ BasicStochasticDifferential(): category == implementation where )set message test on )set message auto off )clear all ---S 1 of 7 + +--S 1 of 8 r := binary(22/7) --R --R @@ -16687,7 +17076,7 @@ r := binary(22/7) --R Type: BinaryExpansion --E 1 ---S 2 of 7 +--S 2 of 8 r + binary(6/7) --R --R @@ -16695,7 +17084,7 @@ r + binary(6/7) --R Type: BinaryExpansion --E 2 ---S 3 of 7 +--S 3 of 8 [binary(1/i) for i in 102..106] --R --R @@ -16709,7 +17098,7 @@ r + binary(6/7) --R Type: List(BinaryExpansion) --E 3 ---S 4 of 7 +--S 4 of 8 binary(1/1007) --R --R @@ -16726,7 +17115,7 @@ binary(1/1007) --R Type: BinaryExpansion --E 4 ---S 5 of 7 +--S 5 of 8 p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9) --R --R @@ -16735,7 +17124,7 @@ p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9) --R Type: Polynomial(BinaryExpansion) --E 5 ---S 6 of 7 +--S 6 of 8 q := D(p, x) --R --R @@ -16744,7 +17133,7 @@ q := D(p, x) --R Type: Polynomial(BinaryExpansion) --E 6 ---S 7 of 7 +--S 7 of 8 g := gcd(p, q) --R --R @@ -16752,6 +17141,126 @@ g := gcd(p, q) --R (7) x + 1.01 --R Type: Polynomial(BinaryExpansion) --E 7 + +--S 8 of 8 +)show BinaryExpansion +--R +--R BinaryExpansion is a domain constructor +--R Abbreviation for BinaryExpansion is BINARY +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for BINARY +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (Integer,Integer) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean D : (%,(Integer -> Integer)) -> % +--R D : % -> % if Integer has DIFRING 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean binary : Fraction(Integer) -> % +--R coerce : % -> RadixExpansion(2) coerce : % -> Fraction(Integer) +--R coerce : Integer -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm denom : % -> Integer +--R denominator : % -> % factor : % -> Factored(%) +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger init : () -> % if Integer has STEP +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R numer : % -> Integer numerator : % -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % random : () -> % if Integer has INS +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R retract : % -> Integer sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R ? Boolean if Integer has ORDSET +--R ?<=? : (%,%) -> Boolean if Integer has ORDSET +--R ?>? : (%,%) -> Boolean if Integer has ORDSET +--R ?>=? : (%,%) -> Boolean if Integer has ORDSET +--R D : (%,(Integer -> Integer),NonNegativeInteger) -> % +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) +--R D : (%,Symbol) -> % if Integer has PDRING(SYMBOL) +--R D : (%,NonNegativeInteger) -> % if Integer has DIFRING +--R abs : % -> % if Integer has OINTDOM +--R ceiling : % -> Integer if Integer has INS +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and Integer has PFECAT or Integer has CHARNZ +--R coerce : Symbol -> % if Integer has RETRACT(SYMBOL) +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and Integer has PFECAT +--R convert : % -> DoubleFloat if Integer has REAL +--R convert : % -> Float if Integer has REAL +--R convert : % -> InputForm if Integer has KONVERT(INFORM) +--R convert : % -> Pattern(Float) if Integer has KONVERT(PATTERN(FLOAT)) +--R convert : % -> Pattern(Integer) if Integer has KONVERT(PATTERN(INT)) +--R differentiate : (%,(Integer -> Integer)) -> % +--R differentiate : (%,(Integer -> Integer),NonNegativeInteger) -> % +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,Symbol) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,NonNegativeInteger) -> % if Integer has DIFRING +--R differentiate : % -> % if Integer has DIFRING +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R ?.? : (%,Integer) -> % if Integer has ELTAB(INT,INT) +--R euclideanSize : % -> NonNegativeInteger +--R eval : (%,Symbol,Integer) -> % if Integer has IEVALAB(SYMBOL,INT) +--R eval : (%,List(Symbol),List(Integer)) -> % if Integer has IEVALAB(SYMBOL,INT) +--R eval : (%,List(Equation(Integer))) -> % if Integer has EVALAB(INT) +--R eval : (%,Equation(Integer)) -> % if Integer has EVALAB(INT) +--R eval : (%,Integer,Integer) -> % if Integer has EVALAB(INT) +--R eval : (%,List(Integer),List(Integer)) -> % if Integer has EVALAB(INT) +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R floor : % -> Integer if Integer has INS +--R fractionPart : % -> Fraction(Integer) +--R fractionPart : % -> % if Integer has EUCDOM +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R map : ((Integer -> Integer),%) -> % +--R max : (%,%) -> % if Integer has ORDSET +--R min : (%,%) -> % if Integer has ORDSET +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R negative? : % -> Boolean if Integer has OINTDOM +--R nextItem : % -> Union(%,"failed") if Integer has STEP +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if Integer has PATMAB(FLOAT) +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if Integer has PATMAB(INT) +--R positive? : % -> Boolean if Integer has OINTDOM +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R reducedSystem : Matrix(%) -> Matrix(Integer) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if Integer has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if Integer has LINEXP(INT) +--R retract : % -> Integer if Integer has RETRACT(INT) +--R retract : % -> Fraction(Integer) if Integer has RETRACT(INT) +--R retract : % -> Symbol if Integer has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(Integer,"failed") if Integer has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if Integer has RETRACT(INT) +--R retractIfCan : % -> Union(Symbol,"failed") if Integer has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(Integer,"failed") +--R sign : % -> Integer if Integer has OINTDOM +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if Integer has PFECAT +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R wholePart : % -> Integer if Integer has EUCDOM +--R +--E 8 + )spool )lisp (bye) \end{chunk} @@ -16971,6 +17480,7 @@ BinaryExpansion(): Exports == Implementation where --S 1 of 1 )show BinaryFile +--R --R BinaryFile is a domain constructor --R Abbreviation for BinaryFile is BINFILE --R This constructor is exposed in this frame. @@ -17147,7 +17657,8 @@ BinaryFile: Cat == Def where )set message test on )set message auto off )clear all ---S 1 of 12 + +--S 1 of 13 lv := [8,3,5,4,6,2,1,5,7] --R --R @@ -17155,7 +17666,7 @@ lv := [8,3,5,4,6,2,1,5,7] --R Type: List(PositiveInteger) --E 1 ---S 2 of 12 +--S 2 of 13 t := binarySearchTree lv --R --R @@ -17163,7 +17674,7 @@ t := binarySearchTree lv --R Type: BinarySearchTree(PositiveInteger) --E 2 ---S 3 of 12 +--S 3 of 13 emptybst := empty()$BSTREE(INT) --R --R @@ -17171,7 +17682,7 @@ emptybst := empty()$BSTREE(INT) --R Type: BinarySearchTree(Integer) --E 3 ---S 4 of 12 +--S 4 of 13 t1 := insert!(8,emptybst) --R --R @@ -17179,7 +17690,7 @@ t1 := insert!(8,emptybst) --R Type: BinarySearchTree(Integer) --E 4 ---S 5 of 12 +--S 5 of 13 insert!(3,t1) --R --R @@ -17187,7 +17698,7 @@ insert!(3,t1) --R Type: BinarySearchTree(Integer) --E 5 ---S 6 of 12 +--S 6 of 13 leaves t --R --R @@ -17195,7 +17706,7 @@ leaves t --R Type: List(PositiveInteger) --E 6 ---S 7 of 12 +--S 7 of 13 split(3,t) --R --R @@ -17203,13 +17714,13 @@ split(3,t) --IType: Record(less: BinarySearchTree(PositiveInteger),greater: ... --E 7 ---S 8 of 12 +--S 8 of 13 insertRoot: (INT,BSTREE INT) -> BSTREE INT --R --R Type: Void --E 8 ---S 9 of 12 +--S 9 of 13 insertRoot(x, t) == a := split(x, t) node(a.less, x, a.greater) @@ -17217,13 +17728,13 @@ insertRoot(x, t) == --R Type: Void --E 9 ---S 10 of 12 +--S 10 of 13 buildFromRoot ls == reduce(insertRoot,ls,emptybst) --R --R Type: Void --E 10 ---S 11 of 12 +--S 11 of 13 rt := buildFromRoot reverse lv --R --R Compiling function buildFromRoot with type List(PositiveInteger) -> @@ -17235,13 +17746,68 @@ rt := buildFromRoot reverse lv --R Type: BinarySearchTree(Integer) --E 11 ---S 12 of 12 +--S 12 of 13 (t = rt)@Boolean --R --R --R (12) true --R Type: Boolean --E 12 + +--S 13 of 13 +)show BinarySearchTree +--R +--R BinarySearchTree(S: OrderedSet) is a domain constructor +--R Abbreviation for BinarySearchTree is BSTREE +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for BSTREE +--R +--R------------------------------- Operations -------------------------------- +--R binarySearchTree : List(S) -> % children : % -> List(%) +--R copy : % -> % cyclic? : % -> Boolean +--R distance : (%,%) -> Integer ?.right : (%,right) -> % +--R ?.left : (%,left) -> % ?.value : (%,value) -> S +--R empty : () -> % empty? : % -> Boolean +--R eq? : (%,%) -> Boolean insert! : (S,%) -> % +--R insertRoot! : (S,%) -> % latex : % -> String if S has SETCAT +--R leaf? : % -> Boolean leaves : % -> List(S) +--R left : % -> % map : ((S -> S),%) -> % +--R node : (%,S,%) -> % nodes : % -> List(%) +--R right : % -> % sample : () -> % +--R value : % -> S +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ?=? : (%,%) -> Boolean if S has SETCAT +--R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate +--R child? : (%,%) -> Boolean if S has SETCAT +--R coerce : % -> OutputForm if S has SETCAT +--R count : (S,%) -> NonNegativeInteger if $ has finiteAggregate and S has SETCAT +--R count : ((S -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,S,S) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,Equation(S)) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT +--R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate +--R hash : % -> SingleInteger if S has SETCAT +--R less? : (%,NonNegativeInteger) -> Boolean +--R map! : ((S -> S),%) -> % if $ has shallowlyMutable +--R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT +--R members : % -> List(S) if $ has finiteAggregate +--R more? : (%,NonNegativeInteger) -> Boolean +--R node? : (%,%) -> Boolean if S has SETCAT +--R parts : % -> List(S) if $ has finiteAggregate +--R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable +--R setelt : (%,right,%) -> % if $ has shallowlyMutable +--R setelt : (%,left,%) -> % if $ has shallowlyMutable +--R setelt : (%,value,S) -> S if $ has shallowlyMutable +--R setleft! : (%,%) -> % if $ has shallowlyMutable +--R setright! : (%,%) -> % if $ has shallowlyMutable +--R setvalue! : (%,S) -> S if $ has shallowlyMutable +--R size? : (%,NonNegativeInteger) -> Boolean +--R split : (S,%) -> Record(less: %,greater: %) +--R ?~=? : (%,%) -> Boolean if S has SETCAT +--R +--E 13 + )spool )lisp (bye) \end{chunk} @@ -17512,11 +18078,11 @@ both BinaryTree(S) --R ?.left : (%,left) -> % ?.value : (%,value) -> S --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean insert! : (S,%) -> % ---R leaf? : % -> Boolean leaves : % -> List(S) ---R left : % -> % map : ((S -> S),%) -> % ---R node : (%,S,%) -> % nodes : % -> List(%) ---R right : % -> % sample : () -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) left : % -> % +--R map : ((S -> S),%) -> % node : (%,S,%) -> % +--R nodes : % -> List(%) right : % -> % +--R sample : () -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -17530,7 +18096,6 @@ both BinaryTree(S) --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -17696,11 +18261,11 @@ BinaryTournament(S: OrderedSet): Exports == Implementation where --R ?.right : (%,right) -> % ?.left : (%,left) -> % --R ?.value : (%,value) -> S empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R left : % -> % map : ((S -> S),%) -> % ---R node : (%,S,%) -> % nodes : % -> List(%) ---R right : % -> % sample : () -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) left : % -> % +--R map : ((S -> S),%) -> % node : (%,S,%) -> % +--R nodes : % -> List(%) right : % -> % +--R sample : () -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -17714,7 +18279,6 @@ BinaryTournament(S: OrderedSet): Exports == Implementation where --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -17903,14 +18467,14 @@ BinaryTree(S: SetCategory): Exports == Implementation where --R empty : () -> % empty? : % -> Boolean --R entries : % -> List(Boolean) eq? : (%,%) -> Boolean --R hash : % -> SingleInteger index? : (Integer,%) -> Boolean ---R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R latex : % -> String max : (%,%) -> % ---R min : (%,%) -> % nand : (%,%) -> % ---R nor : (%,%) -> % not? : % -> % ---R ?or? : (%,%) -> % qelt : (%,Integer) -> Boolean ---R reverse : % -> % sample : () -> % ---R xor : (%,%) -> % ~? : % -> % ---R ?~=? : (%,%) -> Boolean +--R indices : % -> List(Integer) insert : (Boolean,%,Integer) -> % +--R insert : (%,%,Integer) -> % latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R nand : (%,%) -> % nor : (%,%) -> % +--R not? : % -> % ?or? : (%,%) -> % +--R qelt : (%,Integer) -> Boolean reverse : % -> % +--R sample : () -> % xor : (%,%) -> % +--R ~? : % -> % ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R any? : ((Boolean -> Boolean),%) -> Boolean if $ has finiteAggregate --R bits : (NonNegativeInteger,Boolean) -> % @@ -17930,7 +18494,6 @@ BinaryTree(S: SetCategory): Exports == Implementation where --R fill! : (%,Boolean) -> % if $ has shallowlyMutable --R find : ((Boolean -> Boolean),%) -> Union(Boolean,"failed") --R first : % -> Boolean if Integer has ORDSET ---R insert : (Boolean,%,Integer) -> % --R less? : (%,NonNegativeInteger) -> Boolean --R map : ((Boolean -> Boolean),%) -> % --R map : (((Boolean,Boolean) -> Boolean),%,%) -> % @@ -18320,6 +18883,7 @@ BlowUpWithQuadTrans: Exports == Implementation where --S 1 of 1 )show Boolean +--R --R Boolean is a domain constructor --R Abbreviation for Boolean is BOOLEAN --R This constructor is exposed in this frame. @@ -18497,7 +19061,8 @@ Boolean(): Join(OrderedSet, Finite, Logic, ConvertibleTo InputForm) with )set message test on )set message auto off )clear all ---S 1 of 20 + +--S 1 of 21 c0 := 0 :: CardinalNumber --R --R @@ -18505,7 +19070,7 @@ c0 := 0 :: CardinalNumber --R Type: CardinalNumber --E 1 ---S 2 of 20 +--S 2 of 21 c1 := 1 :: CardinalNumber --R --R @@ -18513,7 +19078,7 @@ c1 := 1 :: CardinalNumber --R Type: CardinalNumber --E 2 ---S 3 of 20 +--S 3 of 21 c2 := 2 :: CardinalNumber --R --R @@ -18521,7 +19086,7 @@ c2 := 2 :: CardinalNumber --R Type: CardinalNumber --E 3 ---S 4 of 20 +--S 4 of 21 c3 := 3 :: CardinalNumber --R --R @@ -18529,7 +19094,7 @@ c3 := 3 :: CardinalNumber --R Type: CardinalNumber --E 4 ---S 5 of 20 +--S 5 of 21 A0 := Aleph 0 --R --R @@ -18537,7 +19102,7 @@ A0 := Aleph 0 --R Type: CardinalNumber --E 5 ---S 6 of 20 +--S 6 of 21 A1 := Aleph 1 --R --R @@ -18545,7 +19110,7 @@ A1 := Aleph 1 --R Type: CardinalNumber --E 6 ---S 7 of 20 +--S 7 of 21 finite? c2 --R --R @@ -18553,7 +19118,7 @@ finite? c2 --R Type: Boolean --E 7 ---S 8 of 20 +--S 8 of 21 finite? A0 --R --R @@ -18561,7 +19126,7 @@ finite? A0 --R Type: Boolean --E 8 ---S 9 of 20 +--S 9 of 21 countable? c2 --R --R @@ -18569,7 +19134,7 @@ countable? c2 --R Type: Boolean --E 9 ---S 10 of 20 +--S 10 of 21 countable? A0 --R --R @@ -18577,7 +19142,7 @@ countable? A0 --R Type: Boolean --E 10 ---S 11 of 20 +--S 11 of 21 countable? A1 --R --R @@ -18585,7 +19150,7 @@ countable? A1 --R Type: Boolean --E 11 ---S 12 of 20 +--S 12 of 21 [c2 + c2, c2 + A1] --R --R @@ -18593,7 +19158,7 @@ countable? A1 --R Type: List(CardinalNumber) --E 12 ---S 13 of 20 +--S 13 of 21 [c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1] --R --R @@ -18601,7 +19166,7 @@ countable? A1 --R Type: List(CardinalNumber) --E 13 ---S 14 of 20 +--S 14 of 21 [c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2] --R --R @@ -18609,7 +19174,7 @@ countable? A1 --R Type: List(CardinalNumber) --E 14 ---S 15 of 20 +--S 15 of 21 [c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1] --R --R @@ -18617,7 +19182,7 @@ countable? A1 --R Type: List(Union(CardinalNumber,"failed")) --E 15 ---S 16 of 20 +--S 16 of 21 generalizedContinuumHypothesisAssumed true --R --R @@ -18625,7 +19190,7 @@ generalizedContinuumHypothesisAssumed true --R Type: Boolean --E 16 ---S 17 of 20 +--S 17 of 21 [c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1] --R --R @@ -18633,7 +19198,7 @@ generalizedContinuumHypothesisAssumed true --R Type: List(CardinalNumber) --E 17 ---S 18 of 20 +--S 18 of 21 a := Aleph 0 --R --R @@ -18641,7 +19206,7 @@ a := Aleph 0 --R Type: CardinalNumber --E 18 ---S 19 of 20 +--S 19 of 21 c := 2**a --R --R @@ -18649,13 +19214,45 @@ c := 2**a --R Type: CardinalNumber --E 19 ---S 20 of 20 +--S 20 of 21 f := 2**c --R --R --R (20) Aleph(2) --R Type: CardinalNumber --E 20 + +--S 21 of 21 +)show CardinalNumber +--R +--R CardinalNumber is a domain constructor +--R Abbreviation for CardinalNumber is CARD +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CARD +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?**? : (%,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> Union(%,"failed") +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean Aleph : NonNegativeInteger -> % +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,PositiveInteger) -> % ?^? : (%,NonNegativeInteger) -> % +--R coerce : NonNegativeInteger -> % coerce : % -> OutputForm +--R countable? : % -> Boolean finite? : % -> Boolean +--R hash : % -> SingleInteger latex : % -> String +--R max : (%,%) -> % min : (%,%) -> % +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R retract : % -> NonNegativeInteger sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R generalizedContinuumHypothesisAssumed : Boolean -> Boolean +--R generalizedContinuumHypothesisAssumed? : () -> Boolean +--R retractIfCan : % -> Union(NonNegativeInteger,"failed") +--R +--E 21 + )spool )lisp (bye) \end{chunk} @@ -19040,7 +19637,8 @@ CardinalNumber: Join(OrderedSet, AbelianMonoid, Monoid, )set message test on )set message auto off )clear all ---S 1 of 48 + +--S 1 of 49 CT := CARTEN(i0 := 1, 2, Integer) --R --R @@ -19048,7 +19646,7 @@ CT := CARTEN(i0 := 1, 2, Integer) --R Type: Domain --E 1 ---S 2 of 48 +--S 2 of 49 t0: CT := 8 --R --R @@ -19056,7 +19654,7 @@ t0: CT := 8 --R Type: CartesianTensor(1,2,Integer) --E 2 ---S 3 of 48 +--S 3 of 49 rank t0 --R --R @@ -19064,7 +19662,7 @@ rank t0 --R Type: NonNegativeInteger --E 3 ---S 4 of 48 +--S 4 of 49 v: DirectProduct(2, Integer) := directProduct [3,4] --R --R @@ -19072,7 +19670,7 @@ v: DirectProduct(2, Integer) := directProduct [3,4] --R Type: DirectProduct(2,Integer) --E 4 ---S 5 of 48 +--S 5 of 49 Tv: CT := v --R --R @@ -19080,7 +19678,7 @@ Tv: CT := v --R Type: CartesianTensor(1,2,Integer) --E 5 ---S 6 of 48 +--S 6 of 49 m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ] --R --R @@ -19090,7 +19688,7 @@ m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ] --R Type: SquareMatrix(2,Integer) --E 6 ---S 7 of 48 +--S 7 of 49 Tm: CT := m --R --R @@ -19100,7 +19698,7 @@ Tm: CT := m --R Type: CartesianTensor(1,2,Integer) --E 7 ---S 8 of 48 +--S 8 of 49 n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ] --R --R @@ -19110,7 +19708,7 @@ n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ] --R Type: SquareMatrix(2,Integer) --E 8 ---S 9 of 48 +--S 9 of 49 Tn: CT := n --R --R @@ -19120,7 +19718,7 @@ Tn: CT := n --R Type: CartesianTensor(1,2,Integer) --E 9 ---S 10 of 48 +--S 10 of 49 t1: CT := [2, 3] --R --R @@ -19128,7 +19726,7 @@ t1: CT := [2, 3] --R Type: CartesianTensor(1,2,Integer) --E 10 ---S 11 of 48 +--S 11 of 49 rank t1 --R --R @@ -19136,7 +19734,7 @@ rank t1 --R Type: PositiveInteger --E 11 ---S 12 of 48 +--S 12 of 49 t2: CT := [t1, t1] --R --R @@ -19146,7 +19744,7 @@ t2: CT := [t1, t1] --R Type: CartesianTensor(1,2,Integer) --E 12 ---S 13 of 48 +--S 13 of 49 t3: CT := [t2, t2] --R --R @@ -19156,7 +19754,7 @@ t3: CT := [t2, t2] --R Type: CartesianTensor(1,2,Integer) --E 13 ---S 14 of 48 +--S 14 of 49 tt: CT := [t3, t3]; tt := [tt, tt] --R --R @@ -19170,7 +19768,7 @@ tt: CT := [t3, t3]; tt := [tt, tt] --R Type: CartesianTensor(1,2,Integer) --E 14 ---S 15 of 48 +--S 15 of 49 rank tt --R --R @@ -19178,7 +19776,7 @@ rank tt --R Type: PositiveInteger --E 15 ---S 16 of 48 +--S 16 of 49 Tmn := product(Tm, Tn) --R --R @@ -19192,7 +19790,7 @@ Tmn := product(Tm, Tn) --R Type: CartesianTensor(1,2,Integer) --E 16 ---S 17 of 48 +--S 17 of 49 Tmv := contract(Tm,2,Tv,1) --R --R @@ -19200,7 +19798,7 @@ Tmv := contract(Tm,2,Tv,1) --R Type: CartesianTensor(1,2,Integer) --E 17 ---S 18 of 48 +--S 18 of 49 Tm*Tv --R --R @@ -19208,7 +19806,7 @@ Tm*Tv --R Type: CartesianTensor(1,2,Integer) --E 18 ---S 19 of 48 +--S 19 of 49 Tmv = m * v --R --R @@ -19216,7 +19814,7 @@ Tmv = m * v --R Type: Equation(CartesianTensor(1,2,Integer)) --E 19 ---S 20 of 48 +--S 20 of 49 t0() --R --R @@ -19224,7 +19822,7 @@ t0() --R Type: PositiveInteger --E 20 ---S 21 of 48 +--S 21 of 49 t1(1+1) --R --R @@ -19232,7 +19830,7 @@ t1(1+1) --R Type: PositiveInteger --E 21 ---S 22 of 48 +--S 22 of 49 t2(2,1) --R --R @@ -19240,7 +19838,7 @@ t2(2,1) --R Type: PositiveInteger --E 22 ---S 23 of 48 +--S 23 of 49 t3(2,1,2) --R --R @@ -19248,7 +19846,7 @@ t3(2,1,2) --R Type: PositiveInteger --E 23 ---S 24 of 48 +--S 24 of 49 Tmn(2,1,2,1) --R --R @@ -19256,7 +19854,7 @@ Tmn(2,1,2,1) --R Type: NonNegativeInteger --E 24 ---S 25 of 48 +--S 25 of 49 t0[] --R --R @@ -19264,7 +19862,7 @@ t0[] --R Type: PositiveInteger --E 25 ---S 26 of 48 +--S 26 of 49 t1[2] --R --R @@ -19272,7 +19870,7 @@ t1[2] --R Type: PositiveInteger --E 26 ---S 27 of 48 +--S 27 of 49 t2[2,1] --R --R @@ -19280,7 +19878,7 @@ t2[2,1] --R Type: PositiveInteger --E 27 ---S 28 of 48 +--S 28 of 49 t3[2,1,2] --R --R @@ -19288,7 +19886,7 @@ t3[2,1,2] --R Type: PositiveInteger --E 28 ---S 29 of 48 +--S 29 of 49 Tmn[2,1,2,1] --R --R @@ -19296,7 +19894,7 @@ Tmn[2,1,2,1] --R Type: NonNegativeInteger --E 29 ---S 30 of 48 +--S 30 of 49 cTmn := contract(Tmn,1,2) --R --R @@ -19306,7 +19904,7 @@ cTmn := contract(Tmn,1,2) --R Type: CartesianTensor(1,2,Integer) --E 30 ---S 31 of 48 +--S 31 of 49 trace(m) * n --R --R @@ -19316,7 +19914,7 @@ trace(m) * n --R Type: SquareMatrix(2,Integer) --E 31 ---S 32 of 48 +--S 32 of 49 contract(Tmn,1,2) = trace(m) * n --R --R @@ -19326,7 +19924,7 @@ contract(Tmn,1,2) = trace(m) * n --R Type: Equation(CartesianTensor(1,2,Integer)) --E 32 ---S 33 of 48 +--S 33 of 49 contract(Tmn,1,3) = transpose(m) * n --R --R @@ -19336,7 +19934,7 @@ contract(Tmn,1,3) = transpose(m) * n --R Type: Equation(CartesianTensor(1,2,Integer)) --E 33 ---S 34 of 48 +--S 34 of 49 contract(Tmn,1,4) = transpose(m) * transpose(n) --R --R @@ -19346,7 +19944,7 @@ contract(Tmn,1,4) = transpose(m) * transpose(n) --R Type: Equation(CartesianTensor(1,2,Integer)) --E 34 ---S 35 of 48 +--S 35 of 49 contract(Tmn,2,3) = m * n --R --R @@ -19356,7 +19954,7 @@ contract(Tmn,2,3) = m * n --R Type: Equation(CartesianTensor(1,2,Integer)) --E 35 ---S 36 of 48 +--S 36 of 49 contract(Tmn,2,4) = m * transpose(n) --R --R @@ -19366,7 +19964,7 @@ contract(Tmn,2,4) = m * transpose(n) --R Type: Equation(CartesianTensor(1,2,Integer)) --E 36 ---S 37 of 48 +--S 37 of 49 contract(Tmn,3,4) = trace(n) * m --R --R @@ -19376,7 +19974,7 @@ contract(Tmn,3,4) = trace(n) * m --R Type: Equation(CartesianTensor(1,2,Integer)) --E 37 ---S 38 of 48 +--S 38 of 49 tTmn := transpose(Tmn,1,3) --R --R @@ -19390,7 +19988,7 @@ tTmn := transpose(Tmn,1,3) --R Type: CartesianTensor(1,2,Integer) --E 38 ---S 39 of 48 +--S 39 of 49 transpose Tmn --R --R @@ -19404,7 +20002,7 @@ transpose Tmn --R Type: CartesianTensor(1,2,Integer) --E 39 ---S 40 of 48 +--S 40 of 49 transpose Tm = transpose m --R --R @@ -19414,7 +20012,7 @@ transpose Tm = transpose m --R Type: Equation(CartesianTensor(1,2,Integer)) --E 40 ---S 41 of 48 +--S 41 of 49 rTmn := reindex(Tmn, [1,4,2,3]) --R --R @@ -19428,7 +20026,7 @@ rTmn := reindex(Tmn, [1,4,2,3]) --R Type: CartesianTensor(1,2,Integer) --E 41 ---S 42 of 48 +--S 42 of 49 tt := transpose(Tm)*Tn - Tn*transpose(Tm) --R --R @@ -19438,7 +20036,7 @@ tt := transpose(Tm)*Tn - Tn*transpose(Tm) --R Type: CartesianTensor(1,2,Integer) --E 42 ---S 43 of 48 +--S 43 of 49 Tv*(tt+Tn) --R --R @@ -19446,7 +20044,7 @@ Tv*(tt+Tn) --R Type: CartesianTensor(1,2,Integer) --E 43 ---S 44 of 48 +--S 44 of 49 reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm) --R --R @@ -19460,7 +20058,7 @@ reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm) --R Type: CartesianTensor(1,2,Integer) --E 44 ---S 45 of 48 +--S 45 of 49 delta: CT := kroneckerDelta() --R --R @@ -19470,7 +20068,7 @@ delta: CT := kroneckerDelta() --R Type: CartesianTensor(1,2,Integer) --E 45 ---S 46 of 48 +--S 46 of 49 contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2]) --R --R @@ -19484,7 +20082,7 @@ contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2]) --R Type: Equation(CartesianTensor(1,2,Integer)) --E 46 ---S 47 of 48 +--S 47 of 49 epsilon:CT := leviCivitaSymbol() --R --R @@ -19494,13 +20092,50 @@ epsilon:CT := leviCivitaSymbol() --R Type: CartesianTensor(1,2,Integer) --E 47 ---S 48 of 48 +--S 48 of 49 contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m --R --R --R (48) - 6= - 6 --R Type: Equation(CartesianTensor(1,2,Integer)) --E 48 + +--S 49 of 49 +)show CartesianTensor +--R +--R CartesianTensor(minix: Integer,dim: NonNegativeInteger,R: CommutativeRing) is a domain constructor +--R Abbreviation for CartesianTensor is CARTEN +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CARTEN +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (%,Integer) -> % ?*? : (%,R) -> % +--R ?*? : (R,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % coerce : List(%) -> % +--R coerce : List(R) -> % coerce : SquareMatrix(dim,R) -> % +--R coerce : DirectProduct(dim,R) -> % coerce : R -> % +--R coerce : % -> OutputForm degree : % -> NonNegativeInteger +--R ?.? : (%,List(Integer)) -> R elt : (%,Integer,Integer) -> R +--R ?.? : (%,Integer) -> R elt : % -> R +--R hash : % -> SingleInteger kroneckerDelta : () -> % +--R latex : % -> String leviCivitaSymbol : () -> % +--R product : (%,%) -> % rank : % -> NonNegativeInteger +--R ravel : % -> List(R) reindex : (%,List(Integer)) -> % +--R retract : % -> R sample : () -> % +--R transpose : % -> % unravel : List(R) -> % +--R ?~=? : (%,%) -> Boolean +--R contract : (%,Integer,Integer) -> % +--R contract : (%,Integer,%,Integer) -> % +--R elt : (%,Integer,Integer,Integer,Integer) -> R +--R elt : (%,Integer,Integer,Integer) -> R +--R retractIfCan : % -> Union(R,"failed") +--R transpose : (%,Integer,Integer) -> % +--R +--E 49 + )spool )lisp (bye) \end{chunk} @@ -20632,6 +21267,21 @@ CartesianTensor(minix, dim, R): Exports == Implementation where --S 1 of 1 )show Cell +--R +--R Cell(TheField: RealClosedField) is a domain constructor +--R Abbreviation for Cell is CELL +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CELL +--R +--R------------------------------- Operations -------------------------------- +--R coerce : % -> OutputForm dimension : % -> NonNegativeInteger +--R mainVariableOf : % -> Symbol samplePoint : % -> List(TheField) +--R variablesOf : % -> List(Symbol) +--R hasDimension? : (%,Symbol) -> Boolean +--R makeCell : (SimpleCell(TheField,SparseUnivariatePolynomial(TheField)),%) -> % +--R makeCell : List(SimpleCell(TheField,SparseUnivariatePolynomial(TheField))) -> % +--R projection : % -> Union(%,"failed") +--R --E 1 )spool @@ -20757,7 +21407,7 @@ Cell(TheField) : PUB == PRIV where )set message test on )set message auto off )clear all ---S 1 of 13 +--S 1 of 14 chars := [char "a", char "A", char "X", char "8", char "+"] --R --R @@ -20765,7 +21415,7 @@ chars := [char "a", char "A", char "X", char "8", char "+"] --R Type: List(Character) --E 1 ---S 2 of 13 +--S 2 of 14 space() --R --R @@ -20773,7 +21423,7 @@ space() --R Type: Character --E 2 ---S 3 of 13 +--S 3 of 14 quote() --R --R @@ -20781,7 +21431,7 @@ quote() --R Type: Character --E 3 ---S 4 of 13 +--S 4 of 14 escape() --R --R @@ -20789,7 +21439,7 @@ escape() --R Type: Character --E 4 ---S 5 of 13 +--S 5 of 14 [ord c for c in chars] --R --R @@ -20797,7 +21447,7 @@ escape() --R Type: List(Integer) --E 5 ---S 6 of 13 +--S 6 of 14 [upperCase c for c in chars] --R --R @@ -20805,7 +21455,7 @@ escape() --R Type: List(Character) --E 6 ---S 7 of 13 +--S 7 of 14 [lowerCase c for c in chars] --R --R @@ -20813,7 +21463,7 @@ escape() --R Type: List(Character) --E 7 ---S 8 of 13 +--S 8 of 14 [alphabetic? c for c in chars] --R --R @@ -20821,7 +21471,7 @@ escape() --R Type: List(Boolean) --E 8 ---S 9 of 13 +--S 9 of 14 [upperCase? c for c in chars] --R --R @@ -20829,7 +21479,7 @@ escape() --R Type: List(Boolean) --E 9 ---S 10 of 13 +--S 10 of 14 [lowerCase? c for c in chars] --R --R @@ -20837,7 +21487,7 @@ escape() --R Type: List(Boolean) --E 10 ---S 11 of 13 +--S 11 of 14 [digit? c for c in chars] --R --R @@ -20845,7 +21495,7 @@ escape() --R Type: List(Boolean) --E 11 ---S 12 of 13 +--S 12 of 14 [hexDigit? c for c in chars] --R --R @@ -20853,13 +21503,41 @@ escape() --R Type: List(Boolean) --E 12 ---S 13 of 13 +--S 13 of 14 [alphanumeric? c for c in chars] --R --R --R (13) [true,true,true,true,false] --R Type: List(Boolean) --E 13 + +--S 14 of 14 +)show Character +--R +--R Character is a domain constructor +--R Abbreviation for Character is CHAR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CHAR +--R +--R------------------------------- Operations -------------------------------- +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean alphabetic? : % -> Boolean +--R alphanumeric? : % -> Boolean char : String -> % +--R char : Integer -> % coerce : % -> OutputForm +--R digit? : % -> Boolean enumerate : () -> List(%) +--R escape : () -> % hash : % -> SingleInteger +--R hexDigit? : % -> Boolean index : PositiveInteger -> % +--R latex : % -> String lookup : % -> PositiveInteger +--R lowerCase : % -> % lowerCase? : % -> Boolean +--R max : (%,%) -> % min : (%,%) -> % +--R ord : % -> Integer quote : () -> % +--R random : () -> % size : () -> NonNegativeInteger +--R space : () -> % upperCase : % -> % +--R upperCase? : % -> Boolean ?~=? : (%,%) -> Boolean +--R +--E 14 + )spool )lisp (bye) \end{chunk} @@ -21140,7 +21818,8 @@ Character: OrderedFinite() with )set message test on )set message auto off )clear all ---S 1 of 16 + +--S 1 of 17 cl1:=charClass[char "a",char "e",char "i",char "o",char "u",char "y"] --R --R @@ -21148,7 +21827,7 @@ cl1:=charClass[char "a",char "e",char "i",char "o",char "u",char "y"] --R Type: CharacterClass --E 1 ---S 2 of 16 +--S 2 of 17 cl2 := charClass "bcdfghjklmnpqrstvwxyz" --R --R @@ -21156,7 +21835,7 @@ cl2 := charClass "bcdfghjklmnpqrstvwxyz" --R Type: CharacterClass --E 2 ---S 3 of 16 +--S 3 of 17 digit() --R --R @@ -21164,7 +21843,7 @@ digit() --R Type: CharacterClass --E 3 ---S 4 of 16 +--S 4 of 17 hexDigit() --R --R @@ -21172,7 +21851,7 @@ hexDigit() --R Type: CharacterClass --E 4 ---S 5 of 16 +--S 5 of 17 upperCase() --R --R @@ -21180,7 +21859,7 @@ upperCase() --R Type: CharacterClass --E 5 ---S 6 of 16 +--S 6 of 17 lowerCase() --R --R @@ -21188,7 +21867,7 @@ lowerCase() --R Type: CharacterClass --E 6 ---S 7 of 16 +--S 7 of 17 alphabetic() --R --R @@ -21196,7 +21875,7 @@ alphabetic() --R Type: CharacterClass --E 7 ---S 8 of 16 +--S 8 of 17 alphanumeric() --R --R @@ -21204,7 +21883,7 @@ alphanumeric() --R Type: CharacterClass --E 8 ---S 9 of 16 +--S 9 of 17 member?(char "a", cl1) --R --R @@ -21212,7 +21891,7 @@ member?(char "a", cl1) --R Type: Boolean --E 9 ---S 10 of 16 +--S 10 of 17 member?(char "a", cl2) --R --R @@ -21220,7 +21899,7 @@ member?(char "a", cl2) --R Type: Boolean --E 10 ---S 11 of 16 +--S 11 of 17 intersect(cl1, cl2) --R --R @@ -21228,7 +21907,7 @@ intersect(cl1, cl2) --R Type: CharacterClass --E 11 ---S 12 of 16 +--S 12 of 17 union(cl1,cl2) --R --R @@ -21236,7 +21915,7 @@ union(cl1,cl2) --R Type: CharacterClass --E 12 ---S 13 of 16 +--S 13 of 17 difference(cl1,cl2) --R --R @@ -21244,7 +21923,7 @@ difference(cl1,cl2) --R Type: CharacterClass --E 13 ---S 14 of 16 +--S 14 of 17 intersect(complement(cl1),cl2) --R --R @@ -21252,7 +21931,7 @@ intersect(complement(cl1),cl2) --R Type: CharacterClass --E 14 ---S 15 of 16 +--S 15 of 17 insert!(char "a", cl2) --R --R @@ -21260,13 +21939,85 @@ insert!(char "a", cl2) --R Type: CharacterClass --E 15 ---S 16 of 16 +--S 16 of 17 remove!(char "b", cl2) --R --R --R (16) "acdfghjklmnpqrstvwxyz" --R Type: CharacterClass --E 16 + +--S 17 of 17 +)show CharacterClass +--R +--R CharacterClass is a domain constructor +--R Abbreviation for CharacterClass is CCLASS +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CCLASS +--R +--R------------------------------- Operations -------------------------------- +--R ? Boolean ?=? : (%,%) -> Boolean +--R alphabetic : () -> % alphanumeric : () -> % +--R bag : List(Character) -> % brace : List(Character) -> % +--R brace : () -> % charClass : List(Character) -> % +--R charClass : String -> % coerce : % -> OutputForm +--R construct : List(Character) -> % convert : % -> List(Character) +--R convert : % -> String copy : % -> % +--R dictionary : () -> % dictionary : List(Character) -> % +--R difference : (%,Character) -> % difference : (%,%) -> % +--R digit : () -> % empty : () -> % +--R empty? : % -> Boolean eq? : (%,%) -> Boolean +--R extract! : % -> Character hash : % -> SingleInteger +--R hexDigit : () -> % insert! : (Character,%) -> % +--R inspect : % -> Character intersect : (%,%) -> % +--R latex : % -> String lowerCase : () -> % +--R sample : () -> % set : List(Character) -> % +--R set : () -> % subset? : (%,%) -> Boolean +--R symmetricDifference : (%,%) -> % union : (Character,%) -> % +--R union : (%,Character) -> % union : (%,%) -> % +--R upperCase : () -> % ?~=? : (%,%) -> Boolean +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R any? : ((Character -> Boolean),%) -> Boolean if $ has finiteAggregate +--R cardinality : % -> NonNegativeInteger +--R complement : % -> % if Character has FINITE +--R convert : % -> InputForm if Character has KONVERT(INFORM) +--R count : ((Character -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Character,%) -> NonNegativeInteger if $ has finiteAggregate and Character has SETCAT +--R enumerate : () -> List(%) if Character has FINITE +--R eval : (%,List(Equation(Character))) -> % if Character has EVALAB(CHAR) and Character has SETCAT +--R eval : (%,Equation(Character)) -> % if Character has EVALAB(CHAR) and Character has SETCAT +--R eval : (%,Character,Character) -> % if Character has EVALAB(CHAR) and Character has SETCAT +--R eval : (%,List(Character),List(Character)) -> % if Character has EVALAB(CHAR) and Character has SETCAT +--R every? : ((Character -> Boolean),%) -> Boolean if $ has finiteAggregate +--R find : ((Character -> Boolean),%) -> Union(Character,"failed") +--R index : PositiveInteger -> % if Character has FINITE +--R less? : (%,NonNegativeInteger) -> Boolean +--R lookup : % -> PositiveInteger if Character has FINITE +--R map : ((Character -> Character),%) -> % +--R map! : ((Character -> Character),%) -> % if $ has shallowlyMutable +--R max : % -> Character if Character has ORDSET +--R member? : (Character,%) -> Boolean if $ has finiteAggregate and Character has SETCAT +--R members : % -> List(Character) if $ has finiteAggregate +--R min : % -> Character if Character has ORDSET +--R more? : (%,NonNegativeInteger) -> Boolean +--R parts : % -> List(Character) if $ has finiteAggregate +--R random : () -> % if Character has FINITE +--R reduce : (((Character,Character) -> Character),%,Character,Character) -> Character if $ has finiteAggregate and Character has SETCAT +--R reduce : (((Character,Character) -> Character),%,Character) -> Character if $ has finiteAggregate +--R reduce : (((Character,Character) -> Character),%) -> Character if $ has finiteAggregate +--R remove : (Character,%) -> % if $ has finiteAggregate and Character has SETCAT +--R remove : ((Character -> Boolean),%) -> % if $ has finiteAggregate +--R remove! : (Character,%) -> % if $ has finiteAggregate +--R remove! : ((Character -> Boolean),%) -> % if $ has finiteAggregate +--R removeDuplicates : % -> % if $ has finiteAggregate and Character has SETCAT +--R select : ((Character -> Boolean),%) -> % if $ has finiteAggregate +--R select! : ((Character -> Boolean),%) -> % if $ has finiteAggregate +--R size : () -> NonNegativeInteger if Character has FINITE +--R size? : (%,NonNegativeInteger) -> Boolean +--R universe : () -> % if Character has FINITE +--R +--E 17 + )spool )lisp (bye) \end{chunk} @@ -21746,7 +22497,8 @@ $\mathbb{R}_{m,m}$ & --------$>$ & $\mathbb{R}^{4^m}$ \\ )set message test on )set message auto off )clear all ---S 1 of 36 + +--S 1 of 37 K := Fraction Polynomial Integer --R --R @@ -21754,7 +22506,7 @@ K := Fraction Polynomial Integer --R Type: Domain --E 1 ---S 2 of 36 +--S 2 of 37 m := matrix [ [-1] ] --R --R @@ -21762,7 +22514,7 @@ m := matrix [ [-1] ] --R Type: Matrix(Integer) --E 2 ---S 3 of 36 +--S 3 of 37 C := CliffordAlgebra(1, K, quadraticForm m) --R --R @@ -21770,7 +22522,7 @@ C := CliffordAlgebra(1, K, quadraticForm m) --R Type: Domain --E 3 ---S 4 of 36 +--S 4 of 37 i: C := e(1) --R --R @@ -21779,7 +22531,7 @@ i: C := e(1) --R Type: CliffordAlgebra(1,Fraction(Polynomial(Integer)),[[-1]]) --E 4 ---S 5 of 36 +--S 5 of 37 x := a + b * i --R --R @@ -21788,7 +22540,7 @@ x := a + b * i --R Type: CliffordAlgebra(1,Fraction(Polynomial(Integer)),[[-1]]) --E 5 ---S 6 of 36 +--S 6 of 37 y := c + d * i --R --R @@ -21797,7 +22549,7 @@ y := c + d * i --R Type: CliffordAlgebra(1,Fraction(Polynomial(Integer)),[[-1]]) --E 6 ---S 7 of 36 +--S 7 of 37 x * y --R --R @@ -21807,7 +22559,7 @@ x * y --E 7 )clear all ---S 8 of 36 +--S 8 of 37 K := Fraction Polynomial Integer --R --R @@ -21815,7 +22567,7 @@ K := Fraction Polynomial Integer --R Type: Domain --E 8 ---S 9 of 36 +--S 9 of 37 m := matrix [ [-1,0],[0,-1] ] --R --R @@ -21825,7 +22577,7 @@ m := matrix [ [-1,0],[0,-1] ] --R Type: Matrix(Integer) --E 9 ---S 10 of 36 +--S 10 of 37 H := CliffordAlgebra(2, K, quadraticForm m) --R --R @@ -21833,7 +22585,7 @@ H := CliffordAlgebra(2, K, quadraticForm m) --R Type: Domain --E 10 ---S 11 of 36 +--S 11 of 37 i: H := e(1) --R --R @@ -21842,7 +22594,7 @@ i: H := e(1) --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 11 ---S 12 of 36 +--S 12 of 37 j: H := e(2) --R --R @@ -21851,7 +22603,7 @@ j: H := e(2) --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 12 ---S 13 of 36 +--S 13 of 37 k: H := i * j --R --R @@ -21860,7 +22612,7 @@ k: H := i * j --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 13 ---S 14 of 36 +--S 14 of 37 x := a + b * i + c * j + d * k --R --R @@ -21869,7 +22621,7 @@ x := a + b * i + c * j + d * k --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 14 ---S 15 of 36 +--S 15 of 37 y := e + f * i + g * j + h * k --R --R @@ -21878,7 +22630,7 @@ y := e + f * i + g * j + h * k --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 15 ---S 16 of 36 +--S 16 of 37 x + y --R --R @@ -21887,7 +22639,7 @@ x + y --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 16 ---S 17 of 36 +--S 17 of 37 x * y --R --R @@ -21900,7 +22652,7 @@ x * y --R Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[-1,0],[0,-1]]) --E 17 ---S 18 of 36 +--S 18 of 37 y * x --R --R @@ -21914,7 +22666,7 @@ y * x --E 18 )clear all ---S 19 of 36 +--S 19 of 37 K := Fraction Polynomial Integer --R --R @@ -21922,7 +22674,7 @@ K := Fraction Polynomial Integer --R Type: Domain --E 19 ---S 20 of 36 +--S 20 of 37 Ext := CliffordAlgebra(3, K, quadraticForm 0) --R --R @@ -21931,7 +22683,7 @@ Ext := CliffordAlgebra(3, K, quadraticForm 0) --R Type: Domain --E 20 ---S 21 of 36 +--S 21 of 37 i: Ext := e(1) --R --R @@ -21940,7 +22692,7 @@ i: Ext := e(1) --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 21 ---S 22 of 36 +--S 22 of 37 j: Ext := e(2) --R --R @@ -21949,7 +22701,7 @@ j: Ext := e(2) --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 22 ---S 23 of 36 +--S 23 of 37 k: Ext := e(3) --R --R @@ -21958,7 +22710,7 @@ k: Ext := e(3) --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 23 ---S 24 of 36 +--S 24 of 37 x := x1*i + x2*j + x3*k --R --R @@ -21967,7 +22719,7 @@ x := x1*i + x2*j + x3*k --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 24 ---S 25 of 36 +--S 25 of 37 y := y1*i + y2*j + y3*k --R --R @@ -21976,7 +22728,7 @@ y := y1*i + y2*j + y3*k --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 25 ---S 26 of 36 +--S 26 of 37 x + y --R --R @@ -21985,7 +22737,7 @@ x + y --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 26 ---S 27 of 36 +--S 27 of 37 x * y + y * x --R --R @@ -21993,13 +22745,13 @@ x * y + y * x --RType: CliffordAlgebra(3,Fraction(Polynomial(Integer)),[[0,0,0],[0,0,0],[0,0,0]]) --E 27 ---S 28 of 36 +--S 28 of 37 dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k --R --R Type: Void --E 28 ---S 29 of 36 +--S 29 of 37 dual2(x*y) --R --R Compiling function dual2 with type CliffordAlgebra(3,Fraction( @@ -22013,7 +22765,7 @@ dual2(x*y) --E 29 )clear all ---S 30 of 36 +--S 30 of 37 K := Fraction Integer --R --R @@ -22021,7 +22773,7 @@ K := Fraction Integer --R Type: Domain --E 30 ---S 31 of 36 +--S 31 of 37 g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ] --R --R @@ -22035,7 +22787,7 @@ g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ] --R Type: Matrix(Integer) --E 31 ---S 32 of 36 +--S 32 of 37 D := CliffordAlgebra(4,K, quadraticForm g) --R --R @@ -22045,7 +22797,7 @@ D := CliffordAlgebra(4,K, quadraticForm g) --R Type: Domain --E 32 ---S 33 of 36 +--S 33 of 37 gam := [e(i)$D for i in 1..4] --R --R @@ -22054,14 +22806,14 @@ gam := [e(i)$D for i in 1..4] --RType: List(CliffordAlgebra(4,Fraction(Integer),[[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]])) --E 33 ---S 34 of 36 +--S 34 of 37 m := 1; n:= 2; r := 3; s := 4; --R --R --R Type: PositiveInteger --E 34 ---S 35 of 36 +--S 35 of 37 lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) --R --R @@ -22070,7 +22822,7 @@ lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) f --RType: CliffordAlgebra(4,Fraction(Integer),[[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]]) --E 35 ---S 36 of 36 +--S 36 of 37 rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) --R --R @@ -22078,6 +22830,38 @@ rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) --R 1 2 3 4 --RType: CliffordAlgebra(4,Fraction(Integer),[[1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1]]) --E 36 + +--S 37 of 37 +)show CliffordAlgebra +--R +--R CliffordAlgebra(n: PositiveInteger,K: Field,Q: QuadraticForm(n,K)) is a domain constructor +--R Abbreviation for CliffordAlgebra is CLIF +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CLIF +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,K) -> % ?*? : (K,%) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,K) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coerce : K -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R dimension : () -> CardinalNumber e : PositiveInteger -> % +--R hash : % -> SingleInteger latex : % -> String +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R characteristic : () -> NonNegativeInteger +--R coefficient : (%,List(PositiveInteger)) -> K +--R monomial : (K,List(PositiveInteger)) -> % +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R +--E 37 + )spool )lisp (bye) \end{chunk} @@ -22610,6 +23394,7 @@ CliffordAlgebra(n, K, Q): T == Impl where --S 1 of 1 )show Color +--R --R Color is a domain constructor --R Abbreviation for Color is COLOR --R This constructor is exposed in this frame. @@ -22621,9 +23406,9 @@ CliffordAlgebra(n, K, Q): T == Impl where --R blue : () -> % coerce : % -> OutputForm --R color : Integer -> % green : () -> % --R hash : % -> SingleInteger hue : % -> Integer ---R latex : % -> String red : () -> % ---R yellow : () -> % ?~=? : (%,%) -> Boolean ---R numberOfHues : () -> PositiveInteger +--R latex : % -> String numberOfHues : () -> PositiveInteger +--R red : () -> % yellow : () -> % +--R ?~=? : (%,%) -> Boolean --R --E 1 @@ -22773,6 +23558,7 @@ Color(): Exports == Implementation where --S 1 of 1 )show Commutator +--R --R Commutator is a domain constructor --R Abbreviation for Commutator is COMM --R This constructor is exposed in this frame. @@ -22877,7 +23663,8 @@ Commutator: Export == Implement where )set message test on )set message auto off )clear all ---S 1 of 16 + +--S 1 of 17 a := complex(4/3,5/2) --R --R @@ -22887,7 +23674,7 @@ a := complex(4/3,5/2) --R Type: Complex(Fraction(Integer)) --E 1 ---S 2 of 16 +--S 2 of 17 b := complex(4/3,-5/2) --R --R @@ -22897,7 +23684,7 @@ b := complex(4/3,-5/2) --R Type: Complex(Fraction(Integer)) --E 2 ---S 3 of 16 +--S 3 of 17 a + b --R --R @@ -22907,7 +23694,7 @@ a + b --R Type: Complex(Fraction(Integer)) --E 3 ---S 4 of 16 +--S 4 of 17 a - b --R --R @@ -22915,7 +23702,7 @@ a - b --R Type: Complex(Fraction(Integer)) --E 4 ---S 5 of 16 +--S 5 of 17 a * b --R --R @@ -22925,7 +23712,7 @@ a * b --R Type: Complex(Fraction(Integer)) --E 5 ---S 6 of 16 +--S 6 of 17 a / b --R --R @@ -22935,7 +23722,7 @@ a / b --R Type: Complex(Fraction(Integer)) --E 6 ---S 7 of 16 +--S 7 of 17 % :: Fraction Complex Integer --R --R @@ -22945,7 +23732,7 @@ a / b --R Type: Fraction(Complex(Integer)) --E 7 ---S 8 of 16 +--S 8 of 17 3.4 + 6.7 * %i --R --R @@ -22953,7 +23740,7 @@ a / b --R Type: Complex(Float) --E 8 ---S 9 of 16 +--S 9 of 17 conjugate a --R --R @@ -22963,7 +23750,7 @@ conjugate a --R Type: Complex(Fraction(Integer)) --E 9 ---S 10 of 16 +--S 10 of 17 norm a --R --R @@ -22973,7 +23760,7 @@ norm a --R Type: Fraction(Integer) --E 10 ---S 11 of 16 +--S 11 of 17 real a --R --R @@ -22983,7 +23770,7 @@ real a --R Type: Fraction(Integer) --E 11 ---S 12 of 16 +--S 12 of 17 imag a --R --R @@ -22993,7 +23780,7 @@ imag a --R Type: Fraction(Integer) --E 12 ---S 13 of 16 +--S 13 of 17 gcd(13 - 13*%i,31 + 27*%i) --R --R @@ -23001,7 +23788,7 @@ gcd(13 - 13*%i,31 + 27*%i) --R Type: Complex(Integer) --E 13 ---S 14 of 16 +--S 14 of 17 lcm(13 - 13*%i,31 + 27*%i) --R --R @@ -23009,7 +23796,7 @@ lcm(13 - 13*%i,31 + 27*%i) --R Type: Complex(Integer) --E 14 ---S 15 of 16 +--S 15 of 17 factor(13 - 13*%i) --R --R @@ -23017,7 +23804,7 @@ factor(13 - 13*%i) --R Type: Factored(Complex(Integer)) --E 15 ---S 16 of 16 +--S 16 of 17 factor complex(2,0) --R --R @@ -23025,6 +23812,184 @@ factor complex(2,0) --R (16) - %i (1 + %i) --R Type: Factored(Complex(Integer)) --E 16 + +--S 17 of 17 +)show Complex +--R +--R Complex(R: CommutativeRing) is a domain constructor +--R Abbreviation for Complex is COMPLEX +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for COMPLEX +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (R,%) -> % ?*? : (%,R) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,%) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : % -> % if R has DIFRING +--R D : (%,(R -> R)) -> % OMwrite : % -> String if R has OM +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % if R has RNS acos : % -> % if R has TRANFUN +--R acosh : % -> % if R has TRANFUN acot : % -> % if R has TRANFUN +--R acoth : % -> % if R has TRANFUN acsc : % -> % if R has TRANFUN +--R acsch : % -> % if R has TRANFUN argument : % -> R if R has TRANFUN +--R asec : % -> % if R has TRANFUN asech : % -> % if R has TRANFUN +--R asin : % -> % if R has TRANFUN asinh : % -> % if R has TRANFUN +--R atan : % -> % if R has TRANFUN atanh : % -> % if R has TRANFUN +--R basis : () -> Vector(%) coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complex : (R,R) -> % conjugate : % -> % +--R convert : Vector(R) -> % convert : % -> Vector(R) +--R coordinates : % -> Vector(R) cos : % -> % if R has TRANFUN +--R cosh : % -> % if R has TRANFUN cot : % -> % if R has TRANFUN +--R coth : % -> % if R has TRANFUN csc : % -> % if R has TRANFUN +--R csch : % -> % if R has TRANFUN differentiate : (%,(R -> R)) -> % +--R discriminant : () -> R discriminant : Vector(%) -> R +--R exp : % -> % if R has TRANFUN gcd : (%,%) -> % if R has EUCDOM +--R gcd : List(%) -> % if R has EUCDOM generator : () -> % +--R hash : % -> SingleInteger imag : % -> R +--R imaginary : () -> % init : () -> % if R has FFIELDC +--R inv : % -> % if R has FIELD latex : % -> String +--R lcm : (%,%) -> % if R has EUCDOM lcm : List(%) -> % if R has EUCDOM +--R log : % -> % if R has TRANFUN map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET +--R norm : % -> R one? : % -> Boolean +--R pi : () -> % if R has TRANFUN random : () -> % if R has FINITE +--R rank : () -> PositiveInteger real : % -> R +--R recip : % -> Union(%,"failed") represents : Vector(R) -> % +--R retract : % -> R sample : () -> % +--R sec : % -> % if R has TRANFUN sech : % -> % if R has TRANFUN +--R sin : % -> % if R has TRANFUN sinh : % -> % if R has TRANFUN +--R tan : % -> % if R has TRANFUN tanh : % -> % if R has TRANFUN +--R trace : % -> R traceMatrix : () -> Matrix(R) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R ?*? : (%,Fraction(Integer)) -> % if R has FIELD +--R ?*? : (Fraction(Integer),%) -> % if R has FIELD +--R ?**? : (%,Integer) -> % if R has FIELD +--R ?**? : (%,%) -> % if R has TRANFUN +--R ?**? : (%,Fraction(Integer)) -> % if R has RADCAT and R has TRANFUN +--R ? Boolean if R has ORDSET +--R ?<=? : (%,%) -> Boolean if R has ORDSET +--R ?>? : (%,%) -> Boolean if R has ORDSET +--R ?>=? : (%,%) -> Boolean if R has ORDSET +--R D : (%,NonNegativeInteger) -> % if R has DIFRING +--R D : (%,Symbol) -> % if R has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) +--R D : (%,(R -> R),NonNegativeInteger) -> % +--R OMwrite : (OpenMathDevice,%,Boolean) -> Void if R has OM +--R OMwrite : (OpenMathDevice,%) -> Void if R has OM +--R OMwrite : (%,Boolean) -> String if R has OM +--R ?^? : (%,Integer) -> % if R has FIELD +--R associates? : (%,%) -> Boolean if R has EUCDOM and R has PFECAT or R has INTDOM +--R characteristic : () -> NonNegativeInteger +--R characteristicPolynomial : % -> SparseUnivariatePolynomial(R) +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and R has EUCDOM and R has PFECAT or R has CHARNZ +--R charthRoot : % -> % if R has FFIELDC +--R coerce : % -> % if R has EUCDOM and R has PFECAT or R has INTDOM +--R coerce : Fraction(Integer) -> % if R has FIELD or R has RETRACT(FRAC(INT)) +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and R has EUCDOM and R has PFECAT or R has FFIELDC +--R convert : % -> InputForm if R has KONVERT(INFORM) +--R convert : % -> Complex(DoubleFloat) if R has REAL +--R convert : % -> Complex(Float) if R has REAL +--R convert : % -> Pattern(Float) if R has KONVERT(PATTERN(FLOAT)) +--R convert : % -> Pattern(Integer) if R has KONVERT(PATTERN(INT)) +--R convert : SparseUnivariatePolynomial(R) -> % +--R convert : % -> SparseUnivariatePolynomial(R) +--R coordinates : Vector(%) -> Matrix(R) +--R coordinates : (Vector(%),Vector(%)) -> Matrix(R) +--R coordinates : (%,Vector(%)) -> Vector(R) +--R createPrimitiveElement : () -> % if R has FFIELDC +--R definingPolynomial : () -> SparseUnivariatePolynomial(R) +--R derivationCoordinates : (Vector(%),(R -> R)) -> Matrix(R) if R has FIELD +--R differentiate : % -> % if R has DIFRING +--R differentiate : (%,NonNegativeInteger) -> % if R has DIFRING +--R differentiate : (%,Symbol) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,(R -> R),NonNegativeInteger) -> % +--R discreteLog : (%,%) -> Union(NonNegativeInteger,"failed") if R has FFIELDC +--R discreteLog : % -> NonNegativeInteger if R has FFIELDC +--R divide : (%,%) -> Record(quotient: %,remainder: %) if R has EUCDOM +--R ?.? : (%,R) -> % if R has ELTAB(R,R) +--R enumerate : () -> List(%) if R has FINITE +--R euclideanSize : % -> NonNegativeInteger if R has EUCDOM +--R eval : (%,Symbol,R) -> % if R has IEVALAB(SYMBOL,R) +--R eval : (%,List(Symbol),List(R)) -> % if R has IEVALAB(SYMBOL,R) +--R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) +--R eval : (%,Equation(R)) -> % if R has EVALAB(R) +--R eval : (%,R,R) -> % if R has EVALAB(R) +--R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if R has EUCDOM +--R exquo : (%,%) -> Union(%,"failed") if R has EUCDOM and R has PFECAT or R has INTDOM +--R exquo : (%,R) -> Union(%,"failed") if R has INTDOM +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) if R has EUCDOM +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") if R has EUCDOM +--R factor : % -> Factored(%) if R has EUCDOM and R has PFECAT or R has FIELD +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has EUCDOM and R has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has EUCDOM and R has PFECAT +--R factorsOfCyclicGroupSize : () -> List(Record(factor: Integer,exponent: Integer)) if R has FFIELDC +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has EUCDOM +--R index : PositiveInteger -> % if R has FINITE +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has EUCDOM +--R lift : % -> SparseUnivariatePolynomial(R) +--R lookup : % -> PositiveInteger if R has FINITE +--R minimalPolynomial : % -> SparseUnivariatePolynomial(R) if R has FIELD +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") if R has EUCDOM +--R nextItem : % -> Union(%,"failed") if R has FFIELDC +--R nthRoot : (%,Integer) -> % if R has RADCAT and R has TRANFUN +--R order : % -> OnePointCompletion(PositiveInteger) if R has FFIELDC +--R order : % -> PositiveInteger if R has FFIELDC +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if R has PATMAB(FLOAT) +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if R has PATMAB(INT) +--R polarCoordinates : % -> Record(r: R,phi: R) if R has RNS and R has TRANFUN +--R prime? : % -> Boolean if R has EUCDOM and R has PFECAT or R has FIELD +--R primeFrobenius : % -> % if R has FFIELDC +--R primeFrobenius : (%,NonNegativeInteger) -> % if R has FFIELDC +--R primitive? : % -> Boolean if R has FFIELDC +--R primitiveElement : () -> % if R has FFIELDC +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if R has EUCDOM +--R ?quo? : (%,%) -> % if R has EUCDOM +--R rational : % -> Fraction(Integer) if R has INS +--R rational? : % -> Boolean if R has INS +--R rationalIfCan : % -> Union(Fraction(Integer),"failed") if R has INS +--R reduce : Fraction(SparseUnivariatePolynomial(R)) -> Union(%,"failed") if R has FIELD +--R reduce : SparseUnivariatePolynomial(R) -> % +--R reducedSystem : Matrix(%) -> Matrix(R) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) +--R regularRepresentation : % -> Matrix(R) +--R regularRepresentation : (%,Vector(%)) -> Matrix(R) +--R ?rem? : (%,%) -> % if R has EUCDOM +--R representationType : () -> Union("prime",polynomial,normal,cyclic) if R has FFIELDC +--R represents : (Vector(R),Vector(%)) -> % +--R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) +--R retract : % -> Integer if R has RETRACT(INT) +--R retractIfCan : % -> Union(R,"failed") +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R size : () -> NonNegativeInteger if R has FINITE +--R sizeLess? : (%,%) -> Boolean if R has EUCDOM +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has EUCDOM and R has PFECAT +--R sqrt : % -> % if R has RADCAT and R has TRANFUN +--R squareFree : % -> Factored(%) if R has EUCDOM and R has PFECAT or R has FIELD +--R squareFreePart : % -> % if R has EUCDOM and R has PFECAT or R has FIELD +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has EUCDOM and R has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R tableForDiscreteLogarithm : Integer -> Table(PositiveInteger,NonNegativeInteger) if R has FFIELDC +--R traceMatrix : Vector(%) -> Matrix(R) +--R unit? : % -> Boolean if R has EUCDOM and R has PFECAT or R has INTDOM +--R unitCanonical : % -> % if R has EUCDOM and R has PFECAT or R has INTDOM +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has EUCDOM and R has PFECAT or R has INTDOM +--R +--E 17 + )spool )lisp (bye) \end{chunk} @@ -23398,25 +24363,25 @@ Complex(R:CommutativeRing): ComplexCategory(R) with --R --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,%) -> % ?*? : (%,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % antisymmetric? : % -> Boolean ---R copy : % -> % diagonal? : % -> Boolean ---R diagonalMatrix : List(%) -> % empty : () -> % ---R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R horizConcat : (%,%) -> % maxColIndex : % -> Integer ---R maxRowIndex : % -> Integer minColIndex : % -> Integer ---R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger ---R nrows : % -> NonNegativeInteger qnew : (Integer,Integer) -> % ---R sample : () -> % square? : % -> Boolean ---R squareTop : % -> % symmetric? : % -> Boolean ---R transpose : % -> % vertConcat : (%,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R antisymmetric? : % -> Boolean copy : % -> % +--R diagonal? : % -> Boolean diagonalMatrix : List(%) -> % +--R empty : () -> % empty? : % -> Boolean +--R eq? : (%,%) -> Boolean horizConcat : (%,%) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R qnew : (Integer,Integer) -> % sample : () -> % +--R square? : % -> Boolean squareTop : % -> % +--R symmetric? : % -> Boolean transpose : % -> % +--R vertConcat : (%,%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (ComplexDoubleFloatVector,%) -> ComplexDoubleFloatVector --R ?*? : (%,ComplexDoubleFloatVector) -> ComplexDoubleFloatVector --R ?*? : (%,Complex(DoubleFloat)) -> % --R ?*? : (Complex(DoubleFloat),%) -> % --R ?**? : (%,Integer) -> % if Complex(DoubleFloat) has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Complex(DoubleFloat)) -> % if Complex(DoubleFloat) has FIELD --R ?=? : (%,%) -> Boolean if Complex(DoubleFloat) has SETCAT --R any? : ((Complex(DoubleFloat) -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -24028,7 +24993,8 @@ ComplexDoubleFloatVector : VectorCategory Complex DoubleFloat with )set message test on )set message auto off )clear all ---S 1 of 22 + +--S 1 of 23 c := continuedFraction(314159/100000) --R --R @@ -24038,7 +25004,7 @@ c := continuedFraction(314159/100000) --R Type: ContinuedFraction(Integer) --E 1 ---S 2 of 22 +--S 2 of 23 partialQuotients c --R --R @@ -24046,7 +25012,7 @@ partialQuotients c --R Type: Stream(Integer) --E 2 ---S 3 of 22 +--S 3 of 23 convergents c --R --R @@ -24056,7 +25022,7 @@ convergents c --R Type: Stream(Fraction(Integer)) --E 3 ---S 4 of 22 +--S 4 of 23 approximants c --R --R @@ -24067,7 +25033,7 @@ approximants c --R Type: Stream(Fraction(Integer)) --E 4 ---S 5 of 22 +--S 5 of 23 pq := partialQuotients(1/c) --R --R @@ -24075,7 +25041,7 @@ pq := partialQuotients(1/c) --R Type: Stream(Integer) --E 5 ---S 6 of 22 +--S 6 of 23 continuedFraction(first pq,repeating [1],rest pq) --R --R @@ -24085,7 +25051,7 @@ continuedFraction(first pq,repeating [1],rest pq) --R Type: ContinuedFraction(Integer) --E 6 ---S 7 of 22 +--S 7 of 23 z:=continuedFraction(3,repeating [1],repeating [3,6]) --R --R @@ -24100,7 +25066,7 @@ z:=continuedFraction(3,repeating [1],repeating [3,6]) --R Type: ContinuedFraction(Integer) --E 7 ---S 8 of 22 +--S 8 of 23 dens:Stream Integer := cons(1,generate((x+->x+4),6)) --R --R @@ -24108,7 +25074,7 @@ dens:Stream Integer := cons(1,generate((x+->x+4),6)) --R Type: Stream(Integer) --E 8 ---S 9 of 22 +--S 9 of 23 cf := continuedFraction(0,repeating [1],dens) --R --R @@ -24123,7 +25089,7 @@ cf := continuedFraction(0,repeating [1],dens) --R Type: ContinuedFraction(Integer) --E 9 ---S 10 of 22 +--S 10 of 23 ccf := convergents cf --R --R @@ -24133,7 +25099,7 @@ ccf := convergents cf --R Type: Stream(Fraction(Integer)) --E 10 ---S 11 of 22 +--S 11 of 23 eConvergents := [2*e + 1 for e in ccf] --R --R @@ -24143,7 +25109,7 @@ eConvergents := [2*e + 1 for e in ccf] --R Type: Stream(Fraction(Integer)) --E 11 ---S 12 of 22 +--S 12 of 23 eConvergents :: Stream Float --R --R @@ -24155,7 +25121,7 @@ eConvergents :: Stream Float --R Type: Stream(Float) --E 12 ---S 13 of 22 +--S 13 of 23 exp 1.0 --R --R @@ -24163,7 +25129,7 @@ exp 1.0 --R Type: Float --E 13 ---S 14 of 22 +--S 14 of 23 cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2]) --R --R @@ -24178,7 +25144,7 @@ cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2]) --R Type: ContinuedFraction(Integer) --E 14 ---S 15 of 22 +--S 15 of 23 ccf := convergents cf --R --R @@ -24188,7 +25154,7 @@ ccf := convergents cf --R Type: Stream(Fraction(Integer)) --E 15 ---S 16 of 22 +--S 16 of 23 piConvergents := [4/p for p in ccf] --R --R @@ -24198,7 +25164,7 @@ piConvergents := [4/p for p in ccf] --R Type: Stream(Fraction(Integer)) --E 16 ---S 17 of 22 +--S 17 of 23 piConvergents :: Stream Float --R --R @@ -24210,7 +25176,7 @@ piConvergents :: Stream Float --R Type: Stream(Float) --E 17 ---S 18 of 22 +--S 18 of 23 continuedFraction((- 122 + 597*%i)/(4 - 4*%i)) --R --R @@ -24220,13 +25186,13 @@ continuedFraction((- 122 + 597*%i)/(4 - 4*%i)) --R Type: ContinuedFraction(Complex(Integer)) --E 18 ---S 19 of 22 +--S 19 of 23 r : Fraction UnivariatePolynomial(x,Fraction Integer) --R --R Type: Void --E 19 ---S 20 of 22 +--S 20 of 23 r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) --R --R @@ -24238,7 +25204,7 @@ r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 20 ---S 21 of 22 +--S 21 of 23 continuedFraction r --R --R @@ -24250,7 +25216,7 @@ continuedFraction r --R Type: ContinuedFraction(UnivariatePolynomial(x,Fraction(Integer))) --E 21 ---S 22 of 22 +--S 22 of 23 [i*i for i in convergents(z) :: Stream Float] --R --R @@ -24261,6 +25227,68 @@ continuedFraction r --R ...] --R Type: Stream(Float) --E 22 + +--S 23 of 23 +)show ContinuedFraction +--R +--R ContinuedFraction(R: EuclideanDomain) is a domain constructor +--R Abbreviation for ContinuedFraction is CONTFRAC +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for CONTFRAC +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % +--R ?*? : (%,Fraction(R)) -> % ?*? : (Fraction(R),%) -> % +--R ?*? : (R,%) -> % ?*? : (%,R) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : % -> % coerce : Fraction(Integer) -> % +--R coerce : Fraction(R) -> % coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complete : % -> % denominators : % -> Stream(R) +--R extend : (%,Integer) -> % factor : % -> Factored(%) +--R gcd : (%,%) -> % gcd : List(%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : (%,%) -> % +--R lcm : List(%) -> % numerators : % -> Stream(R) +--R one? : % -> Boolean partialNumerators : % -> Stream(R) +--R partialQuotients : % -> Stream(R) prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R reducedForm : % -> % ?rem? : (%,%) -> % +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R wholePart : % -> R zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R approximants : % -> Stream(Fraction(R)) +--R characteristic : () -> NonNegativeInteger +--R continuedFraction : (R,Stream(R),Stream(R)) -> % +--R continuedFraction : Fraction(R) -> % +--R convergents : % -> Stream(Fraction(R)) +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R euclideanSize : % -> NonNegativeInteger +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R partialDenominators : % -> Stream(R) +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R reducedContinuedFraction : (R,Stream(R)) -> % +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R +--E 23 + )spool )lisp (bye) \end{chunk} @@ -24968,8 +25996,8 @@ ContinuedFraction(R): Exports == Implementation where --S 1 of 1 )show Database --R ---R Database(S) where ---R S: OrderedSet with +--R Database(S)where +--R S: OrderedSetwith --R ?.? : (%,Symbol) -> String --R display : % -> Void --R fullDisplay : % -> Void is a domain constructor @@ -25123,15 +26151,19 @@ Database(S): Exports == Implementation where --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) --R insert : (S,%,Integer) -> % insert : (%,%,Integer) -> % --R insert! : (S,%,Integer) -> % insert! : (%,%,Integer) -> % ---R last : % -> S leaf? : % -> Boolean +--R last : (%,NonNegativeInteger) -> % last : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean --R leaves : % -> List(S) list : S -> % --R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET --R new : (NonNegativeInteger,S) -> % nodes : % -> List(%) --R possiblyInfinite? : % -> Boolean qelt : (%,Integer) -> S +--R remove! : ((S -> Boolean),%) -> % rest : (%,NonNegativeInteger) -> % --R rest : % -> % reverse : % -> % --R sample : () -> % second : % -> S ---R tail : % -> % third : % -> S ---R value : % -> S +--R select! : ((S -> Boolean),%) -> % sort : (((S,S) -> Boolean),%) -> % +--R sort : % -> % if S has ORDSET tail : % -> % +--R third : % -> S value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -25161,11 +26193,8 @@ Database(S): Exports == Implementation where --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R members : % -> List(S) if $ has finiteAggregate @@ -25173,7 +26202,6 @@ Database(S): Exports == Implementation where --R merge : (%,%) -> % if S has ORDSET --R merge! : (((S,S) -> Boolean),%,%) -> % --R merge! : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R node? : (%,%) -> Boolean if S has SETCAT @@ -25187,14 +26215,11 @@ Database(S): Exports == Implementation where --R reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate --R remove : (S,%) -> % if $ has finiteAggregate and S has SETCAT --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R remove! : ((S -> Boolean),%) -> % --R remove! : (S,%) -> % if S has SETCAT --R removeDuplicates : % -> % if $ has finiteAggregate and S has SETCAT --R removeDuplicates! : % -> % if S has SETCAT ---R rest : (%,NonNegativeInteger) -> % --R reverse! : % -> % if $ has shallowlyMutable --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R select! : ((S -> Boolean),%) -> % --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable @@ -25207,8 +26232,6 @@ Database(S): Exports == Implementation where --R setrest! : (%,%) -> % if $ has shallowlyMutable --R setvalue! : (%,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : (((S,S) -> Boolean),%) -> % ---R sort : % -> % if S has ORDSET --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sort! : % -> % if $ has shallowlyMutable and S has ORDSET --R sorted? : (((S,S) -> Boolean),%) -> Boolean @@ -25397,7 +26420,8 @@ DataList(S:OrderedSet) : Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 7 + +--S 1 of 8 r := decimal(22/7) --R --R @@ -25406,7 +26430,7 @@ r := decimal(22/7) --R Type: DecimalExpansion --E 1 ---S 2 of 7 +--S 2 of 8 r + decimal(6/7) --R --R @@ -25414,7 +26438,7 @@ r + decimal(6/7) --R Type: DecimalExpansion --E 2 ---S 3 of 7 +--S 3 of 8 [decimal(1/i) for i in 350..354] --R --R @@ -25426,7 +26450,7 @@ r + decimal(6/7) --R Type: List(DecimalExpansion) --E 3 ---S 4 of 7 +--S 4 of 8 decimal(1/2049) --R --R @@ -25441,7 +26465,7 @@ decimal(1/2049) --R Type: DecimalExpansion --E 4 ---S 5 of 7 +--S 5 of 8 p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9) --R --R @@ -25450,7 +26474,7 @@ p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9) --R Type: Polynomial(DecimalExpansion) --E 5 ---S 6 of 7 +--S 6 of 8 q := differentiate(p, x) --R --R @@ -25459,7 +26483,7 @@ q := differentiate(p, x) --R Type: Polynomial(DecimalExpansion) --E 6 ---S 7 of 7 +--S 7 of 8 g := gcd(p, q) --R --R @@ -25467,6 +26491,125 @@ g := gcd(p, q) --R (7) x + 1.3 --R Type: Polynomial(DecimalExpansion) --E 7 + +--S 8 of 8 +)show DecimalExpansion + DecimalExpansion is a domain constructor + Abbreviation for DecimalExpansion is DECIMAL + This constructor is exposed in this frame. + Issue )edit bookvol10.3.pamphlet to see algebra source code for DECIMAL + +------------------------------- Operations -------------------------------- + ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % + ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % + ?*? : (%,%) -> % ?*? : (Integer,%) -> % + ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % + ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % + ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % + ?-? : (%,%) -> % -? : % -> % + ?/? : (Integer,Integer) -> % ?/? : (%,%) -> % + ?=? : (%,%) -> Boolean D : (%,(Integer -> Integer)) -> % + D : % -> % if Integer has DIFRING 1 : () -> % + 0 : () -> % ?^? : (%,Integer) -> % + ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % + associates? : (%,%) -> Boolean coerce : % -> RadixExpansion(10) + coerce : % -> Fraction(Integer) coerce : Integer -> % + coerce : Fraction(Integer) -> % coerce : % -> % + coerce : Integer -> % coerce : % -> OutputForm + decimal : Fraction(Integer) -> % denom : % -> Integer + denominator : % -> % factor : % -> Factored(%) + gcd : List(%) -> % gcd : (%,%) -> % + hash : % -> SingleInteger init : () -> % if Integer has STEP + inv : % -> % latex : % -> String + lcm : List(%) -> % lcm : (%,%) -> % + numer : % -> Integer numerator : % -> % + one? : % -> Boolean prime? : % -> Boolean + ?quo? : (%,%) -> % random : () -> % if Integer has INS + recip : % -> Union(%,"failed") ?rem? : (%,%) -> % + retract : % -> Integer sample : () -> % + sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) + squareFreePart : % -> % unit? : % -> Boolean + unitCanonical : % -> % zero? : % -> Boolean + ?~=? : (%,%) -> Boolean + ? Boolean if Integer has ORDSET + ?<=? : (%,%) -> Boolean if Integer has ORDSET + ?>? : (%,%) -> Boolean if Integer has ORDSET + ?>=? : (%,%) -> Boolean if Integer has ORDSET + D : (%,(Integer -> Integer),NonNegativeInteger) -> % + D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) + D : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) + D : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) + D : (%,Symbol) -> % if Integer has PDRING(SYMBOL) + D : (%,NonNegativeInteger) -> % if Integer has DIFRING + abs : % -> % if Integer has OINTDOM + ceiling : % -> Integer if Integer has INS + characteristic : () -> NonNegativeInteger + charthRoot : % -> Union(%,"failed") if $ has CHARNZ and Integer has PFECAT or Integer has CHARNZ + coerce : Symbol -> % if Integer has RETRACT(SYMBOL) + conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and Integer has PFECAT + convert : % -> DoubleFloat if Integer has REAL + convert : % -> Float if Integer has REAL + convert : % -> InputForm if Integer has KONVERT(INFORM) + convert : % -> Pattern(Float) if Integer has KONVERT(PATTERN(FLOAT)) + convert : % -> Pattern(Integer) if Integer has KONVERT(PATTERN(INT)) + differentiate : (%,(Integer -> Integer)) -> % + differentiate : (%,(Integer -> Integer),NonNegativeInteger) -> % + differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) + differentiate : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) + differentiate : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) + differentiate : (%,Symbol) -> % if Integer has PDRING(SYMBOL) + differentiate : (%,NonNegativeInteger) -> % if Integer has DIFRING + differentiate : % -> % if Integer has DIFRING + divide : (%,%) -> Record(quotient: %,remainder: %) + ?.? : (%,Integer) -> % if Integer has ELTAB(INT,INT) + euclideanSize : % -> NonNegativeInteger + eval : (%,Symbol,Integer) -> % if Integer has IEVALAB(SYMBOL,INT) + eval : (%,List(Symbol),List(Integer)) -> % if Integer has IEVALAB(SYMBOL,INT) + eval : (%,List(Equation(Integer))) -> % if Integer has EVALAB(INT) + eval : (%,Equation(Integer)) -> % if Integer has EVALAB(INT) + eval : (%,Integer,Integer) -> % if Integer has EVALAB(INT) + eval : (%,List(Integer),List(Integer)) -> % if Integer has EVALAB(INT) + expressIdealMember : (List(%),%) -> Union(List(%),"failed") + exquo : (%,%) -> Union(%,"failed") + extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") + extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) + factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT + factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT + floor : % -> Integer if Integer has INS + fractionPart : % -> Fraction(Integer) + fractionPart : % -> % if Integer has EUCDOM + gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) + lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) + map : ((Integer -> Integer),%) -> % + max : (%,%) -> % if Integer has ORDSET + min : (%,%) -> % if Integer has ORDSET + multiEuclidean : (List(%),%) -> Union(List(%),"failed") + negative? : % -> Boolean if Integer has OINTDOM + nextItem : % -> Union(%,"failed") if Integer has STEP + patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if Integer has PATMAB(FLOAT) + patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if Integer has PATMAB(INT) + positive? : % -> Boolean if Integer has OINTDOM + principalIdeal : List(%) -> Record(coef: List(%),generator: %) + reducedSystem : Matrix(%) -> Matrix(Integer) + reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) + reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if Integer has LINEXP(INT) + reducedSystem : Matrix(%) -> Matrix(Integer) if Integer has LINEXP(INT) + retract : % -> Integer if Integer has RETRACT(INT) + retract : % -> Fraction(Integer) if Integer has RETRACT(INT) + retract : % -> Symbol if Integer has RETRACT(SYMBOL) + retractIfCan : % -> Union(Integer,"failed") if Integer has RETRACT(INT) + retractIfCan : % -> Union(Fraction(Integer),"failed") if Integer has RETRACT(INT) + retractIfCan : % -> Union(Symbol,"failed") if Integer has RETRACT(SYMBOL) + retractIfCan : % -> Union(Integer,"failed") + sign : % -> Integer if Integer has OINTDOM + solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if Integer has PFECAT + squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT + subtractIfCan : (%,%) -> Union(%,"failed") + unitNormal : % -> Record(unit: %,canonical: %,associate: %) + wholePart : % -> Integer if Integer has EUCDOM + +--E 8 + )spool )lisp (bye) \end{chunk} @@ -27264,6 +28407,84 @@ description is in terms of a vector and angle of rotation. --S 1 of 1 )show DenavitHartenbergMatrix +--R +--R DenavitHartenbergMatrix(R: Join(Field,TranscendentalFunctionCategory)) is a domain constructor +--R Abbreviation for DenavitHartenbergMatrix is DHMATRIX +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for DHMATRIX +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,Point(R)) -> Point(R) ?*? : (Vector(R),%) -> Vector(R) +--R ?*? : (%,Vector(R)) -> Vector(R) ?*? : (Integer,%) -> % +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?*? : (%,%) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?+? : (%,%) -> % -? : % -> % +--R ?-? : (%,%) -> % ?/? : (%,R) -> % if R has FIELD +--R antisymmetric? : % -> Boolean coerce : Vector(R) -> % +--R column : (%,Integer) -> Vector(R) copy : % -> % +--R diagonal? : % -> Boolean diagonalMatrix : List(%) -> % +--R diagonalMatrix : List(R) -> % elt : (%,Integer,Integer,R) -> R +--R elt : (%,Integer,Integer) -> R empty : () -> % +--R empty? : % -> Boolean eq? : (%,%) -> Boolean +--R fill! : (%,R) -> % horizConcat : (%,%) -> % +--R identity : () -> % latex : % -> String if R has SETCAT +--R listOfLists : % -> List(List(R)) map : (((R,R) -> R),%,%,R) -> % +--R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R map! : ((R -> R),%) -> % matrix : List(List(R)) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R parts : % -> List(R) pfaffian : % -> R if R has COMRING +--R qelt : (%,Integer,Integer) -> R rotatex : R -> % +--R rotatey : R -> % rotatez : R -> % +--R row : (%,Integer) -> Vector(R) sample : () -> % +--R scale : (R,R,R) -> % setelt : (%,Integer,Integer,R) -> R +--R square? : % -> Boolean squareTop : % -> % +--R symmetric? : % -> Boolean translate : (R,R,R) -> % +--R transpose : % -> % transpose : Vector(R) -> % +--R vertConcat : (%,%) -> % +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ?**? : (%,Integer) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean if R has SETCAT +--R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate +--R coerce : % -> OutputForm if R has SETCAT +--R columnSpace : % -> List(Vector(R)) if R has EUCDOM +--R count : (R,%) -> NonNegativeInteger if $ has finiteAggregate and R has SETCAT +--R count : ((R -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R determinant : % -> R if R has commutative(*) +--R elt : (%,List(Integer),List(Integer)) -> % +--R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT +--R eval : (%,R,R) -> % if R has EVALAB(R) and R has SETCAT +--R eval : (%,Equation(R)) -> % if R has EVALAB(R) and R has SETCAT +--R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) and R has SETCAT +--R every? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate +--R exquo : (%,R) -> Union(%,"failed") if R has INTDOM +--R hash : % -> SingleInteger if R has SETCAT +--R inverse : % -> Union(%,"failed") if R has FIELD +--R less? : (%,NonNegativeInteger) -> Boolean +--R matrix : (NonNegativeInteger,NonNegativeInteger,((Integer,Integer) -> R)) -> % +--R member? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT +--R members : % -> List(R) if $ has finiteAggregate +--R minordet : % -> R if R has commutative(*) +--R more? : (%,NonNegativeInteger) -> Boolean +--R new : (NonNegativeInteger,NonNegativeInteger,R) -> % +--R nullSpace : % -> List(Vector(R)) if R has INTDOM +--R nullity : % -> NonNegativeInteger if R has INTDOM +--R qsetelt! : (%,Integer,Integer,R) -> R +--R rank : % -> NonNegativeInteger if R has INTDOM +--R rowEchelon : % -> % if R has EUCDOM +--R scalarMatrix : (NonNegativeInteger,R) -> % +--R setColumn! : (%,Integer,Vector(R)) -> % +--R setRow! : (%,Integer,Vector(R)) -> % +--R setelt : (%,List(Integer),List(Integer),%) -> % +--R setsubMatrix! : (%,Integer,Integer,%) -> % +--R size? : (%,NonNegativeInteger) -> Boolean +--R subMatrix : (%,Integer,Integer,Integer,Integer) -> % +--R swapColumns! : (%,Integer,Integer) -> % +--R swapRows! : (%,Integer,Integer) -> % +--R zero : (NonNegativeInteger,NonNegativeInteger) -> % +--R ?~=? : (%,%) -> Boolean if R has SETCAT +--R --E 1 )spool @@ -28689,11 +29910,12 @@ latex a --R extractTop! : % -> S front : % -> S --R height : % -> NonNegativeInteger insert! : (S,%) -> % --R insertBottom! : (S,%) -> S insertTop! : (S,%) -> S ---R inspect : % -> S length : % -> NonNegativeInteger ---R map : ((S -> S),%) -> % pop! : % -> S ---R push! : (S,%) -> S reverse! : % -> % ---R rotate! : % -> % sample : () -> % ---R top : % -> S top! : % -> S +--R inspect : % -> S latex : % -> String if S has SETCAT +--R length : % -> NonNegativeInteger map : ((S -> S),%) -> % +--R pop! : % -> S push! : (S,%) -> S +--R reverse! : % -> % rotate! : % -> % +--R sample : () -> % top : % -> S +--R top! : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -28706,7 +29928,6 @@ latex a --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -29377,7 +30598,7 @@ Dequeue(S:SetCategory): DequeueAggregate S with )set message auto off )clear all ---S 1 of 148 +--S 1 of 149 coefRing := Integer --R --R @@ -29385,7 +30606,7 @@ coefRing := Integer --R Type: Domain --E 1 ---S 2 of 148 +--S 2 of 149 lv : List Symbol := [x,y,z] --R --R @@ -29393,7 +30614,7 @@ lv : List Symbol := [x,y,z] --R Type: List(Symbol) --E 2 ---S 3 of 148 +--S 3 of 149 der := DERHAM(coefRing,lv) --R --R @@ -29401,7 +30622,7 @@ der := DERHAM(coefRing,lv) --R Type: Domain --E 3 ---S 4 of 148 +--S 4 of 149 R := Expression coefRing --R --R @@ -29409,7 +30630,7 @@ R := Expression coefRing --R Type: Domain --E 4 ---S 5 of 148 +--S 5 of 149 f : R := x**2*y*z-5*x**3*y**2*z**5 --R --R @@ -29418,7 +30639,7 @@ f : R := x**2*y*z-5*x**3*y**2*z**5 --R Type: Expression(Integer) --E 5 ---S 6 of 148 +--S 6 of 149 g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 --R --R @@ -29427,7 +30648,7 @@ g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 --R Type: Expression(Integer) --E 6 ---S 7 of 148 +--S 7 of 149 h : R :=x*y*z-2*x**3*y*z**2 --R --R @@ -29436,7 +30657,7 @@ h : R :=x*y*z-2*x**3*y*z**2 --R Type: Expression(Integer) --E 7 ---S 8 of 148 +--S 8 of 149 dx : der := generator(1) --R --R @@ -29444,7 +30665,7 @@ dx : der := generator(1) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 8 ---S 9 of 148 +--S 9 of 149 dy : der := generator(2) --R --R @@ -29452,7 +30673,7 @@ dy : der := generator(2) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 9 ---S 10 of 148 +--S 10 of 149 dz : der := generator(3) --R --R @@ -29460,7 +30681,7 @@ dz : der := generator(3) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 10 ---S 11 of 148 +--S 11 of 149 [dx,dy,dz] := [generator(i)$der for i in 1..3] --R --R @@ -29468,7 +30689,7 @@ dz : der := generator(3) --R Type: List(DeRhamComplex(Integer,[x,y,z])) --E 11 ---S 12 of 148 +--S 12 of 149 alpha : der := f*dx + g*dy + h*dz --R --R @@ -29481,7 +30702,7 @@ alpha : der := f*dx + g*dy + h*dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 12 ---S 13 of 148 +--S 13 of 149 beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy --R --R @@ -29489,7 +30710,7 @@ beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy --R Type: DeRhamComplex(Integer,[x,y,z]) --E 13 ---S 14 of 148 +--S 14 of 149 m1:=exteriorDifferential alpha --R --R @@ -29505,7 +30726,7 @@ m1:=exteriorDifferential alpha --R Type: DeRhamComplex(Integer,[x,y,z]) --E 14 ---S 15 of 148 +--S 15 of 149 exteriorDifferential m1 --R --R @@ -29513,7 +30734,7 @@ exteriorDifferential m1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 15 ---S 16 of 148 +--S 16 of 149 gamma := alpha * beta --R --R @@ -29526,7 +30747,7 @@ gamma := alpha * beta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 16 ---S 17 of 148 +--S 17 of 149 exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - _ alpha * exteriorDifferential(beta)) --R @@ -29535,7 +30756,7 @@ exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - _ --R Type: DeRhamComplex(Integer,[x,y,z]) --E 17 ---S 18 of 148 +--S 18 of 149 a : BOP := operator('a) --R --R @@ -29543,7 +30764,7 @@ a : BOP := operator('a) --R Type: BasicOperator --E 18 ---S 19 of 148 +--S 19 of 149 b : BOP := operator('b) --R --R @@ -29551,7 +30772,7 @@ b : BOP := operator('b) --R Type: BasicOperator --E 19 ---S 20 of 148 +--S 20 of 149 c : BOP := operator('c) --R --R @@ -29559,7 +30780,7 @@ c : BOP := operator('c) --R Type: BasicOperator --E 20 ---S 21 of 148 +--S 21 of 149 sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz --R --R @@ -29567,7 +30788,7 @@ sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 21 ---S 22 of 148 +--S 22 of 149 theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R --R @@ -29575,7 +30796,7 @@ theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 22 ---S 23 of 148 +--S 23 of 149 totalDifferential(a(x,y,z))$der --R --R @@ -29584,7 +30805,7 @@ totalDifferential(a(x,y,z))$der --R Type: DeRhamComplex(Integer,[x,y,z]) --E 23 ---S 24 of 148 +--S 24 of 149 exteriorDifferential sigma --R --R @@ -29597,7 +30818,7 @@ exteriorDifferential sigma --R Type: DeRhamComplex(Integer,[x,y,z]) --E 24 ---S 25 of 148 +--S 25 of 149 exteriorDifferential theta --R --R @@ -29606,7 +30827,7 @@ exteriorDifferential theta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 25 ---S 26 of 148 +--S 26 of 149 one : der := 1 --R --R @@ -29614,7 +30835,7 @@ one : der := 1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 26 ---S 27 of 148 +--S 27 of 149 g1 : der := a([x,t,y,u,v,z,e]) * one --R --R @@ -29622,7 +30843,7 @@ g1 : der := a([x,t,y,u,v,z,e]) * one --R Type: DeRhamComplex(Integer,[x,y,z]) --E 27 ---S 28 of 148 +--S 28 of 149 h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one --R --R @@ -29630,7 +30851,7 @@ h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one --R Type: DeRhamComplex(Integer,[x,y,z]) --E 28 ---S 29 of 148 +--S 29 of 149 exteriorDifferential g1 --R --R @@ -29639,7 +30860,7 @@ exteriorDifferential g1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 29 ---S 30 of 148 +--S 30 of 149 exteriorDifferential h1 --R --R @@ -29660,7 +30881,7 @@ exteriorDifferential h1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 30 ---S 31 of 148 +--S 31 of 149 coefficient(gamma, dx*dy) --R --R @@ -29669,7 +30890,7 @@ coefficient(gamma, dx*dy) --R Type: Expression(Integer) --E 31 ---S 32 of 148 +--S 32 of 149 coefficient(gamma, one) --R --R @@ -29677,7 +30898,7 @@ coefficient(gamma, one) --R Type: Expression(Integer) --E 32 ---S 33 of 148 +--S 33 of 149 coefficient(g1,one) --R --R @@ -29685,7 +30906,7 @@ coefficient(g1,one) --R Type: Expression(Integer) --E 33 ---S 34 of 148 +--S 34 of 149 gamma := alpha * beta --R --R @@ -29698,7 +30919,7 @@ gamma := alpha * beta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 34 ---S 35 of 148 +--S 35 of 149 t1:=generator(1)$der --R --R @@ -29706,7 +30927,7 @@ t1:=generator(1)$der --R Type: DeRhamComplex(Integer,[x,y,z]) --E 35 ---S 36 of 148 +--S 36 of 149 degree(t1) --R --R @@ -29716,7 +30937,7 @@ degree(t1) )clear all ---S 37 of 148 +--S 37 of 149 X:=DERHAM(Integer,[x,y,z]) --R --R @@ -29724,7 +30945,7 @@ X:=DERHAM(Integer,[x,y,z]) --R Type: Domain --E 37 ---S 38 of 148 +--S 38 of 149 [dx,dy,dz]:=[generator(i)$X for i in 1..3] --R --R @@ -29732,7 +30953,7 @@ X:=DERHAM(Integer,[x,y,z]) --R Type: List(DeRhamComplex(Integer,[x,y,z])) --E 38 ---S 39 of 148 +--S 39 of 149 f:BOP:=operator('f) --R --R @@ -29740,7 +30961,7 @@ f:BOP:=operator('f) --R Type: BasicOperator --E 39 ---S 40 of 148 +--S 40 of 149 g:BOP:=operator('g) --R --R @@ -29748,7 +30969,7 @@ g:BOP:=operator('g) --R Type: BasicOperator --E 40 ---S 41 of 148 +--S 41 of 149 h:BOP:=operator('h) --R --R @@ -29756,7 +30977,7 @@ h:BOP:=operator('h) --R Type: BasicOperator --E 41 ---S 42 of 148 +--S 42 of 149 a:BOP:=operator('a) --R --R @@ -29764,7 +30985,7 @@ a:BOP:=operator('a) --R Type: BasicOperator --E 42 ---S 43 of 148 +--S 43 of 149 b:BOP:=operator('b) --R --R @@ -29772,7 +30993,7 @@ b:BOP:=operator('b) --R Type: BasicOperator --E 43 ---S 44 of 148 +--S 44 of 149 c:BOP:=operator('c) --R --R @@ -29780,7 +31001,7 @@ c:BOP:=operator('c) --R Type: BasicOperator --E 44 ---S 45 of 148 +--S 45 of 149 sigma:=f(x,y,z)*dx + g(x,y,z)*dy + h(x,y,z)*dz --R --R @@ -29788,7 +31009,7 @@ sigma:=f(x,y,z)*dx + g(x,y,z)*dy + h(x,y,z)*dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 45 ---S 46 of 148 +--S 46 of 149 theta:=a(x,y,z)*dx*dy + b(x,y,z)*dx*dz + c(x,y,z)*dy*dz --R --R @@ -29796,13 +31017,13 @@ theta:=a(x,y,z)*dx*dy + b(x,y,z)*dx*dz + c(x,y,z)*dy*dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 46 ---S 47 of 148 +--S 47 of 149 d ==> exteriorDifferential --R --R Type: Void --E 47 ---S 48 of 148 +--S 48 of 149 leadingCoefficient sigma --R --R @@ -29810,7 +31031,7 @@ leadingCoefficient sigma --R Type: Expression(Integer) --E 48 ---S 49 of 148 +--S 49 of 149 leadingBasisTerm sigma --R --R @@ -29818,7 +31039,7 @@ leadingBasisTerm sigma --R Type: DeRhamComplex(Integer,[x,y,z]) --E 49 ---S 50 of 148 +--S 50 of 149 reductum sigma --R --R @@ -29826,7 +31047,7 @@ reductum sigma --R Type: DeRhamComplex(Integer,[x,y,z]) --E 50 ---S 51 of 148 +--S 51 of 149 coefficient(sigma,dz) --R --R @@ -29834,7 +31055,7 @@ coefficient(sigma,dz) --R Type: Expression(Integer) --E 51 ---S 52 of 148 +--S 52 of 149 homogeneous? sigma --R --R @@ -29842,7 +31063,7 @@ homogeneous? sigma --R Type: Boolean --E 52 ---S 53 of 148 +--S 53 of 149 homogeneous? (sigma+theta) --R --R @@ -29850,7 +31071,7 @@ homogeneous? (sigma+theta) --R Type: Boolean --E 53 ---S 54 of 148 +--S 54 of 149 retractable? sigma --R --R @@ -29858,7 +31079,7 @@ retractable? sigma --R Type: Boolean --E 54 ---S 55 of 148 +--S 55 of 149 retractable? (1::X) --R --R @@ -29866,7 +31087,7 @@ retractable? (1::X) --R Type: Boolean --E 55 ---S 56 of 148 +--S 56 of 149 [degree x for x in [sigma,theta,1::X]] --R --R @@ -29874,7 +31095,7 @@ retractable? (1::X) --R Type: List(NonNegativeInteger) --E 56 ---S 57 of 148 +--S 57 of 149 R:=Expression(Integer) --R --R @@ -29882,19 +31103,19 @@ R:=Expression(Integer) --R Type: Domain --E 57 ---S 58 of 148 +--S 58 of 149 T:R->R --R --R Type: Void --E 58 ---S 59 of 148 +--S 59 of 149 T(x)==x^2 --R --R Type: Void --E 59 ---S 60 of 148 +--S 60 of 149 map(T,sigma) --R --R Compiling function T with type Expression(Integer) -> Expression( @@ -29905,7 +31126,7 @@ map(T,sigma) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 60 ---S 61 of 148 +--S 61 of 149 exteriorDifferential sigma --R --R @@ -29918,7 +31139,7 @@ exteriorDifferential sigma --R Type: DeRhamComplex(Integer,[x,y,z]) --E 61 ---S 62 of 148 +--S 62 of 149 d theta --R --R @@ -29927,7 +31148,7 @@ d theta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 62 ---S 63 of 148 +--S 63 of 149 F:=operator 'F --R --R @@ -29935,7 +31156,7 @@ F:=operator 'F --R Type: BasicOperator --E 63 ---S 64 of 148 +--S 64 of 149 nullForm:=F(x,y,z)*(1::X) --R --R @@ -29943,7 +31164,7 @@ nullForm:=F(x,y,z)*(1::X) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 64 ---S 65 of 148 +--S 65 of 149 d nullForm --R --R @@ -29952,7 +31173,7 @@ d nullForm --R Type: DeRhamComplex(Integer,[x,y,z]) --E 65 ---S 66 of 148 +--S 66 of 149 totalDifferential(x^2+y^2+sin(x)*z^2)$X --R --R @@ -29961,7 +31182,7 @@ totalDifferential(x^2+y^2+sin(x)*z^2)$X --R Type: DeRhamComplex(Integer,[x,y,z]) --E 66 ---S 67 of 148 +--S 67 of 149 d(d sigma) --R --R @@ -29976,7 +31197,7 @@ d(d sigma) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 67 ---S 68 of 148 +--S 68 of 149 d(d theta) --R --R @@ -29984,7 +31205,7 @@ d(d theta) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 68 ---S 69 of 148 +--S 69 of 149 sigma*theta --R --R @@ -29992,7 +31213,7 @@ sigma*theta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 69 ---S 70 of 148 +--S 70 of 149 dx*dy+dy*dx --R --R @@ -30000,7 +31221,7 @@ dx*dy+dy*dx --R Type: DeRhamComplex(Integer,[x,y,z]) --E 70 ---S 71 of 148 +--S 71 of 149 sigma+dz --R --R @@ -30008,7 +31229,7 @@ sigma+dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 71 ---S 72 of 148 +--S 72 of 149 theta-(2/3)*sigma+%pi*dx --R --R @@ -30021,7 +31242,7 @@ theta-(2/3)*sigma+%pi*dx --R Type: DeRhamComplex(Integer,[x,y,z]) --E 72 ---S 73 of 148 +--S 73 of 149 G:=diagonalMatrix([1,1,1]) --R --R @@ -30033,7 +31254,7 @@ G:=diagonalMatrix([1,1,1]) --R Type: Matrix(Integer) --E 73 ---S 74 of 148 +--S 74 of 149 dot(sigma,sigma,G) --R --R @@ -30042,7 +31263,7 @@ dot(sigma,sigma,G) --R Type: Expression(Integer) --E 74 ---S 75 of 148 +--S 75 of 149 hodgeStar(sigma,G) --R --R @@ -30050,7 +31271,7 @@ hodgeStar(sigma,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 75 ---S 76 of 148 +--S 76 of 149 A:=dot(sigma,sigma,G)*sqrt(abs(determinant(G))) --R --R @@ -30059,7 +31280,7 @@ A:=dot(sigma,sigma,G)*sqrt(abs(determinant(G))) --R Type: Expression(Integer) --E 76 ---S 77 of 148 +--S 77 of 149 B:=sigma*hodgeStar(sigma,G) --R --R @@ -30068,7 +31289,7 @@ B:=sigma*hodgeStar(sigma,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 77 ---S 78 of 148 +--S 78 of 149 test(A*dx*dy*dz=B) --R --R @@ -30076,7 +31297,7 @@ test(A*dx*dy*dz=B) --R Type: Boolean --E 78 ---S 79 of 148 +--S 79 of 149 hodgeStar(dx,G) --R --R @@ -30084,7 +31305,7 @@ hodgeStar(dx,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 79 ---S 80 of 148 +--S 80 of 149 hodgeStar(dy,G) --R --R @@ -30092,7 +31313,7 @@ hodgeStar(dy,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 80 ---S 81 of 148 +--S 81 of 149 d hodgeStar(d nullForm,G) --- = Laplace(F) ? --R --R @@ -30101,7 +31322,7 @@ d hodgeStar(d nullForm,G) --- = Laplace(F) ? --R Type: DeRhamComplex(Integer,[x,y,z]) --E 81 ---S 82 of 148 +--S 82 of 149 H:=diagonalMatrix([1+x^2,z^2+exp(-y^2),1/z^2]) --R --R @@ -30119,7 +31340,7 @@ H:=diagonalMatrix([1+x^2,z^2+exp(-y^2),1/z^2]) --R Type: Matrix(Expression(Integer)) --E 82 ---S 83 of 148 +--S 83 of 149 hodgeStar(dy,H) --R --R @@ -30137,7 +31358,7 @@ hodgeStar(dy,H) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 83 ---S 84 of 148 +--S 84 of 149 hodgeStar(sigma,H) --R --R @@ -30175,7 +31396,7 @@ hodgeStar(sigma,H) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 84 ---S 85 of 148 +--S 85 of 149 VolH:=sqrt(abs(determinant(H)))*dx*dy*dz --R --R @@ -30189,7 +31410,7 @@ VolH:=sqrt(abs(determinant(H)))*dx*dy*dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 85 ---S 86 of 148 +--S 86 of 149 AA:=dot(sigma,sigma,H)*VolH --R --R @@ -30217,7 +31438,7 @@ AA:=dot(sigma,sigma,H)*VolH --R Type: DeRhamComplex(Integer,[x,y,z]) --E 86 ---S 87 of 148 +--S 87 of 149 BB:=sigma*hodgeStar(sigma,H) --R --R @@ -30245,7 +31466,7 @@ BB:=sigma*hodgeStar(sigma,H) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 87 ---S 88 of 148 +--S 88 of 149 test(AA=BB) --R --R @@ -30253,7 +31474,7 @@ test(AA=BB) --R Type: Boolean --E 88 ---S 89 of 148 +--S 89 of 149 M:=DERHAM(Integer,[t,x,y,z]) --R --R @@ -30261,7 +31482,7 @@ M:=DERHAM(Integer,[t,x,y,z]) --R Type: Domain --E 89 ---S 90 of 148 +--S 90 of 149 [dt,dx,dy,dz]:=[generator(i)$M for i in 1..4] --R --R @@ -30269,7 +31490,7 @@ M:=DERHAM(Integer,[t,x,y,z]) --R Type: List(DeRhamComplex(Integer,[t,x,y,z])) --E 90 ---S 91 of 148 +--S 91 of 149 L:=diagonalMatrix([1,-1,-1,-1]) --R --R @@ -30283,7 +31504,7 @@ L:=diagonalMatrix([1,-1,-1,-1]) --R Type: Matrix(Integer) --E 91 ---S 92 of 148 +--S 92 of 149 hodgeStar(dt,L) --R --R @@ -30291,7 +31512,7 @@ hodgeStar(dt,L) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 92 ---S 93 of 148 +--S 93 of 149 hodgeStar(dx,L) --R --R @@ -30299,7 +31520,7 @@ hodgeStar(dx,L) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 93 ---S 94 of 148 +--S 94 of 149 hodgeStar(dy,L) --R --R @@ -30307,7 +31528,7 @@ hodgeStar(dy,L) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 94 ---S 95 of 148 +--S 95 of 149 hodgeStar(dz,L) --R --R @@ -30315,7 +31536,7 @@ hodgeStar(dz,L) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 95 ---S 96 of 148 +--S 96 of 149 hodgeStar(dt*dz,L) --R --R @@ -30323,7 +31544,7 @@ hodgeStar(dt*dz,L) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 96 ---S 97 of 148 +--S 97 of 149 P:=a(t,x,y,z)*(dx+dy+dz-dt) --R --R @@ -30331,7 +31552,7 @@ P:=a(t,x,y,z)*(dx+dy+dz-dt) --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 97 ---S 98 of 148 +--S 98 of 149 dot(P,P,L) --R --R @@ -30340,7 +31561,7 @@ dot(P,P,L) --R Type: Expression(Integer) --E 98 ---S 99 of 148 +--S 99 of 149 P*P --R --R @@ -30348,7 +31569,7 @@ P*P --R Type: DeRhamComplex(Integer,[t,x,y,z]) --E 99 ---S 100 of 148 +--S 100 of 149 d P --R --R @@ -30366,7 +31587,7 @@ d P )clear all ---S 101 of 148 +--S 101 of 149 coefRing := Integer --R --R @@ -30374,7 +31595,7 @@ coefRing := Integer --R Type: Domain --E 101 ---S 102 of 148 +--S 102 of 149 R3 : List Symbol := [x,y,z] --R --R @@ -30382,7 +31603,7 @@ R3 : List Symbol := [x,y,z] --R Type: List(Symbol) --E 102 ---S 103 of 148 +--S 103 of 149 X := DERHAM(coefRing,R3) --R --R @@ -30390,7 +31611,7 @@ X := DERHAM(coefRing,R3) --R Type: Domain --E 103 ---S 104 of 148 +--S 104 of 149 [dx,dy,dz] := [generator(i)$X for i in 1..3] --R --R @@ -30398,7 +31619,7 @@ X := DERHAM(coefRing,R3) --R Type: List(DeRhamComplex(Integer,[x,y,z])) --E 104 ---S 105 of 148 +--S 105 of 149 f : BOP := operator('f) --R --R @@ -30406,7 +31627,7 @@ f : BOP := operator('f) --R Type: BasicOperator --E 105 ---S 106 of 148 +--S 106 of 149 g : BOP := operator('g) --R --R @@ -30414,7 +31635,7 @@ g : BOP := operator('g) --R Type: BasicOperator --E 106 ---S 107 of 148 +--S 107 of 149 h : BOP := operator('h) --R --R @@ -30422,7 +31643,7 @@ h : BOP := operator('h) --R Type: BasicOperator --E 107 ---S 108 of 148 +--S 108 of 149 a : BOP := operator('a) --R --R @@ -30430,7 +31651,7 @@ a : BOP := operator('a) --R Type: BasicOperator --E 108 ---S 109 of 148 +--S 109 of 149 b : BOP := operator('b) --R --R @@ -30438,7 +31659,7 @@ b : BOP := operator('b) --R Type: BasicOperator --E 109 ---S 110 of 148 +--S 110 of 149 c : BOP := operator('c) --R --R @@ -30446,7 +31667,7 @@ c : BOP := operator('c) --R Type: BasicOperator --E 110 ---S 111 of 148 +--S 111 of 149 U : BOP := operator('U) --R --R @@ -30454,7 +31675,7 @@ U : BOP := operator('U) --R Type: BasicOperator --E 111 ---S 112 of 148 +--S 112 of 149 V : BOP := operator('V) --R --R @@ -30462,7 +31683,7 @@ V : BOP := operator('V) --R Type: BasicOperator --E 112 ---S 113 of 148 +--S 113 of 149 W : BOP := operator('W) --R --R @@ -30470,7 +31691,7 @@ W : BOP := operator('W) --R Type: BasicOperator --E 113 ---S 114 of 148 +--S 114 of 149 v := vector[U(x,y,z),V(x,y,z),W(x,y,z)] --R --R @@ -30478,7 +31699,7 @@ v := vector[U(x,y,z),V(x,y,z),W(x,y,z)] --R Type: Vector(Expression(Integer)) --E 114 ---S 115 of 148 +--S 115 of 149 sigma := f(x,y,z) * dx + g(x,y,z) * dy + h(x,y,z) * dz --R --R @@ -30486,7 +31707,7 @@ sigma := f(x,y,z) * dx + g(x,y,z) * dy + h(x,y,z) * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 115 ---S 116 of 148 +--S 116 of 149 theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R --R @@ -30494,13 +31715,13 @@ theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 116 ---S 117 of 148 +--S 117 of 149 d ==> exteriorDifferential --R --R Type: Void --E 117 ---S 118 of 148 +--S 118 of 149 G := diagonalMatrix([1,1,1]) --R --R @@ -30512,7 +31733,7 @@ G := diagonalMatrix([1,1,1]) --R Type: Matrix(Integer) --E 118 ---S 119 of 148 +--S 119 of 149 interiorProduct(v,sigma,G) --R --R @@ -30520,7 +31741,7 @@ interiorProduct(v,sigma,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 119 ---S 120 of 148 +--S 120 of 149 interiorProduct(v,theta,G) --R --R @@ -30533,7 +31754,7 @@ interiorProduct(v,theta,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 120 ---S 121 of 148 +--S 121 of 149 t1 := d interiorProduct(v,dz*dy*dz,G) --R --R @@ -30541,7 +31762,7 @@ t1 := d interiorProduct(v,dz*dy*dz,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 121 ---S 122 of 148 +--S 122 of 149 hodgeStar(t1,G) -- should be div(v) --R --R @@ -30549,7 +31770,7 @@ hodgeStar(t1,G) -- should be div(v) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 122 ---S 123 of 148 +--S 123 of 149 eta := lieDerivative(v,theta,G) --R --R @@ -30595,7 +31816,7 @@ eta := lieDerivative(v,theta,G) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 123 ---S 124 of 148 +--S 124 of 149 proj(dx+dy*dz+dx*dy*dz,2) --R --R @@ -30603,7 +31824,7 @@ proj(dx+dy*dz+dx*dy*dz,2) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 124 ---S 125 of 148 +--S 125 of 149 proj(sigma+theta,1) --R --R @@ -30611,7 +31832,7 @@ proj(sigma+theta,1) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 125 ---S 126 of 148 +--S 126 of 149 dim(sigma) --R --R @@ -30619,7 +31840,7 @@ dim(sigma) --R Type: PositiveInteger --E 126 ---S 127 of 148 +--S 127 of 149 degree(sigma) --R --R @@ -30630,7 +31851,7 @@ degree(sigma) -- Laplace Operator for $S^2:\delta F := \star d \star dF$ )clear all ---S 128 of 148 +--S 128 of 149 coefRing := Integer --R --R @@ -30638,7 +31859,7 @@ coefRing := Integer --R Type: Domain --E 128 ---S 129 of 148 +--S 129 of 149 S2 : List Symbol := [r,theta] --R --R @@ -30646,7 +31867,7 @@ S2 : List Symbol := [r,theta] --R Type: List(Symbol) --E 129 ---S 130 of 148 +--S 130 of 149 Y := DERHAM(coefRing,S2) --R --R @@ -30654,7 +31875,7 @@ Y := DERHAM(coefRing,S2) --R Type: Domain --E 130 ---S 131 of 148 +--S 131 of 149 g := diagonalMatrix([1,r^2]) --R --R @@ -30665,7 +31886,7 @@ g := diagonalMatrix([1,r^2]) --R Type: Matrix(Polynomial(Integer)) --E 131 ---S 132 of 148 +--S 132 of 149 F := operator 'F --R --R @@ -30673,13 +31894,13 @@ F := operator 'F --R Type: BasicOperator --E 132 ---S 133 of 148 +--S 133 of 149 d ==> exteriorDifferential --R --R Type: Void --E 133 ---S 134 of 148 +--S 134 of 149 [dr,dtheta] := [generator(i)$Y for i in 1..2] --R --R @@ -30687,7 +31908,7 @@ d ==> exteriorDifferential --R Type: List(DeRhamComplex(Integer,[r,theta])) --E 134 ---S 135 of 148 +--S 135 of 149 F0 := F(r,theta)*1::Y --R --R @@ -30695,7 +31916,7 @@ F0 := F(r,theta)*1::Y --R Type: DeRhamComplex(Integer,[r,theta]) --E 135 ---S 136 of 148 +--S 136 of 149 F1 := d F0 --R --R @@ -30704,7 +31925,7 @@ F1 := d F0 --R Type: DeRhamComplex(Integer,[r,theta]) --E 136 ---S 137 of 148 +--S 137 of 149 F2 := hodgeStar(F1,g) --R --R @@ -30718,7 +31939,7 @@ F2 := hodgeStar(F1,g) --R Type: DeRhamComplex(Integer,[r,theta]) --E 137 ---S 138 of 148 +--S 138 of 149 F3 := d F2 --R --R @@ -30735,7 +31956,7 @@ F3 := d F2 --R Type: DeRhamComplex(Integer,[r,theta]) --E 138 ---S 139 of 148 +--S 139 of 149 LaplaceF := hodgeStar(F3,g) --R --R @@ -30749,7 +31970,7 @@ LaplaceF := hodgeStar(F3,g) --R Type: DeRhamComplex(Integer,[r,theta]) --E 139 ---S 140 of 148 +--S 140 of 149 LapF := LaplaceF::Expression(Integer) --R --R @@ -30763,7 +31984,7 @@ LapF := LaplaceF::Expression(Integer) --R Type: Expression(Integer) --E 140 ---S 141 of 148 +--S 141 of 149 subst(LapF,abs(r^2)=r^2) --R --R @@ -30778,7 +31999,7 @@ subst(LapF,abs(r^2)=r^2) -- Internal representation ---S 142 of 148 +--S 142 of 149 R := Expression(Integer) --R --R @@ -30786,7 +32007,7 @@ R := Expression(Integer) --R Type: Domain --E 142 ---S 143 of 148 +--S 143 of 149 Y := DERHAM(coefRing,S2) --R --R @@ -30794,19 +32015,19 @@ Y := DERHAM(coefRing,S2) --R Type: Domain --E 143 ---S 144 of 148 +--S 144 of 149 terms : Y -> List Record(k : EAB, c : R) --R --R Type: Void --E 144 ---S 145 of 148 +--S 145 of 149 terms(a) == a pretend List Record(k : EAB, c : R) --R --R Type: Void --E 145 ---S 146 of 148 +--S 146 of 149 terms(dr) --R --R Compiling function terms with type DeRhamComplex(Integer,[r,theta]) @@ -30816,7 +32037,7 @@ terms(dr) --R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) --E 146 ---S 147 of 148 +--S 147 of 149 terms(dtheta) --R --R @@ -30824,7 +32045,7 @@ terms(dtheta) --R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) --E 147 ---S 148 of 148 +--S 148 of 149 terms(F1) --R --R @@ -30833,6 +32054,46 @@ terms(F1) --R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) --E 148 +--S 149 of 149 +)show DeRhamComplex +--R +--R DeRhamComplex(CoefRing: Join(Ring,OrderedSet),listIndVar: List(Symbol)) is a domain constructor +--R Abbreviation for DeRhamComplex is DERHAM +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for DERHAM +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : Expression(CoefRing) -> % coerce : Integer -> % +--R coerce : % -> OutputForm degree : % -> NonNegativeInteger +--R dim : % -> NonNegativeInteger exteriorDifferential : % -> % +--R generator : NonNegativeInteger -> % hash : % -> SingleInteger +--R homogeneous? : % -> Boolean latex : % -> String +--R leadingBasisTerm : % -> % one? : % -> Boolean +--R proj : (%,NonNegativeInteger) -> % recip : % -> Union(%,"failed") +--R reductum : % -> % retract : % -> Expression(CoefRing) +--R retractable? : % -> Boolean sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R ?*? : (Expression(CoefRing),%) -> % +--R characteristic : () -> NonNegativeInteger +--R coefficient : (%,%) -> Expression(CoefRing) +--R dot : (%,%,SquareMatrix(#(listIndVar),Expression(CoefRing))) -> Expression(CoefRing) +--R hodgeStar : (%,SquareMatrix(#(listIndVar),Expression(CoefRing))) -> % +--R interiorProduct : (Vector(Expression(CoefRing)),%,SquareMatrix(#(listIndVar),Expression(CoefRing))) -> % +--R leadingCoefficient : % -> Expression(CoefRing) +--R lieDerivative : (Vector(Expression(CoefRing)),%,SquareMatrix(#(listIndVar),Expression(CoefRing))) -> % +--R map : ((Expression(CoefRing) -> Expression(CoefRing)),%) -> % +--R retractIfCan : % -> Union(Expression(CoefRing),"failed") +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R totalDifferential : Expression(CoefRing) -> % +--R +--E 149 )spool )lisp (bye) @@ -32988,11 +34249,11 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where --R empty? : % -> Boolean encode : % -> String --R eq? : (%,%) -> Boolean fullOut : % -> OutputForm --R fullOutput : () -> Boolean fullOutput : Boolean -> Boolean ---R leaf? : % -> Boolean leaves : % -> List(S) ---R map : ((S -> S),%) -> % nodes : % -> List(%) ---R sample : () -> % tree : List(S) -> % ---R tree : S -> % tree : (S,List(%)) -> % ---R value : % -> S +--R latex : % -> String if S has SETCAT leaf? : % -> Boolean +--R leaves : % -> List(S) map : ((S -> S),%) -> % +--R nodes : % -> List(%) sample : () -> % +--R tree : List(S) -> % tree : S -> % +--R tree : (S,List(%)) -> % value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -33006,7 +34267,6 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -33185,27 +34445,36 @@ DesingTree(S: SetCategory): T==C where --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R D : (%,(R -> R)) -> % D : % -> % if R has DIFRING +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean D : (%,(R -> R)) -> % +--R D : % -> % if R has DIFRING D : (%,V,NonNegativeInteger) -> % --R D : (%,List(V)) -> % D : (%,V) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coefficients : % -> List(R) +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coefficients : % -> List(R) coerce : % -> % if R has INTDOM --R coerce : S -> % coerce : V -> % --R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm degree : % -> IndexedExponents(V) +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM +--R degree : % -> IndexedExponents(V) differentiate : (%,(R -> R)) -> % --R differentiate : (%,List(V)) -> % differentiate : (%,V) -> % --R eval : (%,List(V),List(%)) -> % eval : (%,V,%) -> % --R eval : (%,List(V),List(R)) -> % eval : (%,V,R) -> % ---R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(%),List(%)) -> % eval : (%,%,%) -> % +--R eval : (%,Equation(%)) -> % eval : (%,List(Equation(%))) -> % +--R gcd : (%,%) -> % if R has GCDDOM gcd : List(%) -> % if R has GCDDOM --R ground : % -> R ground? : % -> Boolean --R hash : % -> SingleInteger initial : % -> % --R isobaric? : % -> Boolean latex : % -> String +--R lcm : (%,%) -> % if R has GCDDOM lcm : List(%) -> % if R has GCDDOM --R leader : % -> V leadingCoefficient : % -> R --R leadingMonomial : % -> % map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET --R monomial? : % -> Boolean monomials : % -> List(%) --R one? : % -> Boolean order : % -> NonNegativeInteger +--R order : (%,S) -> NonNegativeInteger primitiveMonomials : % -> List(%) --R recip : % -> Union(%,"failed") reductum : % -> % --R retract : % -> S retract : % -> V --R retract : % -> R sample : () -> % @@ -33214,9 +34483,6 @@ DesingTree(S: SetCategory): T==C where --R ?~=? : (%,%) -> Boolean --R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) --R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?>? : (%,%) -> Boolean if R has ORDSET @@ -33228,8 +34494,6 @@ DesingTree(S: SetCategory): T==C where --R D : (%,Symbol) -> % if R has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if R has DIFRING --R D : (%,List(V),List(NonNegativeInteger)) -> % ---R D : (%,V,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if R has INTDOM --R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING --R characteristic : () -> NonNegativeInteger @@ -33237,12 +34501,10 @@ DesingTree(S: SetCategory): T==C where --R coefficient : (%,List(V),List(NonNegativeInteger)) -> % --R coefficient : (%,V,NonNegativeInteger) -> % --R coefficient : (%,IndexedExponents(V)) -> R ---R coerce : % -> % if R has INTDOM --R coerce : Fraction(Integer) -> % if R has ALGEBRA(FRAC(INT)) or R has RETRACT(FRAC(INT)) --R coerce : SparseMultivariatePolynomial(R,S) -> % --R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and R has PFECAT --R content : (%,V) -> % if R has GCDDOM ---R content : % -> R if R has GCDDOM --R convert : % -> InputForm if R has KONVERT(INFORM) and V has KONVERT(INFORM) --R convert : % -> Pattern(Integer) if R has KONVERT(PATTERN(INT)) and V has KONVERT(PATTERN(INT)) --R convert : % -> Pattern(Float) if R has KONVERT(PATTERN(FLOAT)) and V has KONVERT(PATTERN(FLOAT)) @@ -33250,7 +34512,6 @@ DesingTree(S: SetCategory): T==C where --R degree : (%,List(V)) -> List(NonNegativeInteger) --R degree : (%,V) -> NonNegativeInteger --R differentialVariables : % -> List(S) ---R differentiate : (%,(R -> R)) -> % --R differentiate : (%,(R -> R),NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) --R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) @@ -33265,28 +34526,20 @@ DesingTree(S: SetCategory): T==C where --R eval : (%,S,R) -> % if R has DIFRING --R eval : (%,List(S),List(%)) -> % if R has DIFRING --R eval : (%,S,%) -> % if R has DIFRING ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R exquo : (%,%) -> Union(%,"failed") if R has INTDOM --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R factor : % -> Factored(%) if R has PFECAT --R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT --R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT ---R gcd : (%,%) -> % if R has GCDDOM ---R gcd : List(%) -> % if R has GCDDOM --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM --R isExpt : % -> Union(Record(var: V,exponent: NonNegativeInteger),"failed") --R isPlus : % -> Union(List(%),"failed") --R isTimes : % -> Union(List(%),"failed") ---R lcm : (%,%) -> % if R has GCDDOM ---R lcm : List(%) -> % if R has GCDDOM --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM --R mainVariable : % -> Union(V,"failed") --R makeVariable : % -> (NonNegativeInteger -> %) if R has DIFRING --R makeVariable : S -> (NonNegativeInteger -> %) --R mapExponents : ((IndexedExponents(V) -> IndexedExponents(V)),%) -> % ---R max : (%,%) -> % if R has ORDSET ---R min : (%,%) -> % if R has ORDSET --R minimumDegree : (%,List(V)) -> List(NonNegativeInteger) --R minimumDegree : (%,V) -> NonNegativeInteger --R minimumDegree : % -> IndexedExponents(V) @@ -33297,12 +34550,10 @@ DesingTree(S: SetCategory): T==C where --R multivariate : (SparseUnivariatePolynomial(%),V) -> % --R multivariate : (SparseUnivariatePolynomial(R),V) -> % --R numberOfMonomials : % -> NonNegativeInteger ---R order : (%,S) -> NonNegativeInteger --R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if R has PATMAB(INT) and V has PATMAB(INT) --R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if R has PATMAB(FLOAT) and V has PATMAB(FLOAT) --R pomopo! : (%,R,IndexedExponents(V),%) -> % --R prime? : % -> Boolean if R has PFECAT ---R primitiveMonomials : % -> List(%) --R primitivePart : (%,V) -> % if R has GCDDOM --R primitivePart : % -> % if R has GCDDOM --R reducedSystem : Matrix(%) -> Matrix(R) @@ -33516,33 +34767,35 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --S 1 of 1 )show DirectProduct +--R --R DirectProduct(dim: NonNegativeInteger,R: Type) is a domain constructor --R Abbreviation for DirectProduct is DIRPROD --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for DIRPROD --R --R------------------------------- Operations -------------------------------- ---R -? : % -> % if R has RING 1 : () -> % if R has MONOID ---R 0 : () -> % if R has CABMON coerce : % -> Vector(R) +--R ?*? : (R,%) -> % if R has MONOID ?*? : (%,R) -> % if R has MONOID +--R ?*? : (%,%) -> % if R has MONOID ?+? : (%,%) -> % if R has ABELSG +--R -? : % -> % if R has RING ?-? : (%,%) -> % if R has RING +--R ?/? : (%,R) -> % if R has FIELD 1 : () -> % if R has MONOID +--R 0 : () -> % if R has CABMON abs : % -> % if R has ORDRING +--R coerce : R -> % if R has SETCAT coerce : % -> Vector(R) --R copy : % -> % directProduct : Vector(R) -> % ---R ?.? : (%,Integer) -> R elt : (%,Integer,R) -> R ---R empty : () -> % empty? : % -> Boolean ---R entries : % -> List(R) eq? : (%,%) -> Boolean ---R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R map : ((R -> R),%) -> % qelt : (%,Integer) -> R ---R sample : () -> % +--R dot : (%,%) -> R if R has RING ?.? : (%,Integer) -> R +--R elt : (%,Integer,R) -> R empty : () -> % +--R empty? : % -> Boolean entries : % -> List(R) +--R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean +--R indices : % -> List(Integer) latex : % -> String if R has SETCAT +--R map : ((R -> R),%) -> % one? : % -> Boolean if R has MONOID +--R qelt : (%,Integer) -> R random : () -> % if R has FINITE +--R retract : % -> R if R has SETCAT sample : () -> % +--R sup : (%,%) -> % if R has OAMONS --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (PositiveInteger,%) -> % if R has ABELSG --R ?*? : (NonNegativeInteger,%) -> % if R has CABMON ---R ?*? : (R,%) -> % if R has MONOID ---R ?*? : (%,R) -> % if R has MONOID ---R ?*? : (%,%) -> % if R has MONOID --R ?*? : (Integer,%) -> % if R has RING --R ?**? : (%,PositiveInteger) -> % if R has MONOID --R ?**? : (%,NonNegativeInteger) -> % if R has MONOID ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has RING ---R ?/? : (%,R) -> % if R has FIELD --R ? Boolean if R has OAMONS or R has ORDRING --R ?<=? : (%,%) -> Boolean if R has OAMONS or R has ORDRING --R ?=? : (%,%) -> Boolean if R has SETCAT @@ -33558,10 +34811,8 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --R D : % -> % if R has DIFRING and R has RING --R ?^? : (%,PositiveInteger) -> % if R has MONOID --R ?^? : (%,NonNegativeInteger) -> % if R has MONOID ---R abs : % -> % if R has ORDRING --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger if R has RING ---R coerce : R -> % if R has SETCAT --R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) and R has SETCAT --R coerce : Integer -> % if R has RETRACT(INT) and R has SETCAT or R has RING --R coerce : % -> OutputForm if R has SETCAT @@ -33576,7 +34827,6 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --R differentiate : (%,NonNegativeInteger) -> % if R has DIFRING and R has RING --R differentiate : % -> % if R has DIFRING and R has RING --R dimension : () -> CardinalNumber if R has FIELD ---R dot : (%,%) -> R if R has RING --R entry? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R enumerate : () -> List(%) if R has FINITE --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -33588,7 +34838,6 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --R first : % -> R if Integer has ORDSET --R hash : % -> SingleInteger if R has SETCAT --R index : PositiveInteger -> % if R has FINITE ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R lookup : % -> PositiveInteger if R has FINITE --R map! : ((R -> R),%) -> % if $ has shallowlyMutable @@ -33600,17 +34849,14 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R negative? : % -> Boolean if R has ORDRING ---R one? : % -> Boolean if R has MONOID --R parts : % -> List(R) if $ has finiteAggregate --R positive? : % -> Boolean if R has ORDRING --R qsetelt! : (%,Integer,R) -> R if $ has shallowlyMutable ---R random : () -> % if R has FINITE --R recip : % -> Union(%,"failed") if R has MONOID --R reducedSystem : Matrix(%) -> Matrix(R) if R has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) if R has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) and R has RING --R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) and R has RING ---R retract : % -> R if R has SETCAT --R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) and R has SETCAT --R retract : % -> Integer if R has RETRACT(INT) and R has SETCAT --R retractIfCan : % -> Union(R,"failed") if R has SETCAT @@ -33621,7 +34867,6 @@ DifferentialSparseMultivariatePolynomial(R, S, V): --R size : () -> NonNegativeInteger if R has FINITE --R size? : (%,NonNegativeInteger) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") if R has CABMON ---R sup : (%,%) -> % if R has OAMONS --R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable --R unitVector : PositiveInteger -> % if R has RING --R zero? : % -> Boolean if R has CABMON @@ -33846,34 +35091,36 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --S 1 of 1 )show DirectProductMatrixModule +--R --R DirectProductMatrixModule(n: PositiveInteger,R: Ring,M: SquareMatrixCategory(n,R,DirectProduct(n,R),DirectProduct(n,R)),S: LeftModule(R)) is a domain constructor --R Abbreviation for DirectProductMatrixModule is DPMM --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for DPMM --R --R------------------------------- Operations -------------------------------- +--R ?*? : (S,%) -> % if S has MONOID ?*? : (%,S) -> % if S has MONOID --R ?*? : (M,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (R,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (R,%) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R ?/? : (%,S) -> % if S has FIELD ?=? : (%,%) -> Boolean +--R 0 : () -> % abs : % -> % if S has ORDRING +--R coerce : S -> % if S has SETCAT coerce : % -> OutputForm --R coerce : % -> Vector(S) copy : % -> % ---R directProduct : Vector(S) -> % ?.? : (%,Integer) -> S ---R elt : (%,Integer,S) -> S empty : () -> % ---R empty? : % -> Boolean entries : % -> List(S) ---R eq? : (%,%) -> Boolean hash : % -> SingleInteger ---R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R latex : % -> String map : ((S -> S),%) -> % ---R qelt : (%,Integer) -> S sample : () -> % +--R directProduct : Vector(S) -> % dot : (%,%) -> S if S has RING +--R ?.? : (%,Integer) -> S elt : (%,Integer,S) -> S +--R empty : () -> % empty? : % -> Boolean +--R entries : % -> List(S) eq? : (%,%) -> Boolean +--R hash : % -> SingleInteger index? : (Integer,%) -> Boolean +--R indices : % -> List(Integer) latex : % -> String +--R map : ((S -> S),%) -> % qelt : (%,Integer) -> S +--R random : () -> % if S has FINITE retract : % -> S if S has SETCAT +--R sample : () -> % sup : (%,%) -> % if S has OAMONS --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (%,%) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R ?*? : (S,%) -> % if S has MONOID ---R ?*? : (%,S) -> % if S has MONOID ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,PositiveInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?**? : (%,NonNegativeInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R ?/? : (%,S) -> % if S has FIELD --R ? Boolean if S has OAMONS or S has ORDRING --R ?<=? : (%,%) -> Boolean if S has OAMONS or S has ORDRING --R ?>? : (%,%) -> Boolean if S has OAMONS or S has ORDRING @@ -33889,12 +35136,10 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --R 1 : () -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?^? : (%,PositiveInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?^? : (%,NonNegativeInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R abs : % -> % if S has ORDRING --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger if S has RING --R coerce : Fraction(Integer) -> % if S has RETRACT(FRAC(INT)) and S has SETCAT --R coerce : Integer -> % if S has RETRACT(INT) and S has SETCAT or S has RING ---R coerce : S -> % if S has SETCAT --R count : (S,%) -> NonNegativeInteger if $ has finiteAggregate and S has SETCAT --R count : ((S -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate --R differentiate : % -> % if S has DIFRING and S has RING @@ -33906,7 +35151,6 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --R differentiate : (%,(S -> S)) -> % if S has RING --R differentiate : (%,(S -> S),NonNegativeInteger) -> % if S has RING --R dimension : () -> CardinalNumber if S has FIELD ---R dot : (%,%) -> S if S has RING --R entry? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R enumerate : () -> List(%) if S has FINITE --R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT @@ -33932,7 +35176,6 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --R parts : % -> List(S) if $ has finiteAggregate --R positive? : % -> Boolean if S has ORDRING --R qsetelt! : (%,Integer,S) -> S if $ has shallowlyMutable ---R random : () -> % if S has FINITE --R recip : % -> Union(%,"failed") if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R reducedSystem : Matrix(%) -> Matrix(Integer) if S has LINEXP(INT) and S has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if S has LINEXP(INT) and S has RING @@ -33940,7 +35183,6 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(S),vec: Vector(S)) if S has RING --R retract : % -> Fraction(Integer) if S has RETRACT(FRAC(INT)) and S has SETCAT --R retract : % -> Integer if S has RETRACT(INT) and S has SETCAT ---R retract : % -> S if S has SETCAT --R retractIfCan : % -> Union(Fraction(Integer),"failed") if S has RETRACT(FRAC(INT)) and S has SETCAT --R retractIfCan : % -> Union(Integer,"failed") if S has RETRACT(INT) and S has SETCAT --R retractIfCan : % -> Union(S,"failed") if S has SETCAT @@ -33949,7 +35191,6 @@ DirectProduct(dim:NonNegativeInteger, R:Type): --R size : () -> NonNegativeInteger if S has FINITE --R size? : (%,NonNegativeInteger) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") ---R sup : (%,%) -> % if S has OAMONS --R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable --R unitVector : PositiveInteger -> % if S has RING --R @@ -34101,34 +35342,36 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --S 1 of 1 )show DirectProductModule +--R --R DirectProductModule(n: NonNegativeInteger,R: Ring,S: LeftModule(R)) is a domain constructor --R Abbreviation for DirectProductModule is DPMO --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for DPMO --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (R,%) -> % ?+? : (%,%) -> % ---R -? : % -> % ?-? : (%,%) -> % +--R ?*? : (S,%) -> % if S has MONOID ?*? : (%,S) -> % if S has MONOID +--R ?*? : (PositiveInteger,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (R,%) -> % +--R ?+? : (%,%) -> % -? : % -> % +--R ?-? : (%,%) -> % ?/? : (%,S) -> % if S has FIELD --R ?=? : (%,%) -> Boolean 0 : () -> % +--R abs : % -> % if S has ORDRING coerce : S -> % if S has SETCAT --R coerce : % -> OutputForm coerce : % -> Vector(S) --R copy : % -> % directProduct : Vector(S) -> % ---R ?.? : (%,Integer) -> S elt : (%,Integer,S) -> S ---R empty : () -> % empty? : % -> Boolean ---R entries : % -> List(S) eq? : (%,%) -> Boolean ---R hash : % -> SingleInteger index? : (Integer,%) -> Boolean ---R indices : % -> List(Integer) latex : % -> String ---R map : ((S -> S),%) -> % qelt : (%,Integer) -> S ---R sample : () -> % zero? : % -> Boolean +--R dot : (%,%) -> S if S has RING ?.? : (%,Integer) -> S +--R elt : (%,Integer,S) -> S empty : () -> % +--R empty? : % -> Boolean entries : % -> List(S) +--R eq? : (%,%) -> Boolean hash : % -> SingleInteger +--R index? : (Integer,%) -> Boolean indices : % -> List(Integer) +--R latex : % -> String map : ((S -> S),%) -> % +--R qelt : (%,Integer) -> S random : () -> % if S has FINITE +--R retract : % -> S if S has SETCAT sample : () -> % +--R sup : (%,%) -> % if S has OAMONS zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (%,%) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R ?*? : (S,%) -> % if S has MONOID ---R ?*? : (%,S) -> % if S has MONOID ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,PositiveInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?**? : (%,NonNegativeInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R ?/? : (%,S) -> % if S has FIELD --R ? Boolean if S has OAMONS or S has ORDRING --R ?<=? : (%,%) -> Boolean if S has OAMONS or S has ORDRING --R ?>? : (%,%) -> Boolean if S has OAMONS or S has ORDRING @@ -34144,12 +35387,10 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --R 1 : () -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?^? : (%,PositiveInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R ?^? : (%,NonNegativeInteger) -> % if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING ---R abs : % -> % if S has ORDRING --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger if S has RING --R coerce : Fraction(Integer) -> % if S has RETRACT(FRAC(INT)) and S has SETCAT --R coerce : Integer -> % if S has RETRACT(INT) and S has SETCAT or S has RING ---R coerce : S -> % if S has SETCAT --R count : (S,%) -> NonNegativeInteger if $ has finiteAggregate and S has SETCAT --R count : ((S -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate --R differentiate : % -> % if S has DIFRING and S has RING @@ -34161,7 +35402,6 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --R differentiate : (%,(S -> S)) -> % if S has RING --R differentiate : (%,(S -> S),NonNegativeInteger) -> % if S has RING --R dimension : () -> CardinalNumber if S has FIELD ---R dot : (%,%) -> S if S has RING --R entry? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R enumerate : () -> List(%) if S has FINITE --R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT @@ -34187,7 +35427,6 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --R parts : % -> List(S) if $ has finiteAggregate --R positive? : % -> Boolean if S has ORDRING --R qsetelt! : (%,Integer,S) -> S if $ has shallowlyMutable ---R random : () -> % if S has FINITE --R recip : % -> Union(%,"failed") if S has DIFRING and S has RING or S has LINEXP(INT) and S has RING or S has MONOID or S has PDRING(SYMBOL) and S has RING --R reducedSystem : Matrix(%) -> Matrix(Integer) if S has LINEXP(INT) and S has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if S has LINEXP(INT) and S has RING @@ -34195,7 +35434,6 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(S),vec: Vector(S)) if S has RING --R retract : % -> Fraction(Integer) if S has RETRACT(FRAC(INT)) and S has SETCAT --R retract : % -> Integer if S has RETRACT(INT) and S has SETCAT ---R retract : % -> S if S has SETCAT --R retractIfCan : % -> Union(Fraction(Integer),"failed") if S has RETRACT(FRAC(INT)) and S has SETCAT --R retractIfCan : % -> Union(Integer,"failed") if S has RETRACT(INT) and S has SETCAT --R retractIfCan : % -> Union(S,"failed") if S has SETCAT @@ -34204,7 +35442,6 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where --R size : () -> NonNegativeInteger if S has FINITE --R size? : (%,NonNegativeInteger) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") ---R sup : (%,%) -> % if S has OAMONS --R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable --R unitVector : PositiveInteger -> % if S has RING --R @@ -34542,22 +35779,21 @@ reduce(_and,[(x = y)@Boolean for x in t9 for y in t10]) --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : % -> Stream(Coef) ---R coerce : Stream(Coef) -> % coerce : Integer -> % ---R coerce : % -> OutputForm ?.? : (%,PositiveInteger) -> Coef ---R hash : % -> SingleInteger latex : % -> String ---R one? : % -> Boolean recip : % -> Union(%,"failed") ---R sample : () -> % zero? : % -> Boolean ---R zeta : () -> % ?~=? : (%,%) -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : % -> Stream(Coef) coerce : Stream(Coef) -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R ?.? : (%,PositiveInteger) -> Coef hash : % -> SingleInteger +--R latex : % -> String one? : % -> Boolean +--R recip : % -> Union(%,"failed") sample : () -> % +--R zero? : % -> Boolean zeta : () -> % +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,Coef) -> % if Coef has COMRING --R ?*? : (Coef,%) -> % if Coef has COMRING ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R additive? : (%,PositiveInteger) -> Boolean --R associates? : (%,%) -> Boolean if Coef has COMRING --R characteristic : () -> NonNegativeInteger @@ -34775,13 +36011,14 @@ DirichletRing(Coef: Ring): )set message test on )set message auto off )clear all ---S 1 of 10 + +--S 1 of 11 (d1,d2,d3) : DMP([z,y,x],FRAC INT) --R --R Type: Void --E 1 ---S 2 of 10 +--S 2 of 11 d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R --R @@ -34790,7 +36027,7 @@ d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 2 ---S 3 of 10 +--S 3 of 11 d2 := 2*z*y**2 + 4*x + 1 --R --R @@ -34799,7 +36036,7 @@ d2 := 2*z*y**2 + 4*x + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 3 ---S 4 of 10 +--S 4 of 11 d3 := 2*z*x**2 - 2*y**2 - x --R --R @@ -34808,7 +36045,7 @@ d3 := 2*z*x**2 - 2*y**2 - x --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 4 ---S 5 of 10 +--S 5 of 11 groebner [d1,d2,d3] --R --R @@ -34825,13 +36062,13 @@ groebner [d1,d2,d3] --R Type: List(DistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 5 ---S 6 of 10 +--S 6 of 11 (n1,n2,n3) : HDMP([z,y,x],FRAC INT) --R --R Type: Void --E 6 ---S 7 of 10 +--S 7 of 11 n1 := d1 --R --R @@ -34840,7 +36077,7 @@ n1 := d1 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 7 ---S 8 of 10 +--S 8 of 11 n2 := d2 --R --R @@ -34849,7 +36086,7 @@ n2 := d2 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 8 ---S 9 of 10 +--S 9 of 11 n3 := d3 --R --R @@ -34858,7 +36095,7 @@ n3 := d3 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 9 ---S 10 of 10 +--S 10 of 11 groebner [n1,n2,n3] --R --R @@ -34874,6 +36111,134 @@ groebner [n1,n2,n3] --R 4 2 --RType: List(HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 10 + +--S 11 of 11 +)show DistributedMultivariatePolynomial +--R +--R DistributedMultivariatePolynomial(vl: List(Symbol),R: Ring) is a domain constructor +--R Abbreviation for DistributedMultivariatePolynomial is DMP +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for DMP +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficients : % -> List(R) +--R coerce : % -> % if R has INTDOM coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R content : % -> R if R has GCDDOM eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R +--R ground? : % -> Boolean hash : % -> SingleInteger +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leadingCoefficient : % -> R +--R leadingMonomial : % -> % map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET +--R monomial? : % -> Boolean monomials : % -> List(%) +--R one? : % -> Boolean primitiveMonomials : % -> List(%) +--R recip : % -> Union(%,"failed") reductum : % -> % +--R reorder : (%,List(Integer)) -> % retract : % -> R +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) +--R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) +--R ? Boolean if R has ORDSET +--R ?<=? : (%,%) -> Boolean if R has ORDSET +--R ?>? : (%,%) -> Boolean if R has ORDSET +--R ?>=? : (%,%) -> Boolean if R has ORDSET +--R D : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R D : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R D : (%,List(OrderedVariableList(vl))) -> % +--R D : (%,OrderedVariableList(vl)) -> % +--R associates? : (%,%) -> Boolean if R has INTDOM +--R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and R has PFECAT or R has CHARNZ +--R coefficient : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R coefficient : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R coefficient : (%,DirectProduct(#(vl),NonNegativeInteger)) -> R +--R coerce : Fraction(Integer) -> % if R has ALGEBRA(FRAC(INT)) or R has RETRACT(FRAC(INT)) +--R coerce : OrderedVariableList(vl) -> % +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and R has PFECAT +--R content : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R convert : % -> InputForm if OrderedVariableList(vl) has KONVERT(INFORM) and R has KONVERT(INFORM) +--R convert : % -> Pattern(Integer) if OrderedVariableList(vl) has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) +--R convert : % -> Pattern(Float) if OrderedVariableList(vl) has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) +--R degree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R degree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R degree : % -> DirectProduct(#(vl),NonNegativeInteger) +--R differentiate : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R differentiate : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R differentiate : (%,List(OrderedVariableList(vl))) -> % +--R differentiate : (%,OrderedVariableList(vl)) -> % +--R discriminant : (%,OrderedVariableList(vl)) -> % if R has COMRING +--R eval : (%,List(OrderedVariableList(vl)),List(%)) -> % +--R eval : (%,OrderedVariableList(vl),%) -> % +--R eval : (%,List(OrderedVariableList(vl)),List(R)) -> % +--R eval : (%,OrderedVariableList(vl),R) -> % +--R exquo : (%,%) -> Union(%,"failed") if R has INTDOM +--R exquo : (%,R) -> Union(%,"failed") if R has INTDOM +--R factor : % -> Factored(%) if R has PFECAT +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM +--R isExpt : % -> Union(Record(var: OrderedVariableList(vl),exponent: NonNegativeInteger),"failed") +--R isPlus : % -> Union(List(%),"failed") +--R isTimes : % -> Union(List(%),"failed") +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM +--R mainVariable : % -> Union(OrderedVariableList(vl),"failed") +--R mapExponents : ((DirectProduct(#(vl),NonNegativeInteger) -> DirectProduct(#(vl),NonNegativeInteger)),%) -> % +--R minimumDegree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R minimumDegree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R minimumDegree : % -> DirectProduct(#(vl),NonNegativeInteger) +--R monicDivide : (%,%,OrderedVariableList(vl)) -> Record(quotient: %,remainder: %) +--R monomial : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R monomial : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R monomial : (R,DirectProduct(#(vl),NonNegativeInteger)) -> % +--R multivariate : (SparseUnivariatePolynomial(%),OrderedVariableList(vl)) -> % +--R multivariate : (SparseUnivariatePolynomial(R),OrderedVariableList(vl)) -> % +--R numberOfMonomials : % -> NonNegativeInteger +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if OrderedVariableList(vl) has PATMAB(INT) and R has PATMAB(INT) +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if OrderedVariableList(vl) has PATMAB(FLOAT) and R has PATMAB(FLOAT) +--R pomopo! : (%,R,DirectProduct(#(vl),NonNegativeInteger),%) -> % +--R prime? : % -> Boolean if R has PFECAT +--R primitivePart : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R primitivePart : % -> % if R has GCDDOM +--R reducedSystem : Matrix(%) -> Matrix(R) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) +--R resultant : (%,%,OrderedVariableList(vl)) -> % if R has COMRING +--R retract : % -> OrderedVariableList(vl) +--R retract : % -> Integer if R has RETRACT(INT) +--R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(OrderedVariableList(vl),"failed") +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(R,"failed") +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has PFECAT +--R squareFree : % -> Factored(%) if R has GCDDOM +--R squareFreePart : % -> % if R has GCDDOM +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R totalDegree : (%,List(OrderedVariableList(vl))) -> NonNegativeInteger +--R totalDegree : % -> NonNegativeInteger +--R unit? : % -> Boolean if R has INTDOM +--R unitCanonical : % -> % if R has INTDOM +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has INTDOM +--R univariate : % -> SparseUnivariatePolynomial(R) +--R univariate : (%,OrderedVariableList(vl)) -> SparseUnivariatePolynomial(%) +--R variables : % -> List(OrderedVariableList(vl)) +--R +--E 11 + )spool )lisp (bye) \end{chunk} @@ -35127,24 +36492,23 @@ DistributedMultivariatePolynomial(vl,R): public == private where --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,S) -> % ?*? : (Integer,%) -> % --R ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (S,%) -> Integer coerce : S -> % ---R coerce : % -> OutputForm collect : % -> % ---R concat : (%,%) -> % degree : % -> Integer ---R divOfPole : % -> % divOfZero : % -> % ---R effective? : % -> Boolean hash : % -> SingleInteger ---R incr : % -> % latex : % -> String ---R mapGen : ((S -> S),%) -> % nthCoef : (%,Integer) -> Integer ---R nthFactor : (%,Integer) -> S reductum : % -> % ---R retract : % -> S sample : () -> % ---R size : % -> NonNegativeInteger split : % -> List(%) ---R supp : % -> List(S) suppOfPole : % -> List(S) ---R suppOfZero : % -> List(S) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (S,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (S,%) -> Integer +--R coerce : S -> % coerce : % -> OutputForm +--R collect : % -> % concat : (%,%) -> % +--R degree : % -> Integer divOfPole : % -> % +--R divOfZero : % -> % effective? : % -> Boolean +--R hash : % -> SingleInteger incr : % -> % +--R latex : % -> String mapGen : ((S -> S),%) -> % +--R nthCoef : (%,Integer) -> Integer nthFactor : (%,Integer) -> S +--R reductum : % -> % retract : % -> S +--R sample : () -> % size : % -> NonNegativeInteger +--R split : % -> List(%) supp : % -> List(S) +--R suppOfPole : % -> List(S) suppOfZero : % -> List(S) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R head : % -> Record(gen: S,exp: Integer) --R highCommonTerms : (%,%) -> % if Integer has OAMON --R mapCoef : ((Integer -> Integer),%) -> % @@ -35526,7 +36890,7 @@ which just returns a list of the multiple values from the )set message auto off )clear all ---S 1 of 13 +--S 1 of 14 2.71828 --R --R @@ -35534,7 +36898,7 @@ which just returns a list of the multiple values from the --R Type: Float --E 1 ---S 2 of 13 +--S 2 of 14 2.71828@DoubleFloat --R --R @@ -35542,7 +36906,7 @@ which just returns a list of the multiple values from the --R Type: DoubleFloat --E 2 ---S 3 of 13 +--S 3 of 14 2.71828 :: DoubleFloat --R --R @@ -35550,7 +36914,7 @@ which just returns a list of the multiple values from the --R Type: DoubleFloat --E 3 ---S 4 of 13 +--S 4 of 14 eApprox : DoubleFloat := 2.71828 --R --R @@ -35558,13 +36922,13 @@ eApprox : DoubleFloat := 2.71828 --R Type: DoubleFloat --E 4 ---S 5 of 13 +--S 5 of 14 avg : List DoubleFloat -> DoubleFloat --R --R Type: Void --E 5 ---S 6 of 13 +--S 6 of 14 avg l == empty? l => 0 :: DoubleFloat reduce(_+,l) / #l @@ -35572,7 +36936,7 @@ avg l == --R Type: Void --E 6 ---S 7 of 13 +--S 7 of 14 avg [] --R --R Compiling function avg with type List(DoubleFloat) -> DoubleFloat @@ -35581,7 +36945,7 @@ avg [] --R Type: DoubleFloat --E 7 ---S 8 of 13 +--S 8 of 14 avg [3.4,9.7,-6.8] --R --R @@ -35589,7 +36953,7 @@ avg [3.4,9.7,-6.8] --R Type: DoubleFloat --E 8 ---S 9 of 13 +--S 9 of 14 cos(3.1415926)$DoubleFloat --R --R @@ -35597,7 +36961,7 @@ cos(3.1415926)$DoubleFloat --R Type: DoubleFloat --E 9 ---S 10 of 13 +--S 10 of 14 cos(3.1415926 :: DoubleFloat) --R --R @@ -35605,7 +36969,7 @@ cos(3.1415926 :: DoubleFloat) --R Type: DoubleFloat --E 10 ---S 11 of 13 +--S 11 of 14 a:DFLOAT := -1.0/3.0 --R --R @@ -35613,7 +36977,7 @@ a:DFLOAT := -1.0/3.0 --R Type: DoubleFloat --E 11 ---S 12 of 13 +--S 12 of 14 integerDecode a --R --R @@ -35621,7 +36985,7 @@ integerDecode a --R Type: List(Integer) --E 12 ---S 13 of 13 +--S 13 of 14 machineFraction a --R --R @@ -35631,6 +36995,115 @@ machineFraction a --R Type: Fraction(Integer) --E 13 +--S 14 of 14 +)show DoubleFloat +--R +--R DoubleFloat is a domain constructor +--R Abbreviation for DoubleFloat is DFLOAT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for DFLOAT +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,%) -> % ?**? : (%,Fraction(Integer)) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,Integer) -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean Beta : (%,%) -> % +--R D : % -> % D : (%,NonNegativeInteger) -> % +--R Gamma : (%,%) -> % Gamma : % -> % +--R OMwrite : (%,Boolean) -> String OMwrite : % -> String +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % abs : % -> % +--R acos : % -> % acosh : % -> % +--R acot : % -> % acoth : % -> % +--R acsc : % -> % acsch : % -> % +--R airyAi : % -> % airyBi : % -> % +--R asec : % -> % asech : % -> % +--R asin : % -> % asinh : % -> % +--R associates? : (%,%) -> Boolean atan : (%,%) -> % +--R atan : % -> % atanh : % -> % +--R base : () -> PositiveInteger besselI : (%,%) -> % +--R besselJ : (%,%) -> % besselK : (%,%) -> % +--R besselY : (%,%) -> % bits : () -> PositiveInteger +--R ceiling : % -> % coerce : Fraction(Integer) -> % +--R coerce : Integer -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm convert : % -> InputForm +--R convert : % -> Pattern(Float) convert : % -> DoubleFloat +--R convert : % -> Float cos : % -> % +--R cosh : % -> % cot : % -> % +--R coth : % -> % csc : % -> % +--R csch : % -> % differentiate : % -> % +--R digamma : % -> % digits : () -> PositiveInteger +--R doubleFloatFormat : String -> String exp : % -> % +--R exp1 : () -> % exponent : % -> Integer +--R factor : % -> Factored(%) float : (Integer,Integer) -> % +--R floor : % -> % fractionPart : % -> % +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> Integer hash : % -> SingleInteger +--R integerDecode : % -> List(Integer) inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % log : % -> % +--R log10 : % -> % log2 : % -> % +--R mantissa : % -> Integer max : (%,%) -> % +--R min : (%,%) -> % negative? : % -> Boolean +--R norm : % -> % nthRoot : (%,Integer) -> % +--R one? : % -> Boolean order : % -> Integer +--R pi : () -> % polygamma : (%,%) -> % +--R positive? : % -> Boolean precision : () -> PositiveInteger +--R prime? : % -> Boolean ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Integer +--R round : % -> % sample : () -> % +--R sec : % -> % sech : % -> % +--R sign : % -> Integer sin : % -> % +--R sinh : % -> % sizeLess? : (%,%) -> Boolean +--R sqrt : % -> % squareFree : % -> Factored(%) +--R squareFreePart : % -> % tan : % -> % +--R tanh : % -> % truncate : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R wholePart : % -> Integer zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R OMwrite : (OpenMathDevice,%,Boolean) -> Void +--R OMwrite : (OpenMathDevice,%) -> Void +--R bits : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R characteristic : () -> NonNegativeInteger +--R decreasePrecision : Integer -> PositiveInteger if $ has arbitraryPrecision +--R differentiate : (%,NonNegativeInteger) -> % +--R digits : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R euclideanSize : % -> NonNegativeInteger +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R float : (Integer,Integer,PositiveInteger) -> % +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R increasePrecision : Integer -> PositiveInteger if $ has arbitraryPrecision +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R machineFraction : % -> Fraction(Integer) +--R max : () -> % if not(has($,arbitraryExponent)) and not(has($,arbitraryPrecision)) +--R min : () -> % if not(has($,arbitraryExponent)) and not(has($,arbitraryPrecision)) +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) +--R precision : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R rationalApproximation : (%,NonNegativeInteger,NonNegativeInteger) -> Fraction(Integer) +--R rationalApproximation : (%,NonNegativeInteger) -> Fraction(Integer) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") +--R retractIfCan : % -> Union(Integer,"failed") +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R +--E 14 + )spool )lisp (bye) @@ -36255,25 +37728,25 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath, --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,%) -> % ?*? : (%,DoubleFloat) -> % --R ?*? : (DoubleFloat,%) -> % ?*? : (%,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % antisymmetric? : % -> Boolean ---R coerce : DoubleFloatVector -> % copy : % -> % ---R diagonal? : % -> Boolean diagonalMatrix : List(%) -> % ---R empty : () -> % empty? : % -> Boolean ---R eq? : (%,%) -> Boolean fill! : (%,DoubleFloat) -> % ---R horizConcat : (%,%) -> % maxColIndex : % -> Integer ---R maxRowIndex : % -> Integer minColIndex : % -> Integer ---R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger ---R nrows : % -> NonNegativeInteger parts : % -> List(DoubleFloat) ---R qnew : (Integer,Integer) -> % sample : () -> % ---R square? : % -> Boolean squareTop : % -> % ---R symmetric? : % -> Boolean transpose : % -> % +--R ?**? : (%,NonNegativeInteger) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R antisymmetric? : % -> Boolean coerce : DoubleFloatVector -> % +--R copy : % -> % diagonal? : % -> Boolean +--R diagonalMatrix : List(%) -> % empty : () -> % +--R empty? : % -> Boolean eq? : (%,%) -> Boolean +--R fill! : (%,DoubleFloat) -> % horizConcat : (%,%) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R parts : % -> List(DoubleFloat) qnew : (Integer,Integer) -> % +--R sample : () -> % square? : % -> Boolean +--R squareTop : % -> % symmetric? : % -> Boolean +--R transpose : % -> % transpose : DoubleFloatVector -> % --R vertConcat : (%,%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (DoubleFloatVector,%) -> DoubleFloatVector --R ?*? : (%,DoubleFloatVector) -> DoubleFloatVector --R ?**? : (%,Integer) -> % if DoubleFloat has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,DoubleFloat) -> % if DoubleFloat has FIELD --R ?=? : (%,%) -> Boolean if DoubleFloat has SETCAT --R any? : ((DoubleFloat -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -36327,7 +37800,6 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath, --R subMatrix : (%,Integer,Integer,Integer,Integer) -> % --R swapColumns! : (%,Integer,Integer) -> % --R swapRows! : (%,Integer,Integer) -> % ---R transpose : DoubleFloatVector -> % --R zero : (NonNegativeInteger,NonNegativeInteger) -> % --R ?~=? : (%,%) -> Boolean if DoubleFloat has SETCAT --R @@ -36581,13 +38053,14 @@ DoubleFloatMatrix : MatrixCategory(DoubleFloat, --R------------------------------- Operations -------------------------------- --R concat : List(%) -> % concat : (%,%) -> % --R concat : (DoubleFloat,%) -> % concat : (%,DoubleFloat) -> % ---R copy : % -> % delete : (%,Integer) -> % ---R ?.? : (%,Integer) -> DoubleFloat empty : () -> % ---R empty? : % -> Boolean entries : % -> List(DoubleFloat) ---R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean ---R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R qelt : (%,Integer) -> DoubleFloat qnew : Integer -> % ---R reverse : % -> % sample : () -> % +--R construct : List(DoubleFloat) -> % copy : % -> % +--R delete : (%,Integer) -> % ?.? : (%,Integer) -> DoubleFloat +--R empty : () -> % empty? : % -> Boolean +--R entries : % -> List(DoubleFloat) eq? : (%,%) -> Boolean +--R index? : (Integer,%) -> Boolean indices : % -> List(Integer) +--R insert : (%,%,Integer) -> % qelt : (%,Integer) -> DoubleFloat +--R qnew : Integer -> % reverse : % -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (%,DoubleFloat) -> % if DoubleFloat has MONOID --R ?*? : (DoubleFloat,%) -> % if DoubleFloat has MONOID @@ -36602,7 +38075,6 @@ DoubleFloatMatrix : MatrixCategory(DoubleFloat, --R ?>=? : (%,%) -> Boolean if DoubleFloat has ORDSET --R any? : ((DoubleFloat -> Boolean),%) -> Boolean if $ has finiteAggregate --R coerce : % -> OutputForm if DoubleFloat has SETCAT ---R construct : List(DoubleFloat) -> % --R convert : % -> InputForm if DoubleFloat has KONVERT(INFORM) --R copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable --R count : (DoubleFloat,%) -> NonNegativeInteger if $ has finiteAggregate and DoubleFloat has SETCAT @@ -36896,11 +38368,11 @@ DoubleFloatVector : VectorCategory DoubleFloat with --R curveColor : Float -> % hash : % -> SingleInteger --R latex : % -> String pointColor : Palette -> % --R pointColor : Float -> % range : List(Segment(Float)) -> % ---R style : String -> % title : String -> % ---R toScale : Boolean -> % tubePoints : PositiveInteger -> % ---R tubeRadius : Float -> % unit : List(Float) -> % ---R var1Steps : PositiveInteger -> % var2Steps : PositiveInteger -> % ---R ?~=? : (%,%) -> Boolean +--R ranges : List(Segment(Float)) -> % style : String -> % +--R title : String -> % toScale : Boolean -> % +--R tubePoints : PositiveInteger -> % tubeRadius : Float -> % +--R unit : List(Float) -> % var1Steps : PositiveInteger -> % +--R var2Steps : PositiveInteger -> % ?~=? : (%,%) -> Boolean --R colorFunction : ((DoubleFloat,DoubleFloat,DoubleFloat) -> DoubleFloat) -> % --R colorFunction : ((DoubleFloat,DoubleFloat) -> DoubleFloat) -> % --R colorFunction : (DoubleFloat -> DoubleFloat) -> % @@ -36909,7 +38381,6 @@ DoubleFloatVector : VectorCategory DoubleFloat with --R option : (List(%),Symbol) -> Union(Any,"failed") --R option? : (List(%),Symbol) -> Boolean --R range : List(Segment(Fraction(Integer))) -> % ---R ranges : List(Segment(Float)) -> % --R space : ThreeSpace(DoubleFloat) -> % --R viewpoint : Record(theta: DoubleFloat,phi: DoubleFloat,scale: DoubleFloat,scaleX: DoubleFloat,scaleY: DoubleFloat,scaleZ: DoubleFloat,deltaX: DoubleFloat,deltaY: DoubleFloat) -> % --R @@ -39675,6 +41146,55 @@ d03fafAnnaType():PartialDifferentialEquationsSolverCategory == Result add --S 1 of 1 )show ElementaryFunctionsUnivariateLaurentSeries +--R +--R ElementaryFunctionsUnivariateLaurentSeries(Coef: Algebra(Fraction(Integer)),UTS: UnivariateTaylorSeriesCategory(Coef),ULS: UnivariateLaurentSeriesConstructorCategory(Coef,UTS)) is a domain constructor +--R Abbreviation for ElementaryFunctionsUnivariateLaurentSeries is EFULS +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EFULS +--R +--R------------------------------- Operations -------------------------------- +--R acos : ULS -> ULS acosh : ULS -> ULS +--R acot : ULS -> ULS acoth : ULS -> ULS +--R acsc : ULS -> ULS acsch : ULS -> ULS +--R asec : ULS -> ULS asech : ULS -> ULS +--R asin : ULS -> ULS asinh : ULS -> ULS +--R atan : ULS -> ULS atanh : ULS -> ULS +--R cos : ULS -> ULS cosh : ULS -> ULS +--R cot : ULS -> ULS coth : ULS -> ULS +--R csc : ULS -> ULS csch : ULS -> ULS +--R exp : ULS -> ULS log : ULS -> ULS +--R sec : ULS -> ULS sech : ULS -> ULS +--R sin : ULS -> ULS sinh : ULS -> ULS +--R tan : ULS -> ULS tanh : ULS -> ULS +--R ?**? : (ULS,Fraction(Integer)) -> ULS if Coef has FIELD +--R acosIfCan : ULS -> Union(ULS,"failed") +--R acoshIfCan : ULS -> Union(ULS,"failed") +--R acotIfCan : ULS -> Union(ULS,"failed") +--R acothIfCan : ULS -> Union(ULS,"failed") +--R acscIfCan : ULS -> Union(ULS,"failed") +--R acschIfCan : ULS -> Union(ULS,"failed") +--R asecIfCan : ULS -> Union(ULS,"failed") +--R asechIfCan : ULS -> Union(ULS,"failed") +--R asinIfCan : ULS -> Union(ULS,"failed") +--R asinhIfCan : ULS -> Union(ULS,"failed") +--R atanIfCan : ULS -> Union(ULS,"failed") +--R atanhIfCan : ULS -> Union(ULS,"failed") +--R cosIfCan : ULS -> Union(ULS,"failed") +--R coshIfCan : ULS -> Union(ULS,"failed") +--R cotIfCan : ULS -> Union(ULS,"failed") +--R cothIfCan : ULS -> Union(ULS,"failed") +--R cscIfCan : ULS -> Union(ULS,"failed") +--R cschIfCan : ULS -> Union(ULS,"failed") +--R expIfCan : ULS -> Union(ULS,"failed") +--R logIfCan : ULS -> Union(ULS,"failed") +--R nthRootIfCan : (ULS,NonNegativeInteger) -> Union(ULS,"failed") +--R secIfCan : ULS -> Union(ULS,"failed") +--R sechIfCan : ULS -> Union(ULS,"failed") +--R sinIfCan : ULS -> Union(ULS,"failed") +--R sinhIfCan : ULS -> Union(ULS,"failed") +--R tanIfCan : ULS -> Union(ULS,"failed") +--R tanhIfCan : ULS -> Union(ULS,"failed") +--R --E 1 )spool @@ -40120,6 +41640,55 @@ ElementaryFunctionsUnivariateLaurentSeries(Coef,UTS,ULS):_ --S 1 of 1 )show ElementaryFunctionsUnivariatePuiseuxSeries +--R +--R ElementaryFunctionsUnivariatePuiseuxSeries(Coef: Algebra(Fraction(Integer)),ULS: UnivariateLaurentSeriesCategory(Coef),UPXS: UnivariatePuiseuxSeriesConstructorCategory(Coef,ULS),EFULS: PartialTranscendentalFunctions(ULS)) is a domain constructor +--R Abbreviation for ElementaryFunctionsUnivariatePuiseuxSeries is EFUPXS +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EFUPXS +--R +--R------------------------------- Operations -------------------------------- +--R acos : UPXS -> UPXS acosh : UPXS -> UPXS +--R acot : UPXS -> UPXS acoth : UPXS -> UPXS +--R acsc : UPXS -> UPXS acsch : UPXS -> UPXS +--R asec : UPXS -> UPXS asech : UPXS -> UPXS +--R asin : UPXS -> UPXS asinh : UPXS -> UPXS +--R atan : UPXS -> UPXS atanh : UPXS -> UPXS +--R cos : UPXS -> UPXS cosh : UPXS -> UPXS +--R cot : UPXS -> UPXS coth : UPXS -> UPXS +--R csc : UPXS -> UPXS csch : UPXS -> UPXS +--R exp : UPXS -> UPXS log : UPXS -> UPXS +--R sec : UPXS -> UPXS sech : UPXS -> UPXS +--R sin : UPXS -> UPXS sinh : UPXS -> UPXS +--R tan : UPXS -> UPXS tanh : UPXS -> UPXS +--R ?**? : (UPXS,Fraction(Integer)) -> UPXS if Coef has FIELD +--R acosIfCan : UPXS -> Union(UPXS,"failed") +--R acoshIfCan : UPXS -> Union(UPXS,"failed") +--R acotIfCan : UPXS -> Union(UPXS,"failed") +--R acothIfCan : UPXS -> Union(UPXS,"failed") +--R acscIfCan : UPXS -> Union(UPXS,"failed") +--R acschIfCan : UPXS -> Union(UPXS,"failed") +--R asecIfCan : UPXS -> Union(UPXS,"failed") +--R asechIfCan : UPXS -> Union(UPXS,"failed") +--R asinIfCan : UPXS -> Union(UPXS,"failed") +--R asinhIfCan : UPXS -> Union(UPXS,"failed") +--R atanIfCan : UPXS -> Union(UPXS,"failed") +--R atanhIfCan : UPXS -> Union(UPXS,"failed") +--R cosIfCan : UPXS -> Union(UPXS,"failed") +--R coshIfCan : UPXS -> Union(UPXS,"failed") +--R cotIfCan : UPXS -> Union(UPXS,"failed") +--R cothIfCan : UPXS -> Union(UPXS,"failed") +--R cscIfCan : UPXS -> Union(UPXS,"failed") +--R cschIfCan : UPXS -> Union(UPXS,"failed") +--R expIfCan : UPXS -> Union(UPXS,"failed") +--R logIfCan : UPXS -> Union(UPXS,"failed") +--R nthRootIfCan : (UPXS,NonNegativeInteger) -> Union(UPXS,"failed") +--R secIfCan : UPXS -> Union(UPXS,"failed") +--R sechIfCan : UPXS -> Union(UPXS,"failed") +--R sinIfCan : UPXS -> Union(UPXS,"failed") +--R sinhIfCan : UPXS -> Union(UPXS,"failed") +--R tanIfCan : UPXS -> Union(UPXS,"failed") +--R tanhIfCan : UPXS -> Union(UPXS,"failed") +--R --E 1 )spool @@ -40478,7 +42047,8 @@ ElementaryFunctionsUnivariatePuiseuxSeries(Coef,ULS,UPXS,EFULS):_ )set message test on )set message auto off )clear all ---S 1 of 12 + +--S 1 of 13 eq1 := 3*x + 4*y = 5 --R --R @@ -40486,7 +42056,7 @@ eq1 := 3*x + 4*y = 5 --R Type: Equation(Polynomial(Integer)) --E 1 ---S 2 of 12 +--S 2 of 13 eq2 := 2*x + 2*y = 3 --R --R @@ -40494,7 +42064,7 @@ eq2 := 2*x + 2*y = 3 --R Type: Equation(Polynomial(Integer)) --E 2 ---S 3 of 12 +--S 3 of 13 lhs eq1 --R --R @@ -40502,7 +42072,7 @@ lhs eq1 --R Type: Polynomial(Integer) --E 3 ---S 4 of 12 +--S 4 of 13 rhs eq1 --R --R @@ -40510,7 +42080,7 @@ rhs eq1 --R Type: Polynomial(Integer) --E 4 ---S 5 of 12 +--S 5 of 13 eq1 + eq2 --R --R @@ -40518,7 +42088,7 @@ eq1 + eq2 --R Type: Equation(Polynomial(Integer)) --E 5 ---S 6 of 12 +--S 6 of 13 eq1 * eq2 --R --R @@ -40527,7 +42097,7 @@ eq1 * eq2 --R Type: Equation(Polynomial(Integer)) --E 6 ---S 7 of 12 +--S 7 of 13 2*eq2 - eq1 --R --R @@ -40535,7 +42105,7 @@ eq1 * eq2 --R Type: Equation(Polynomial(Integer)) --E 7 ---S 8 of 12 +--S 8 of 13 eq1**2 --R --R @@ -40544,7 +42114,7 @@ eq1**2 --R Type: Equation(Polynomial(Integer)) --E 8 ---S 9 of 12 +--S 9 of 13 if x+1 = y then "equal" else "unequal" --R --R @@ -40552,7 +42122,7 @@ if x+1 = y then "equal" else "unequal" --R Type: String --E 9 ---S 10 of 12 +--S 10 of 13 eqpol := x+1 = y --R --R @@ -40560,7 +42130,7 @@ eqpol := x+1 = y --R Type: Equation(Polynomial(Integer)) --E 10 ---S 11 of 12 +--S 11 of 13 if eqpol then "equal" else "unequal" --R --R @@ -40568,13 +42138,78 @@ if eqpol then "equal" else "unequal" --R Type: String --E 11 ---S 12 of 12 +--S 12 of 13 eqpol::Boolean --R --R --R (12) false --R Type: Boolean --E 12 + +--S 13 of 13 +)show Equation +--R +--R Equation(S: Type) is a domain constructor +--R Abbreviation for Equation is EQ +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EQ +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,%) -> % if S has SGROUP ?*? : (S,%) -> % if S has SGROUP +--R ?*? : (%,S) -> % if S has SGROUP ?+? : (%,%) -> % if S has ABELSG +--R ?+? : (%,S) -> % if S has ABELSG ?+? : (S,%) -> % if S has ABELSG +--R -? : % -> % if S has ABELGRP ?-? : (%,%) -> % if S has ABELGRP +--R ?-? : (%,S) -> % if S has ABELGRP ?-? : (S,%) -> % if S has ABELGRP +--R ?/? : (%,S) -> % if S has FIELD ?=? : (S,S) -> % +--R 1 : () -> % if S has MONOID 0 : () -> % if S has ABELGRP +--R coerce : Integer -> % if S has RING equation : (S,S) -> % +--R latex : % -> String if S has SETCAT leftZero : % -> % if S has ABELGRP +--R lhs : % -> S map : ((S -> S),%) -> % +--R one? : % -> Boolean if S has MONOID rhs : % -> S +--R subst : (%,%) -> % if S has ES swap : % -> % +--R ?*? : (PositiveInteger,%) -> % if S has ABELSG +--R ?*? : (NonNegativeInteger,%) -> % if S has ABELGRP +--R ?*? : (Integer,%) -> % if S has ABELGRP +--R ?**? : (%,PositiveInteger) -> % if S has SGROUP +--R ?**? : (%,NonNegativeInteger) -> % if S has MONOID +--R ?**? : (%,Integer) -> % if S has GROUP +--R ?/? : (%,%) -> % if S has FIELD or S has GROUP +--R ?=? : (%,%) -> Boolean if S has SETCAT +--R D : (%,Symbol) -> % if S has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if S has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if S has PDRING(SYMBOL) +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if S has PDRING(SYMBOL) +--R ?^? : (%,PositiveInteger) -> % if S has SGROUP +--R ?^? : (%,NonNegativeInteger) -> % if S has MONOID +--R ?^? : (%,Integer) -> % if S has GROUP +--R characteristic : () -> NonNegativeInteger if S has RING +--R coerce : % -> OutputForm if S has SETCAT +--R coerce : % -> Boolean if S has SETCAT +--R commutator : (%,%) -> % if S has GROUP +--R conjugate : (%,%) -> % if S has GROUP +--R differentiate : (%,Symbol) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if S has PDRING(SYMBOL) +--R dimension : () -> CardinalNumber if S has FIELD +--R eval : (%,List(Symbol),List(S)) -> % if S has IEVALAB(SYMBOL,S) +--R eval : (%,Symbol,S) -> % if S has IEVALAB(SYMBOL,S) +--R eval : (%,List(%)) -> % if S has EVALAB(S) and S has SETCAT +--R eval : (%,%) -> % if S has EVALAB(S) and S has SETCAT +--R factorAndSplit : % -> List(%) if S has INTDOM +--R hash : % -> SingleInteger if S has SETCAT +--R inv : % -> % if S has FIELD or S has GROUP +--R leftOne : % -> Union(%,"failed") if S has MONOID +--R recip : % -> Union(%,"failed") if S has MONOID +--R rightOne : % -> Union(%,"failed") if S has MONOID +--R rightZero : % -> % if S has ABELGRP +--R sample : () -> % if S has ABELGRP or S has MONOID +--R subtractIfCan : (%,%) -> Union(%,"failed") if S has ABELGRP +--R zero? : % -> Boolean if S has ABELGRP +--R ?~=? : (%,%) -> Boolean if S has SETCAT +--R +--E 13 + )spool )lisp (bye) \end{chunk} @@ -40929,7 +42564,8 @@ Equation(S: Type): public == private where )set message test on )set message auto off )clear all ---S 1 of 6 + +--S 1 of 7 e: EqTable(List Integer, Integer) := table() --R --R @@ -40937,7 +42573,7 @@ e: EqTable(List Integer, Integer) := table() --R Type: EqTable(List(Integer),Integer) --E 1 ---S 2 of 6 +--S 2 of 7 l1 := [1,2,3] --R --R @@ -40945,7 +42581,7 @@ l1 := [1,2,3] --R Type: List(PositiveInteger) --E 2 ---S 3 of 6 +--S 3 of 7 l2 := [1,2,3] --R --R @@ -40953,7 +42589,7 @@ l2 := [1,2,3] --R Type: List(PositiveInteger) --E 3 ---S 4 of 6 +--S 4 of 7 e.l1 := 111 --R --R @@ -40961,7 +42597,7 @@ e.l1 := 111 --R Type: PositiveInteger --E 4 ---S 5 of 6 +--S 5 of 7 e.l2 := 222 --R --R @@ -40969,13 +42605,98 @@ e.l2 := 222 --R Type: PositiveInteger --E 5 ---S 6 of 6 +--S 6 of 7 e.l1 --R --R --R (6) 111 --R Type: PositiveInteger --E 6 + +--S 7 of 7 +)show EqTable +--R +--R EqTable(Key: SetCategory,Entry: SetCategory) is a domain constructor +--R Abbreviation for EqTable is EQTBL +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EQTBL +--R +--R------------------------------- Operations -------------------------------- +--R copy : % -> % dictionary : () -> % +--R elt : (%,Key,Entry) -> Entry ?.? : (%,Key) -> Entry +--R empty : () -> % empty? : % -> Boolean +--R entries : % -> List(Entry) eq? : (%,%) -> Boolean +--R index? : (Key,%) -> Boolean indices : % -> List(Key) +--R key? : (Key,%) -> Boolean keys : % -> List(Key) +--R map : ((Entry -> Entry),%) -> % qelt : (%,Key) -> Entry +--R sample : () -> % setelt : (%,Key,Entry) -> Entry +--R table : () -> % +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ?=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R any? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R any? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R bag : List(Record(key: Key,entry: Entry)) -> % +--R coerce : % -> OutputForm if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R construct : List(Record(key: Key,entry: Entry)) -> % +--R convert : % -> InputForm if Record(key: Key,entry: Entry) has KONVERT(INFORM) +--R count : ((Entry -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Entry,%) -> NonNegativeInteger if $ has finiteAggregate and Entry has SETCAT +--R count : (Record(key: Key,entry: Entry),%) -> NonNegativeInteger if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R count : ((Record(key: Key,entry: Entry) -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R dictionary : List(Record(key: Key,entry: Entry)) -> % +--R entry? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R eval : (%,List(Equation(Entry))) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Equation(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Entry,Entry) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Entry),List(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Record(key: Key,entry: Entry)),List(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Equation(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,List(Equation(Record(key: Key,entry: Entry)))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R every? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R every? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R extract! : % -> Record(key: Key,entry: Entry) +--R fill! : (%,Entry) -> % if $ has shallowlyMutable +--R find : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Union(Record(key: Key,entry: Entry),"failed") +--R first : % -> Entry if Key has ORDSET +--R hash : % -> SingleInteger if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R insert! : (Record(key: Key,entry: Entry),%) -> % +--R inspect : % -> Record(key: Key,entry: Entry) +--R latex : % -> String if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R less? : (%,NonNegativeInteger) -> Boolean +--R map : (((Entry,Entry) -> Entry),%,%) -> % +--R map : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % +--R map! : ((Entry -> Entry),%) -> % if $ has shallowlyMutable +--R map! : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % if $ has shallowlyMutable +--R maxIndex : % -> Key if Key has ORDSET +--R member? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R member? : (Record(key: Key,entry: Entry),%) -> Boolean if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R members : % -> List(Entry) if $ has finiteAggregate +--R members : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R minIndex : % -> Key if Key has ORDSET +--R more? : (%,NonNegativeInteger) -> Boolean +--R parts : % -> List(Entry) if $ has finiteAggregate +--R parts : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R qsetelt! : (%,Key,Entry) -> Entry if $ has shallowlyMutable +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove! : (Key,%) -> Union(Entry,"failed") +--R remove! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove! : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate +--R removeDuplicates : % -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R search : (Key,%) -> Union(Entry,"failed") +--R select : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R select! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R size? : (%,NonNegativeInteger) -> Boolean +--R swap! : (%,Key,Key) -> Void if $ has shallowlyMutable +--R table : List(Record(key: Key,entry: Entry)) -> % +--R ?~=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R +--E 7 + )spool )lisp (bye) \end{chunk} @@ -41136,26 +42857,25 @@ EqTable(Key: SetCategory, Entry: SetCategory) == --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> R coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R ?.? : (%,R) -> R gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R modulus : % -> Mod one? : % -> Boolean ---R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") ---R reduce : (R,Mod) -> % ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> R +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm ?.? : (%,R) -> R +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % modulus : % -> Mod +--R one? : % -> Boolean ?quo? : (%,%) -> % +--R recip : % -> Union(%,"failed") reduce : (R,Mod) -> % +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) --R euclideanSize : % -> NonNegativeInteger @@ -41364,7 +43084,8 @@ EuclideanModularRing(S,R,Mod,reduction:(R,Mod) -> R, )set message test on )set message auto off )clear all ---S 1 + +--S 1 of 7 n := 0 --R --R @@ -41372,7 +43093,7 @@ n := 0 --R Type: NonNegativeInteger --E 1 ---S 2 +--S 2 of 7 gasp(): Exit == free n n := n + 1 @@ -41382,7 +43103,7 @@ gasp(): Exit == --R Type: Void --E 2 ---S 3 +--S 3 of 7 half(k) == if odd? k then gasp() else k quo 2 @@ -41390,7 +43111,7 @@ half(k) == --R Type: Void --E 3 ---S 4 +--S 4 of 7 half 4 --R --R Compiling function gasp with type () -> Exit @@ -41400,7 +43121,7 @@ half 4 --R Type: PositiveInteger --E 4 ---S 5 +--S 5 of 7 half 3 --R --R @@ -41409,13 +43130,29 @@ half 3 --R Oh no! --E 5 ---S 6 +--S 6 of 7 n --R --R --R (5) 1 --R Type: NonNegativeInteger --E 6 + +--S 7 of 7 +)show Exit +--R +--R Exit is a domain constructor +--R Abbreviation for Exit is EXIT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EXIT +--R +--R------------------------------- Operations -------------------------------- +--R ?=? : (%,%) -> Boolean coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R ?~=? : (%,%) -> Boolean +--R +--E 7 + )spool )lisp (bye) \end{chunk} @@ -41518,30 +43255,30 @@ Exit: SetCategory == add --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R associates? : (%,%) -> Boolean coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm denominator : % -> % ---R factor : % -> Factored(%) gcd : List(%) -> % ---R gcd : (%,%) -> % hash : % -> SingleInteger ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R numerator : % -> % one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R denominator : % -> % factor : % -> Factored(%) +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % numerator : % -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R ?*? : (%,UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen)) -> % --R ?*? : (UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen),%) -> % ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen),UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen)) -> % --R ? Boolean if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has ORDSET --R ?<=? : (%,%) -> Boolean if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has ORDSET @@ -41555,7 +43292,6 @@ Exit: SetCategory == add --R D : (%,Symbol) -> % if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has PDRING(SYMBOL) --R D : (%,NonNegativeInteger) -> % if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has DIFRING --R D : % -> % if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has DIFRING ---R ?^? : (%,NonNegativeInteger) -> % --R abs : % -> % if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has OINTDOM --R ceiling : % -> UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) if UnivariatePuiseuxSeriesWithExponentialSingularity(R,FE,var,cen) has INS --R characteristic : () -> NonNegativeInteger @@ -41899,7 +43635,8 @@ ExponentialExpansion(R,FE,var,cen): Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 23 + +--S 1 of 24 sin(x) + 3*cos(x)**2 --R --R @@ -41908,7 +43645,7 @@ sin(x) + 3*cos(x)**2 --R Type: Expression(Integer) --E 1 ---S 2 of 23 +--S 2 of 24 tan(x) - 3.45*x --R --R @@ -41916,7 +43653,7 @@ tan(x) - 3.45*x --R Type: Expression(Float) --E 2 ---S 3 of 23 +--S 3 of 24 (tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y)) --R --R @@ -41927,7 +43664,7 @@ tan(x) - 3.45*x --R Type: Expression(Integer) --E 3 ---S 4 of 23 +--S 4 of 24 log(exp x)@Expression(Integer) --R --R @@ -41935,7 +43672,7 @@ log(exp x)@Expression(Integer) --R Type: Expression(Integer) --E 4 ---S 5 of 23 +--S 5 of 24 log(exp x)@Expression(Complex Integer) --R --R @@ -41944,7 +43681,7 @@ log(exp x)@Expression(Complex Integer) --R Type: Expression(Complex(Integer)) --E 5 ---S 6 of 23 +--S 6 of 24 sqrt 3 + sqrt(2 + sqrt(-5)) --R --R @@ -41954,7 +43691,7 @@ sqrt 3 + sqrt(2 + sqrt(-5)) --R Type: AlgebraicNumber --E 6 ---S 7 of 23 +--S 7 of 24 % :: Expression Integer --R --R @@ -41964,7 +43701,7 @@ sqrt 3 + sqrt(2 + sqrt(-5)) --R Type: Expression(Integer) --E 7 ---S 8 of 23 +--S 8 of 24 height mainKernel sin(x + 4) --R --R @@ -41972,7 +43709,7 @@ height mainKernel sin(x + 4) --R Type: PositiveInteger --E 8 ---S 9 of 23 +--S 9 of 24 e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) --R --R @@ -41984,7 +43721,7 @@ e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) --R Type: Expression(Integer) --E 9 ---S 10 of 23 +--S 10 of 24 numer e --R --R @@ -41993,7 +43730,7 @@ numer e --R Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))) --E 10 ---S 11 of 23 +--S 11 of 24 denom e --R --R @@ -42002,7 +43739,7 @@ denom e --R Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))) --E 11 ---S 12 of 23 +--S 12 of 24 D(e, x) --R --R @@ -42014,7 +43751,7 @@ D(e, x) --R Type: Expression(Integer) --E 12 ---S 13 of 23 +--S 13 of 24 D(e, [x, y], [1, 2]) --R --R @@ -42036,7 +43773,7 @@ D(e, [x, y], [1, 2]) --R Type: Expression(Integer) --E 13 ---S 14 of 23 +--S 14 of 24 complexNumeric(cos(2 - 3*%i)) --R --R @@ -42044,7 +43781,7 @@ complexNumeric(cos(2 - 3*%i)) --R Type: Complex(Float) --E 14 ---S 15 of 23 +--S 15 of 24 numeric(tan 3.8) --R --R @@ -42052,7 +43789,7 @@ numeric(tan 3.8) --R Type: Float --E 15 ---S 16 of 23 +--S 16 of 24 e2 := cos(x**2 - y + 3) --R --R @@ -42061,7 +43798,7 @@ e2 := cos(x**2 - y + 3) --R Type: Expression(Integer) --E 16 ---S 17 of 23 +--S 17 of 24 e3 := asin(e2) - %pi/2 --R --R @@ -42070,7 +43807,7 @@ e3 := asin(e2) - %pi/2 --R Type: Expression(Integer) --E 17 ---S 18 of 23 +--S 18 of 24 e3 :: Polynomial Integer --R --R @@ -42079,7 +43816,7 @@ e3 :: Polynomial Integer --R Type: Polynomial(Integer) --E 18 ---S 19 of 23 +--S 19 of 24 e3 :: DMP([x, y], Integer) --R --R @@ -42088,7 +43825,7 @@ e3 :: DMP([x, y], Integer) --R Type: DistributedMultivariatePolynomial([x,y],Integer) --E 19 ---S 20 of 23 +--S 20 of 24 sin %pi --R --R @@ -42096,7 +43833,7 @@ sin %pi --R Type: Expression(Integer) --E 20 ---S 21 of 23 +--S 21 of 24 cos(%pi / 4) --R --R @@ -42107,7 +43844,7 @@ cos(%pi / 4) --R Type: Expression(Integer) --E 21 ---S 22 of 23 +--S 22 of 24 tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 --R --R @@ -42116,7 +43853,7 @@ tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 --R Type: Expression(Integer) --E 22 ---S 23 of 23 +--S 23 of 24 simplify % --R --R @@ -42126,6 +43863,242 @@ simplify % --R cos(x) --R Type: Expression(Integer) --E 23 + +--S 24 of 24 +)show Expression +--R +--R Expression(R: OrderedSet) is a domain constructor +--R Abbreviation for Expression is EXPR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for EXPR +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % if R has COMRING ?*? : (R,%) -> % if R has COMRING +--R ?**? : (%,%) -> % if R has INTDOM ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R Beta : (%,%) -> % if R has INTDOM Ci : % -> % if R has INTDOM +--R D : (%,Symbol) -> % if R has RING Ei : % -> % if R has INTDOM +--R Gamma : % -> % if R has INTDOM Si : % -> % if R has INTDOM +--R abs : % -> % if R has INTDOM acos : % -> % if R has INTDOM +--R acosh : % -> % if R has INTDOM acot : % -> % if R has INTDOM +--R acoth : % -> % if R has INTDOM acsc : % -> % if R has INTDOM +--R acsch : % -> % if R has INTDOM airyAi : % -> % if R has INTDOM +--R airyBi : % -> % if R has INTDOM applyQuote : (Symbol,List(%)) -> % +--R applyQuote : (Symbol,%,%,%) -> % applyQuote : (Symbol,%,%) -> % +--R applyQuote : (Symbol,%) -> % asec : % -> % if R has INTDOM +--R asech : % -> % if R has INTDOM asin : % -> % if R has INTDOM +--R asinh : % -> % if R has INTDOM atan : % -> % if R has INTDOM +--R atanh : % -> % if R has INTDOM belong? : BasicOperator -> Boolean +--R box : List(%) -> % box : % -> % +--R coerce : % -> % if R has INTDOM coerce : R -> % +--R coerce : Symbol -> % coerce : Kernel(%) -> % +--R coerce : % -> OutputForm cos : % -> % if R has INTDOM +--R cosh : % -> % if R has INTDOM cot : % -> % if R has INTDOM +--R coth : % -> % if R has INTDOM csc : % -> % if R has INTDOM +--R csch : % -> % if R has INTDOM digamma : % -> % if R has INTDOM +--R dilog : % -> % if R has INTDOM distribute : (%,%) -> % +--R distribute : % -> % elt : (BasicOperator,List(%)) -> % +--R elt : (BasicOperator,%,%,%) -> % elt : (BasicOperator,%,%) -> % +--R elt : (BasicOperator,%) -> % erf : % -> % if R has INTDOM +--R eval : (%,Symbol,(% -> %)) -> % eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % eval : (%,Kernel(%),%) -> % +--R exp : % -> % if R has INTDOM factorial : % -> % if R has INTDOM +--R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean +--R fresnelC : % -> % if R has INTDOM fresnelS : % -> % if R has INTDOM +--R gcd : (%,%) -> % if R has INTDOM gcd : List(%) -> % if R has INTDOM +--R ground : % -> R ground? : % -> Boolean +--R hash : % -> SingleInteger height : % -> NonNegativeInteger +--R is? : (%,Symbol) -> Boolean is? : (%,BasicOperator) -> Boolean +--R kernel : (BasicOperator,%) -> % kernels : % -> List(Kernel(%)) +--R latex : % -> String lcm : (%,%) -> % if R has INTDOM +--R lcm : List(%) -> % if R has INTDOM li : % -> % if R has INTDOM +--R log : % -> % if R has INTDOM map : ((% -> %),Kernel(%)) -> % +--R max : (%,%) -> % min : (%,%) -> % +--R numerator : % -> % if R has RING paren : List(%) -> % +--R paren : % -> % pi : () -> % if R has INTDOM +--R reduce : % -> % if R has INTDOM retract : % -> R +--R retract : % -> Symbol retract : % -> Kernel(%) +--R rootOf : % -> % if R has INTDOM sec : % -> % if R has INTDOM +--R sech : % -> % if R has INTDOM sin : % -> % if R has INTDOM +--R sinh : % -> % if R has INTDOM sqrt : % -> % if R has INTDOM +--R subst : (%,Equation(%)) -> % tan : % -> % if R has INTDOM +--R tanh : % -> % if R has INTDOM tower : % -> List(Kernel(%)) +--R variables : % -> List(Symbol) zeroOf : % -> % if R has INTDOM +--R ?~=? : (%,%) -> Boolean +--R ?*? : (PositiveInteger,%) -> % if R has ABELSG or R has LINEXP(INT) and R has RING +--R ?*? : (NonNegativeInteger,%) -> % if R has ABELSG or R has LINEXP(INT) and R has RING +--R ?*? : (Integer,%) -> % if R has ABELGRP or R has LINEXP(INT) and R has RING +--R ?*? : (%,%) -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R ?*? : (%,Fraction(Integer)) -> % if R has INTDOM +--R ?*? : (Fraction(Integer),%) -> % if R has INTDOM +--R ?**? : (%,PositiveInteger) -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R ?**? : (%,NonNegativeInteger) -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R ?**? : (%,Integer) -> % if R has GROUP or R has INTDOM +--R ?**? : (%,Fraction(Integer)) -> % if R has INTDOM +--R ?+? : (%,%) -> % if R has ABELSG or R has LINEXP(INT) and R has RING +--R -? : % -> % if R has ABELGRP or R has LINEXP(INT) and R has RING +--R ?-? : (%,%) -> % if R has ABELGRP or R has LINEXP(INT) and R has RING +--R ?/? : (%,%) -> % if R has GROUP or R has INTDOM +--R ?/? : (SparseMultivariatePolynomial(R,Kernel(%)),SparseMultivariatePolynomial(R,Kernel(%))) -> % if R has INTDOM +--R D : (%,List(Symbol)) -> % if R has RING +--R D : (%,Symbol,NonNegativeInteger) -> % if R has RING +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has RING +--R Gamma : (%,%) -> % if R has INTDOM +--R 1 : () -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R 0 : () -> % if R has ABELSG or R has LINEXP(INT) and R has RING +--R ?^? : (%,PositiveInteger) -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R ?^? : (%,NonNegativeInteger) -> % if R has LINEXP(INT) and R has RING or R has SGROUP +--R ?^? : (%,Integer) -> % if R has GROUP or R has INTDOM +--R applyQuote : (Symbol,%,%,%,%) -> % +--R associates? : (%,%) -> Boolean if R has INTDOM +--R besselI : (%,%) -> % if R has INTDOM +--R besselJ : (%,%) -> % if R has INTDOM +--R besselK : (%,%) -> % if R has INTDOM +--R besselY : (%,%) -> % if R has INTDOM +--R binomial : (%,%) -> % if R has INTDOM +--R characteristic : () -> NonNegativeInteger if R has RING +--R charthRoot : % -> Union(%,"failed") if R has CHARNZ +--R coerce : Integer -> % if R has RETRACT(INT) or R has RING +--R coerce : Fraction(Integer) -> % if R has INTDOM or R has RETRACT(FRAC(INT)) +--R coerce : Polynomial(R) -> % if R has RING +--R coerce : Fraction(Polynomial(R)) -> % if R has INTDOM +--R coerce : Fraction(Polynomial(Fraction(R))) -> % if R has INTDOM +--R coerce : Polynomial(Fraction(R)) -> % if R has INTDOM +--R coerce : Fraction(R) -> % if R has INTDOM +--R coerce : SparseMultivariatePolynomial(R,Kernel(%)) -> % if R has RING +--R coerce : AlgebraicNumber -> % if R has INTDOM and R has RETRACT(INT) +--R commutator : (%,%) -> % if R has GROUP +--R conjugate : (%,%) -> % if R has GROUP +--R convert : % -> InputForm if R has KONVERT(INFORM) +--R convert : Factored(%) -> % if R has INTDOM +--R convert : % -> Pattern(Float) if R has KONVERT(PATTERN(FLOAT)) +--R convert : % -> Pattern(Integer) if R has KONVERT(PATTERN(INT)) +--R definingPolynomial : % -> % if $ has RING +--R denom : % -> SparseMultivariatePolynomial(R,Kernel(%)) if R has INTDOM +--R denominator : % -> % if R has INTDOM +--R differentiate : (%,Symbol) -> % if R has RING +--R differentiate : (%,List(Symbol)) -> % if R has RING +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has RING +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has RING +--R divide : (%,%) -> Record(quotient: %,remainder: %) if R has INTDOM +--R elt : (BasicOperator,%,%,%,%) -> % +--R euclideanSize : % -> NonNegativeInteger if R has INTDOM +--R eval : (%,Symbol,NonNegativeInteger,(% -> %)) -> % if R has RING +--R eval : (%,Symbol,NonNegativeInteger,(List(%) -> %)) -> % if R has RING +--R eval : (%,List(Symbol),List(NonNegativeInteger),List((List(%) -> %))) -> % if R has RING +--R eval : (%,List(Symbol),List(NonNegativeInteger),List((% -> %))) -> % if R has RING +--R eval : (%,List(BasicOperator),List(%),Symbol) -> % if R has KONVERT(INFORM) +--R eval : (%,BasicOperator,%,Symbol) -> % if R has KONVERT(INFORM) +--R eval : % -> % if R has KONVERT(INFORM) +--R eval : (%,List(Symbol)) -> % if R has KONVERT(INFORM) +--R eval : (%,Symbol) -> % if R has KONVERT(INFORM) +--R eval : (%,BasicOperator,(% -> %)) -> % +--R eval : (%,BasicOperator,(List(%) -> %)) -> % +--R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % +--R eval : (%,List(BasicOperator),List((% -> %))) -> % +--R eval : (%,Symbol,(List(%) -> %)) -> % +--R eval : (%,List(Symbol),List((List(%) -> %))) -> % +--R eval : (%,List(Symbol),List((% -> %))) -> % +--R eval : (%,List(Kernel(%)),List(%)) -> % +--R even? : % -> Boolean if $ has RETRACT(INT) +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") if R has INTDOM +--R exquo : (%,%) -> Union(%,"failed") if R has INTDOM +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) if R has INTDOM +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") if R has INTDOM +--R factor : % -> Factored(%) if R has INTDOM +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has GCDDOM and R has INTDOM +--R factorials : % -> % if R has INTDOM +--R factorials : (%,Symbol) -> % if R has INTDOM +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has INTDOM +--R integral : (%,SegmentBinding(%)) -> % if R has INTDOM +--R integral : (%,Symbol) -> % if R has INTDOM +--R inv : % -> % if R has GROUP or R has INTDOM +--R isExpt : (%,Symbol) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING +--R isExpt : (%,BasicOperator) -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has RING +--R isExpt : % -> Union(Record(var: Kernel(%),exponent: Integer),"failed") if R has SGROUP +--R isMult : % -> Union(Record(coef: Integer,var: Kernel(%)),"failed") if R has ABELSG +--R isPlus : % -> Union(List(%),"failed") if R has ABELSG +--R isPower : % -> Union(Record(val: %,exponent: Integer),"failed") if R has RING +--R isTimes : % -> Union(List(%),"failed") if R has SGROUP +--R kernel : (BasicOperator,List(%)) -> % +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has INTDOM +--R mainKernel : % -> Union(Kernel(%),"failed") +--R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") if R has INTDOM +--R nthRoot : (%,Integer) -> % if R has INTDOM +--R number? : % -> Boolean if R has INTDOM +--R numer : % -> SparseMultivariatePolynomial(R,Kernel(%)) if R has RING +--R odd? : % -> Boolean if $ has RETRACT(INT) +--R one? : % -> Boolean if R has LINEXP(INT) and R has RING or R has SGROUP +--R operator : BasicOperator -> BasicOperator +--R operators : % -> List(BasicOperator) +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if R has PATMAB(FLOAT) +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if R has PATMAB(INT) +--R permutation : (%,%) -> % if R has INTDOM +--R polygamma : (%,%) -> % if R has INTDOM +--R prime? : % -> Boolean if R has INTDOM +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) if R has INTDOM +--R product : (%,Symbol) -> % if R has INTDOM +--R product : (%,SegmentBinding(%)) -> % if R has INTDOM +--R ?quo? : (%,%) -> % if R has INTDOM +--R recip : % -> Union(%,"failed") if R has LINEXP(INT) and R has RING or R has SGROUP +--R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) and R has RING +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) and R has RING +--R reducedSystem : Matrix(%) -> Matrix(R) if R has RING +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) if R has RING +--R ?rem? : (%,%) -> % if R has INTDOM +--R retract : % -> Fraction(Integer) if R has INTDOM and R has RETRACT(INT) or R has RETRACT(FRAC(INT)) +--R retract : % -> Polynomial(R) if R has RING +--R retract : % -> Fraction(Polynomial(R)) if R has INTDOM +--R retract : % -> Integer if R has RETRACT(INT) +--R retract : % -> AlgebraicNumber if R has INTDOM and R has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has INTDOM and R has RETRACT(INT) or R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(Polynomial(R),"failed") if R has RING +--R retractIfCan : % -> Union(Fraction(Polynomial(R)),"failed") if R has INTDOM +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R retractIfCan : % -> Union(AlgebraicNumber,"failed") if R has INTDOM and R has RETRACT(INT) +--R retractIfCan : % -> Union(R,"failed") +--R retractIfCan : % -> Union(Symbol,"failed") +--R retractIfCan : % -> Union(Kernel(%),"failed") +--R rootOf : Polynomial(%) -> % if R has INTDOM +--R rootOf : SparseUnivariatePolynomial(%) -> % if R has INTDOM +--R rootOf : (SparseUnivariatePolynomial(%),Symbol) -> % if R has INTDOM +--R rootOf : (%,Symbol) -> % if R has INTDOM +--R rootsOf : Polynomial(%) -> List(%) if R has INTDOM +--R rootsOf : SparseUnivariatePolynomial(%) -> List(%) if R has INTDOM +--R rootsOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) if R has INTDOM +--R rootsOf : % -> List(%) if R has INTDOM +--R rootsOf : (%,Symbol) -> List(%) if R has INTDOM +--R sample : () -> % if R has ABELSG or R has LINEXP(INT) and R has RING or R has SGROUP +--R simplifyPower : (%,Integer) -> % if R has INTDOM +--R sizeLess? : (%,%) -> Boolean if R has INTDOM +--R squareFree : % -> Factored(%) if R has INTDOM +--R squareFreePart : % -> % if R has INTDOM +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has GCDDOM and R has INTDOM +--R subst : (%,List(Kernel(%)),List(%)) -> % +--R subst : (%,List(Equation(%))) -> % +--R subtractIfCan : (%,%) -> Union(%,"failed") if R has ABELGRP or R has LINEXP(INT) and R has RING +--R summation : (%,Symbol) -> % if R has INTDOM +--R summation : (%,SegmentBinding(%)) -> % if R has INTDOM +--R unit? : % -> Boolean if R has INTDOM +--R unitCanonical : % -> % if R has INTDOM +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has INTDOM +--R univariate : (%,Kernel(%)) -> Fraction(SparseUnivariatePolynomial(%)) if R has INTDOM +--R zero? : % -> Boolean if R has ABELSG or R has LINEXP(INT) and R has RING +--R zeroOf : Polynomial(%) -> % if R has INTDOM +--R zeroOf : SparseUnivariatePolynomial(%) -> % if R has INTDOM +--R zeroOf : (SparseUnivariatePolynomial(%),Symbol) -> % if R has INTDOM +--R zeroOf : (%,Symbol) -> % if R has INTDOM +--R zerosOf : Polynomial(%) -> List(%) if R has INTDOM +--R zerosOf : SparseUnivariatePolynomial(%) -> List(%) if R has INTDOM +--R zerosOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) if R has INTDOM +--R zerosOf : % -> List(%) if R has INTDOM +--R zerosOf : (%,Symbol) -> List(%) if R has INTDOM +--R +--E 24 + )spool )lisp (bye) \end{chunk} @@ -42989,34 +44962,35 @@ Expression(R:OrderedSet): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (FE,%) -> % ?*? : (%,FE) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ? Boolean +--R -? : % -> % ?/? : (%,%) -> % if FE has FIELD +--R ?/? : (%,FE) -> % if FE has FIELD ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> FE ---R coerce : Integer -> % coerce : % -> OutputForm ---R complete : % -> % degree : % -> Fraction(Integer) ---R ?.? : (%,Fraction(Integer)) -> FE hash : % -> SingleInteger ---R inv : % -> % if FE has FIELD latex : % -> String ---R leadingCoefficient : % -> FE leadingMonomial : % -> % ---R map : ((FE -> FE),%) -> % max : (%,%) -> % ---R min : (%,%) -> % monomial? : % -> Boolean ---R one? : % -> Boolean order : % -> Fraction(Integer) ---R pole? : % -> Boolean recip : % -> Union(%,"failed") ---R reductum : % -> % sample : () -> % ---R variable : % -> Symbol zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> FE coerce : % -> % if FE has INTDOM +--R coerce : FE -> % if FE has COMRING coerce : Integer -> % +--R coerce : % -> OutputForm complete : % -> % +--R degree : % -> Fraction(Integer) ?.? : (%,Fraction(Integer)) -> FE +--R gcd : (%,%) -> % if FE has FIELD gcd : List(%) -> % if FE has FIELD +--R hash : % -> SingleInteger inv : % -> % if FE has FIELD +--R latex : % -> String lcm : (%,%) -> % if FE has FIELD +--R lcm : List(%) -> % if FE has FIELD leadingCoefficient : % -> FE +--R leadingMonomial : % -> % map : ((FE -> FE),%) -> % +--R max : (%,%) -> % min : (%,%) -> % +--R monomial? : % -> Boolean one? : % -> Boolean +--R order : % -> Fraction(Integer) pole? : % -> Boolean +--R recip : % -> Union(%,"failed") reductum : % -> % +--R sample : () -> % variable : % -> Symbol +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if FE has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if FE has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if FE has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if FE has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if FE has FIELD ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,%) -> % if FE has FIELD ---R ?/? : (%,FE) -> % if FE has FIELD --R D : % -> % if FE has *: (Fraction(Integer),FE) -> FE --R D : (%,NonNegativeInteger) -> % if FE has *: (Fraction(Integer),FE) -> FE --R D : (%,Symbol) -> % if FE has *: (Fraction(Integer),FE) -> FE and FE has PDRING(SYMBOL) @@ -43024,7 +44998,6 @@ Expression(R:OrderedSet): Exports == Implementation where --R D : (%,Symbol,NonNegativeInteger) -> % if FE has *: (Fraction(Integer),FE) -> FE and FE has PDRING(SYMBOL) --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if FE has *: (Fraction(Integer),FE) -> FE and FE has PDRING(SYMBOL) --R ?^? : (%,Integer) -> % if FE has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if FE has ALGEBRA(FRAC(INT)) --R acosh : % -> % if FE has ALGEBRA(FRAC(INT)) --R acot : % -> % if FE has ALGEBRA(FRAC(INT)) @@ -43042,9 +45015,7 @@ Expression(R:OrderedSet): Exports == Implementation where --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if FE has CHARNZ --R coefficient : (%,Fraction(Integer)) -> FE ---R coerce : % -> % if FE has INTDOM --R coerce : Fraction(Integer) -> % if FE has ALGEBRA(FRAC(INT)) ---R coerce : FE -> % if FE has COMRING --R cos : % -> % if FE has ALGEBRA(FRAC(INT)) --R cosh : % -> % if FE has ALGEBRA(FRAC(INT)) --R cot : % -> % if FE has ALGEBRA(FRAC(INT)) @@ -43071,13 +45042,9 @@ Expression(R:OrderedSet): Exports == Implementation where --R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) if FE has FIELD --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") if FE has FIELD --R factor : % -> Factored(%) if FE has FIELD ---R gcd : (%,%) -> % if FE has FIELD ---R gcd : List(%) -> % if FE has FIELD --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if FE has FIELD --R integrate : (%,Symbol) -> % if FE has integrate: (FE,Symbol) -> FE and FE has variables: FE -> List(Symbol) and FE has ALGEBRA(FRAC(INT)) or FE has ACFS(INT) and FE has ALGEBRA(FRAC(INT)) and FE has PRIMCAT and FE has TRANFUN --R integrate : % -> % if FE has ALGEBRA(FRAC(INT)) ---R lcm : (%,%) -> % if FE has FIELD ---R lcm : List(%) -> % if FE has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if FE has FIELD --R log : % -> % if FE has ALGEBRA(FRAC(INT)) --R monomial : (%,List(SingletonAsOrderedSet),List(Fraction(Integer))) -> % @@ -44659,7 +46626,8 @@ e04ucfAnnaType(): NumericalOptimizationCategory == Result add )set message test on )set message auto off )clear all ---S 1 of 38 + +--S 1 of 39 g := factor(4312) --R --R @@ -44668,7 +46636,7 @@ g := factor(4312) --R Type: Factored(Integer) --E 1 ---S 2 of 38 +--S 2 of 39 unit(g) --R --R @@ -44676,7 +46644,7 @@ unit(g) --R Type: PositiveInteger --E 2 ---S 3 of 38 +--S 3 of 39 numberOfFactors(g) --R --R @@ -44684,7 +46652,7 @@ numberOfFactors(g) --R Type: PositiveInteger --E 3 ---S 4 of 38 +--S 4 of 39 [nthFactor(g,i) for i in 1..numberOfFactors(g)] --R --R @@ -44692,7 +46660,7 @@ numberOfFactors(g) --R Type: List(Integer) --E 4 ---S 5 of 38 +--S 5 of 39 [nthExponent(g,i) for i in 1..numberOfFactors(g)] --R --R @@ -44700,7 +46668,7 @@ numberOfFactors(g) --R Type: List(Integer) --E 5 ---S 6 of 38 +--S 6 of 39 [nthFlag(g,i) for i in 1..numberOfFactors(g)] --R --R @@ -44708,7 +46676,7 @@ numberOfFactors(g) --R Type: List(Union("nil","sqfr","irred","prime")) --E 6 ---S 7 of 38 +--S 7 of 39 factorList(g) --R --R @@ -44718,7 +46686,7 @@ factorList(g) --RType: List(Record(flg: Union("nil","sqfr","irred","prime"),fctr: Integer,xpnt: Integer)) --E 7 ---S 8 of 38 +--S 8 of 39 factors(g) --R --R @@ -44727,7 +46695,7 @@ factors(g) --R Type: List(Record(factor: Integer,exponent: Integer)) --E 8 ---S 9 of 38 +--S 9 of 39 first(%).factor --R --R @@ -44735,7 +46703,7 @@ first(%).factor --R Type: PositiveInteger --E 9 ---S 10 of 38 +--S 10 of 39 g := factor(4312) --R --R @@ -44744,7 +46712,7 @@ g := factor(4312) --R Type: Factored(Integer) --E 10 ---S 11 of 38 +--S 11 of 39 expand(g) --R --R @@ -44752,7 +46720,7 @@ expand(g) --R Type: PositiveInteger --E 11 ---S 12 of 38 +--S 12 of 39 reduce(*,[t.factor for t in factors(g)]) --R --R @@ -44760,7 +46728,7 @@ reduce(*,[t.factor for t in factors(g)]) --R Type: PositiveInteger --E 12 ---S 13 of 38 +--S 13 of 39 g := factor(4312) --R --R @@ -44769,7 +46737,7 @@ g := factor(4312) --R Type: Factored(Integer) --E 13 ---S 14 of 38 +--S 14 of 39 f := factor(246960) --R --R @@ -44778,7 +46746,7 @@ f := factor(246960) --R Type: Factored(Integer) --E 14 ---S 15 of 38 +--S 15 of 39 f * g --R --R @@ -44787,7 +46755,7 @@ f * g --R Type: Factored(Integer) --E 15 ---S 16 of 38 +--S 16 of 39 f**500 --R --R @@ -44796,7 +46764,7 @@ f**500 --R Type: Factored(Integer) --E 16 ---S 17 of 38 +--S 17 of 39 gcd(f,g) --R --R @@ -44805,7 +46773,7 @@ gcd(f,g) --R Type: Factored(Integer) --E 17 ---S 18 of 38 +--S 18 of 39 lcm(f,g) --R --R @@ -44814,7 +46782,7 @@ lcm(f,g) --R Type: Factored(Integer) --E 18 ---S 19 of 38 +--S 19 of 39 f + g --R --R @@ -44823,7 +46791,7 @@ f + g --R Type: Factored(Integer) --E 19 ---S 20 of 38 +--S 20 of 39 f - g --R --R @@ -44832,7 +46800,7 @@ f - g --R Type: Factored(Integer) --E 20 ---S 21 of 38 +--S 21 of 39 zero?(factor(0)) --R --R @@ -44840,7 +46808,7 @@ zero?(factor(0)) --R Type: Boolean --E 21 ---S 22 of 38 +--S 22 of 39 zero?(g) --R --R @@ -44848,7 +46816,7 @@ zero?(g) --R Type: Boolean --E 22 ---S 23 of 38 +--S 23 of 39 one?(factor(1)) --R --R @@ -44856,7 +46824,7 @@ one?(factor(1)) --R Type: Boolean --E 23 ---S 24 of 38 +--S 24 of 39 one?(f) --R --R @@ -44864,7 +46832,7 @@ one?(f) --R Type: Boolean --E 24 ---S 25 of 38 +--S 25 of 39 0$Factored(Integer) --R --R @@ -44872,7 +46840,7 @@ one?(f) --R Type: Factored(Integer) --E 25 ---S 26 of 38 +--S 26 of 39 1$Factored(Integer) --R --R @@ -44880,7 +46848,7 @@ one?(f) --R Type: Factored(Integer) --E 26 ---S 27 of 38 +--S 27 of 39 nilFactor(24,2) --R --R @@ -44889,7 +46857,7 @@ nilFactor(24,2) --R Type: Factored(Integer) --E 27 ---S 28 of 38 +--S 28 of 39 nthFlag(%,1) --R --R @@ -44897,7 +46865,7 @@ nthFlag(%,1) --R Type: Union("nil",...) --E 28 ---S 29 of 38 +--S 29 of 39 sqfrFactor(30,2) --R --R @@ -44906,7 +46874,7 @@ sqfrFactor(30,2) --R Type: Factored(Integer) --E 29 ---S 30 of 38 +--S 30 of 39 irreducibleFactor(13,10) --R --R @@ -44915,7 +46883,7 @@ irreducibleFactor(13,10) --R Type: Factored(Integer) --E 30 ---S 31 of 38 +--S 31 of 39 primeFactor(11,5) --R --R @@ -44924,7 +46892,7 @@ primeFactor(11,5) --R Type: Factored(Integer) --E 31 ---S 32 of 38 +--S 32 of 39 h := factor(-720) --R --R @@ -44933,7 +46901,7 @@ h := factor(-720) --R Type: Factored(Integer) --E 32 ---S 33 of 38 +--S 33 of 39 h - makeFR(unit(h),factorList(h)) --R --R @@ -44941,7 +46909,7 @@ h - makeFR(unit(h),factorList(h)) --R Type: Factored(Integer) --E 33 ---S 34 of 38 +--S 34 of 39 p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 --R --R @@ -44950,7 +46918,7 @@ p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 --R Type: Polynomial(Integer) --E 34 ---S 35 of 38 +--S 35 of 39 fp := factor(p) --R --R @@ -44959,7 +46927,7 @@ fp := factor(p) --R Type: Factored(Polynomial(Integer)) --E 35 ---S 36 of 38 +--S 36 of 39 D(p,x) --R --R @@ -44968,7 +46936,7 @@ D(p,x) --R Type: Polynomial(Integer) --E 36 ---S 37 of 38 +--S 37 of 39 D(fp,x) --R --R @@ -44977,13 +46945,106 @@ D(fp,x) --R Type: Factored(Polynomial(Integer)) --E 37 ---S 38 of 38 +--S 38 of 39 numberOfFactors(%) --R --R --R (38) 3 --R Type: PositiveInteger --E 38 + +--S 39 of 39 +)show Factored +--R +--R Factored(R: IntegralDomain) is a domain constructor +--R Abbreviation for Factored is FR +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FR +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R D : (%,(R -> R)) -> % D : % -> % if R has DIFRING +--R 1 : () -> % 0 : () -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : R -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm convert : % -> Float if R has REAL +--R differentiate : (%,(R -> R)) -> % expand : % -> R +--R exponent : % -> Integer gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM hash : % -> SingleInteger +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM map : ((R -> R),%) -> % +--R nilFactor : (R,Integer) -> % nthFactor : (%,Integer) -> R +--R one? : % -> Boolean prime? : % -> Boolean if R has UFD +--R primeFactor : (R,Integer) -> % recip : % -> Union(%,"failed") +--R retract : % -> R sample : () -> % +--R sqfrFactor : (R,Integer) -> % unit : % -> R +--R unit? : % -> Boolean unitCanonical : % -> % +--R unitNormalize : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R D : (%,(R -> R),NonNegativeInteger) -> % +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) +--R D : (%,Symbol) -> % if R has PDRING(SYMBOL) +--R D : (%,NonNegativeInteger) -> % if R has DIFRING +--R characteristic : () -> NonNegativeInteger +--R coerce : Fraction(Integer) -> % if R has RETRACT(FRAC(INT)) +--R convert : % -> DoubleFloat if R has REAL +--R convert : % -> InputForm if R has KONVERT(INFORM) +--R differentiate : (%,(R -> R),NonNegativeInteger) -> % +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,Symbol) -> % if R has PDRING(SYMBOL) +--R differentiate : (%,NonNegativeInteger) -> % if R has DIFRING +--R differentiate : % -> % if R has DIFRING +--R ?.? : (%,%) -> % if R has ELTAB($,$) +--R ?.? : (%,R) -> % if R has ELTAB(R,R) +--R eval : (%,List(%),List(%)) -> % if R has EVALAB($) +--R eval : (%,%,%) -> % if R has EVALAB($) +--R eval : (%,Equation(%)) -> % if R has EVALAB($) +--R eval : (%,List(Equation(%))) -> % if R has EVALAB($) +--R eval : (%,List(Symbol),List(%)) -> % if R has IEVALAB(SYMBOL,$) +--R eval : (%,Symbol,%) -> % if R has IEVALAB(SYMBOL,$) +--R eval : (%,Symbol,R) -> % if R has IEVALAB(SYMBOL,R) +--R eval : (%,List(Symbol),List(R)) -> % if R has IEVALAB(SYMBOL,R) +--R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) +--R eval : (%,Equation(R)) -> % if R has EVALAB(R) +--R eval : (%,R,R) -> % if R has EVALAB(R) +--R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) +--R exquo : (%,%) -> Union(%,"failed") +--R factor : % -> Factored(%) if R has UFD +--R factorList : % -> List(Record(flg: Union("nil","sqfr","irred","prime"),fctr: R,xpnt: Integer)) +--R factors : % -> List(Record(factor: R,exponent: Integer)) +--R flagFactor : (R,Integer,Union("nil","sqfr","irred","prime")) -> % +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM +--R irreducibleFactor : (R,Integer) -> % +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM +--R makeFR : (R,List(Record(flg: Union("nil","sqfr","irred","prime"),fctr: R,xpnt: Integer))) -> % +--R nthExponent : (%,Integer) -> Integer +--R nthFlag : (%,Integer) -> Union("nil","sqfr","irred","prime") +--R numberOfFactors : % -> NonNegativeInteger +--R rational : % -> Fraction(Integer) if R has INS +--R rational? : % -> Boolean if R has INS +--R rationalIfCan : % -> Union(Fraction(Integer),"failed") if R has INS +--R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) +--R retract : % -> Integer if R has RETRACT(INT) +--R retractIfCan : % -> Union(R,"failed") +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R squareFree : % -> Factored(%) if R has UFD +--R squareFreePart : % -> % if R has UFD +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R +--E 39 + )spool )lisp (bye) \end{chunk} @@ -45890,7 +47951,8 @@ Factored(R: IntegralDomain): Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 12 + +--S 1 of 13 ifile:File List Integer:=open("jazz1","output") --R --R @@ -45898,7 +47960,7 @@ ifile:File List Integer:=open("jazz1","output") --R Type: File(List(Integer)) --E 1 ---S 2 of 12 +--S 2 of 13 write!(ifile, [-1,2,3]) --R --R @@ -45906,7 +47968,7 @@ write!(ifile, [-1,2,3]) --R Type: List(Integer) --E 2 ---S 3 of 12 +--S 3 of 13 write!(ifile, [10,-10,0,111]) --R --R @@ -45914,7 +47976,7 @@ write!(ifile, [10,-10,0,111]) --R Type: List(Integer) --E 3 ---S 4 of 12 +--S 4 of 13 write!(ifile, [7]) --R --R @@ -45922,7 +47984,7 @@ write!(ifile, [7]) --R Type: List(Integer) --E 4 ---S 5 of 12 +--S 5 of 13 reopen!(ifile, "input") --R --R @@ -45930,7 +47992,7 @@ reopen!(ifile, "input") --R Type: File(List(Integer)) --E 5 ---S 6 of 12 +--S 6 of 13 read! ifile --R --R @@ -45938,7 +48000,7 @@ read! ifile --R Type: List(Integer) --E 6 ---S 7 of 12 +--S 7 of 13 read! ifile --R --R @@ -45946,7 +48008,7 @@ read! ifile --R Type: List(Integer) --E 7 ---S 8 of 12 +--S 8 of 13 readIfCan! ifile --R --R @@ -45954,7 +48016,7 @@ readIfCan! ifile --R Type: Union(List(Integer),...) --E 8 ---S 9 of 12 +--S 9 of 13 readIfCan! ifile --R --R @@ -45962,7 +48024,7 @@ readIfCan! ifile --R Type: Union("failed",...) --E 9 ---S 10 of 12 +--S 10 of 13 iomode ifile --R --R @@ -45970,7 +48032,7 @@ iomode ifile --R Type: String --E 10 ---S 11 of 12 +--S 11 of 13 name ifile --R --R @@ -45978,13 +48040,34 @@ name ifile --R Type: FileName --E 11 ---S 12 of 12 +--S 12 of 13 close! ifile --R --R --R (12) "jazz1" --R Type: File(List(Integer)) --E 12 + +--S 13 of 13 +)show File +--R +--R File(S: SetCategory) is a domain constructor +--R Abbreviation for File is FILE +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FILE +--R +--R------------------------------- Operations -------------------------------- +--R ?=? : (%,%) -> Boolean close! : % -> % +--R coerce : % -> OutputForm flush : % -> Void +--R hash : % -> SingleInteger iomode : % -> String +--R latex : % -> String name : % -> FileName +--R open : (FileName,String) -> % open : FileName -> % +--R read! : % -> S readIfCan! : % -> Union(S,"failed") +--R reopen! : (%,String) -> % write! : (%,S) -> S +--R ?~=? : (%,%) -> Boolean +--R +--E 13 + )system rm jazz1 )spool )lisp (bye) @@ -46201,13 +48284,14 @@ File(S:SetCategory): FileCategory(FileName, S) with )set message test on )set message auto off )clear all ---S 1 of 18 + +--S 1 of 19 fn: FileName --R --R Type: Void --E 1 ---S 2 of 18 +--S 2 of 19 fn := "fname.input" --R --R @@ -46215,7 +48299,7 @@ fn := "fname.input" --R Type: FileName --E 2 ---S 3 of 18 +--S 3 of 19 directory fn --R --R @@ -46223,7 +48307,7 @@ directory fn --R Type: String --E 3 ---S 4 of 18 +--S 4 of 19 name fn --R --R @@ -46231,7 +48315,7 @@ name fn --R Type: String --E 4 ---S 5 of 18 +--S 5 of 19 extension fn --R --R @@ -46239,7 +48323,7 @@ extension fn --R Type: String --E 5 ---S 6 of 18 +--S 6 of 19 fn := filename("/tmp", "fname", "input") --R --R @@ -46247,7 +48331,7 @@ fn := filename("/tmp", "fname", "input") --R Type: FileName --E 6 ---S 7 of 18 +--S 7 of 19 objdir := "/tmp" --R --R @@ -46255,7 +48339,7 @@ objdir := "/tmp" --R Type: String --E 7 ---S 8 of 18 +--S 8 of 19 fn := filename(objdir, "table", "spad") --R --R @@ -46263,7 +48347,7 @@ fn := filename(objdir, "table", "spad") --R Type: FileName --E 8 ---S 9 of 18 +--S 9 of 19 fn := filename("", "letter", "") --R --R @@ -46271,7 +48355,7 @@ fn := filename("", "letter", "") --R Type: FileName --E 9 ---S 10 of 18 +--S 10 of 19 exists? "/etc/passwd" --R --R @@ -46279,7 +48363,7 @@ exists? "/etc/passwd" --R Type: Boolean --E 10 ---S 11 of 18 +--S 11 of 19 readable? "/etc/passwd" --R --R @@ -46287,7 +48371,7 @@ readable? "/etc/passwd" --R Type: Boolean --E 11 ---S 12 of 18 +--S 12 of 19 readable? "/etc/security/passwd" --R --R @@ -46295,7 +48379,7 @@ readable? "/etc/security/passwd" --R Type: Boolean --E 12 ---S 13 of 18 +--S 13 of 19 readable? "/ect/passwd" --R --R @@ -46303,7 +48387,7 @@ readable? "/ect/passwd" --R Type: Boolean --E 13 ---S 14 of 18 +--S 14 of 19 writable? "/etc/passwd" --R --R @@ -46311,7 +48395,7 @@ writable? "/etc/passwd" --R Type: Boolean --E 14 ---S 15 of 18 +--S 15 of 19 writable? "/dev/null" --R --R @@ -46319,7 +48403,7 @@ writable? "/dev/null" --R Type: Boolean --E 15 ---S 16 of 18 +--S 16 of 19 writable? "/etc/DoesNotExist" --R --R @@ -46327,7 +48411,7 @@ writable? "/etc/DoesNotExist" --R Type: Boolean --E 16 ---S 17 of 18 +--S 17 of 19 writable? "/tmp/DoesNotExist" --R --R @@ -46335,13 +48419,34 @@ writable? "/tmp/DoesNotExist" --R Type: Boolean --E 17 ---S 18 of 18 +--S 18 of 19 fn := new(objdir, "xxx", "yy") --R --R --I (18) "/tmp/xxx1419.yy" --R Type: FileName --E 18 + +--S 19 of 19 +)show FileName +--R +--R FileName is a domain constructor +--R Abbreviation for FileName is FNAME +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FNAME +--R +--R------------------------------- Operations -------------------------------- +--R ?=? : (%,%) -> Boolean coerce : % -> String +--R coerce : String -> % coerce : % -> OutputForm +--R directory : % -> String exists? : % -> Boolean +--R extension : % -> String hash : % -> SingleInteger +--R latex : % -> String name : % -> String +--R new : (String,String,String) -> % readable? : % -> Boolean +--R writable? : % -> Boolean ?~=? : (%,%) -> Boolean +--R filename : (String,String,String) -> % +--R +--E 19 + )spool )lisp (bye) \end{chunk} @@ -46548,21 +48653,20 @@ FileName(): FileNameCategory == add --R Issue )edit bookvol10.3.pamphlet to see algebra source code for FDIV --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R divisor : (R,UP,UP,UP,F) -> % divisor : (F,F,Integer) -> % ---R divisor : (F,F) -> % divisor : R -> % ---R finiteBasis : % -> Vector(R) hash : % -> SingleInteger +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm divisor : (R,UP,UP,UP,F) -> % +--R divisor : (F,F,Integer) -> % divisor : (F,F) -> % +--R divisor : R -> % finiteBasis : % -> Vector(R) +--R generator : % -> Union(R,"failed") hash : % -> SingleInteger --R lSpaceBasis : % -> Vector(R) latex : % -> String --R principal? : % -> Boolean reduce : % -> % --R sample : () -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % --R decompose : % -> Record(id: FractionalIdeal(UP,Fraction(UP),UPUP,R),principalPart: R) --R divisor : FractionalIdeal(UP,Fraction(UP),UPUP,R) -> % ---R generator : % -> Union(R,"failed") --R ideal : % -> FractionalIdeal(UP,Fraction(UP),UPUP,R) --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -46773,38 +48877,36 @@ FiniteDivisor(F, UP, UPUP, R): Exports == Implementation where --R ?*? : (PrimeField(p),%) -> % ?*? : (%,PrimeField(p)) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,PrimeField(p)) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean ---R associates? : (%,%) -> Boolean basis : () -> Vector(%) ---R coerce : PrimeField(p) -> % coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R norm : % -> PrimeField(p) one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R retract : % -> PrimeField(p) sample : () -> % ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % trace : % -> PrimeField(p) ---R transcendent? : % -> Boolean unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R algebraic? : % -> Boolean associates? : (%,%) -> Boolean +--R basis : () -> Vector(%) coerce : PrimeField(p) -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % norm : % -> PrimeField(p) +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % retract : % -> PrimeField(p) +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R trace : % -> PrimeField(p) transcendent? : % -> Boolean +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R D : % -> % if PrimeField(p) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R Frobenius : % -> % if PrimeField(p) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if PrimeField(p) has CHARNZ or PrimeField(p) has FINITE @@ -47034,38 +49136,36 @@ FiniteField(p:PositiveInteger, n:PositiveInteger): _ --R ?*? : (PrimeField(p),%) -> % ?*? : (%,PrimeField(p)) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,PrimeField(p)) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean ---R associates? : (%,%) -> Boolean basis : () -> Vector(%) ---R coerce : PrimeField(p) -> % coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R norm : % -> PrimeField(p) one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R retract : % -> PrimeField(p) sample : () -> % ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % trace : % -> PrimeField(p) ---R transcendent? : % -> Boolean unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R algebraic? : % -> Boolean associates? : (%,%) -> Boolean +--R basis : () -> Vector(%) coerce : PrimeField(p) -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % norm : % -> PrimeField(p) +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % retract : % -> PrimeField(p) +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R trace : % -> PrimeField(p) transcendent? : % -> Boolean +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R D : % -> % if PrimeField(p) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R Frobenius : % -> % if PrimeField(p) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if PrimeField(p) has CHARNZ or PrimeField(p) has FINITE @@ -47309,38 +49409,38 @@ FiniteFieldCyclicGroup(p,extdeg):_ --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -47369,7 +49469,6 @@ FiniteFieldCyclicGroup(p,extdeg):_ --R generator : () -> % if GF has FINITE --R getZechTable : () -> PrimitiveArray(SingleInteger) --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -47390,7 +49489,6 @@ FiniteFieldCyclicGroup(p,extdeg):_ --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -47584,38 +49682,38 @@ FiniteFieldCyclicGroupExtension(GF,extdeg):_ --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -47644,7 +49742,6 @@ FiniteFieldCyclicGroupExtension(GF,extdeg):_ --R generator : () -> % if GF has FINITE --R getZechTable : () -> PrimitiveArray(SingleInteger) --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -47665,7 +49762,6 @@ FiniteFieldCyclicGroupExtension(GF,extdeg):_ --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -48126,38 +50222,38 @@ FiniteFieldCyclicGroupExtensionByPolynomial(GF,defpol):_ --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -48185,7 +50281,6 @@ FiniteFieldCyclicGroupExtensionByPolynomial(GF,defpol):_ --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) --R generator : () -> % if GF has FINITE --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -48206,7 +50301,6 @@ FiniteFieldCyclicGroupExtensionByPolynomial(GF,defpol):_ --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -48388,38 +50482,38 @@ FiniteFieldExtension(GF, n): Exports == Implementation where --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -48447,7 +50541,6 @@ FiniteFieldExtension(GF, n): Exports == Implementation where --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) --R generator : () -> % if GF has FINITE --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -48468,7 +50561,6 @@ FiniteFieldExtension(GF, n): Exports == Implementation where --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -48850,38 +50942,36 @@ FiniteFieldExtensionByPolynomial(GF:FiniteFieldCategory,_ --R ?*? : (PrimeField(p),%) -> % ?*? : (%,PrimeField(p)) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,PrimeField(p)) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % --R 0 : () -> % ?^? : (%,Integer) -> % ---R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean ---R associates? : (%,%) -> Boolean basis : () -> Vector(%) ---R coerce : PrimeField(p) -> % coerce : Fraction(Integer) -> % ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R norm : % -> PrimeField(p) one? : % -> Boolean ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R retract : % -> PrimeField(p) sample : () -> % ---R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) ---R squareFreePart : % -> % trace : % -> PrimeField(p) ---R transcendent? : % -> Boolean unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R algebraic? : % -> Boolean associates? : (%,%) -> Boolean +--R basis : () -> Vector(%) coerce : PrimeField(p) -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % norm : % -> PrimeField(p) +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % retract : % -> PrimeField(p) +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R trace : % -> PrimeField(p) transcendent? : % -> Boolean +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R D : % -> % if PrimeField(p) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if PrimeField(p) has FINITE --R Frobenius : % -> % if PrimeField(p) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if PrimeField(p) has CHARNZ or PrimeField(p) has FINITE @@ -49137,38 +51227,38 @@ FiniteFieldNormalBasis(p,extdeg):_ --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -49198,7 +51288,6 @@ FiniteFieldNormalBasis(p,extdeg):_ --R getMultiplicationMatrix : () -> Matrix(GF) --R getMultiplicationTable : () -> Vector(List(Record(value: GF,index: SingleInteger))) --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -49219,7 +51308,6 @@ FiniteFieldNormalBasis(p,extdeg):_ --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -49423,38 +51511,38 @@ FiniteFieldNormalBasisExtension(GF,extdeg):_ --R ?*? : (GF,%) -> % ?*? : (%,GF) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,GF) -> % ?/? : (%,%) -> % --R ?=? : (%,%) -> Boolean D : % -> % if GF has FINITE --R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % ---R algebraic? : % -> Boolean associates? : (%,%) -> Boolean ---R basis : () -> Vector(%) coerce : GF -> % ---R coerce : Fraction(Integer) -> % coerce : % -> % ---R coerce : Integer -> % coerce : % -> OutputForm ---R coordinates : % -> Vector(GF) degree : % -> PositiveInteger ---R dimension : () -> CardinalNumber factor : % -> Factored(%) ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % algebraic? : % -> Boolean +--R associates? : (%,%) -> Boolean basis : () -> Vector(%) +--R coerce : GF -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm coordinates : % -> Vector(GF) +--R degree : % -> PositiveInteger dimension : () -> CardinalNumber +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R inGroundField? : % -> Boolean init : () -> % if GF has FINITE --R inv : % -> % latex : % -> String --R lcm : List(%) -> % lcm : (%,%) -> % --R norm : % -> GF one? : % -> Boolean --R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R represents : Vector(GF) -> % retract : % -> GF ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R squareFree : % -> Factored(%) squareFreePart : % -> % ---R trace : % -> GF transcendent? : % -> Boolean ---R unit? : % -> Boolean unitCanonical : % -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R random : () -> % if GF has FINITE recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % represents : Vector(GF) -> % +--R retract : % -> GF sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % trace : % -> GF +--R transcendent? : % -> Boolean unit? : % -> Boolean +--R unitCanonical : % -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R D : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if GF has FINITE --R Frobenius : % -> % if GF has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if GF has CHARNZ or GF has FINITE @@ -49484,7 +51572,6 @@ FiniteFieldNormalBasisExtension(GF,extdeg):_ --R getMultiplicationMatrix : () -> Matrix(GF) --R getMultiplicationTable : () -> Vector(List(Record(value: GF,index: SingleInteger))) --R index : PositiveInteger -> % if GF has FINITE ---R init : () -> % if GF has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(GF)) -> % if GF has FINITE --R linearAssociatedLog : (%,%) -> Union(SparseUnivariatePolynomial(GF),"failed") if GF has FINITE @@ -49505,7 +51592,6 @@ FiniteFieldNormalBasisExtension(GF,extdeg):_ --R primitive? : % -> Boolean if GF has FINITE --R primitiveElement : () -> % if GF has FINITE --R principalIdeal : List(%) -> Record(coef: List(%),generator: %) ---R random : () -> % if GF has FINITE --R representationType : () -> Union("prime",polynomial,normal,cyclic) if GF has FINITE --R retractIfCan : % -> Union(GF,"failed") --R size : () -> NonNegativeInteger if GF has FINITE @@ -50007,7 +52093,8 @@ divisor of the order of the multiplicative group" )set message test on )set message auto off )clear all ---S 1 of 16 + +--S 1 of 17 flexibleArray [i for i in 1..6] --R --R @@ -50015,7 +52102,7 @@ flexibleArray [i for i in 1..6] --R Type: FlexibleArray(PositiveInteger) --E 1 ---S 2 of 16 +--S 2 of 17 f : FARRAY INT := new(6,0) --R --R @@ -50023,7 +52110,7 @@ f : FARRAY INT := new(6,0) --R Type: FlexibleArray(Integer) --E 2 ---S 3 of 16 +--S 3 of 17 for i in 1..6 repeat f.i := i; f --R --R @@ -50031,7 +52118,7 @@ for i in 1..6 repeat f.i := i; f --R Type: FlexibleArray(Integer) --E 3 ---S 4 of 16 +--S 4 of 17 physicalLength f --R --R @@ -50039,7 +52126,7 @@ physicalLength f --R Type: PositiveInteger --E 4 ---S 5 of 16 +--S 5 of 17 concat!(f,11) --R --R @@ -50047,7 +52134,7 @@ concat!(f,11) --R Type: FlexibleArray(Integer) --E 5 ---S 6 of 16 +--S 6 of 17 physicalLength f --R --R @@ -50055,7 +52142,7 @@ physicalLength f --R Type: PositiveInteger --E 6 ---S 7 of 16 +--S 7 of 17 physicalLength!(f,15) --R --R @@ -50063,7 +52150,7 @@ physicalLength!(f,15) --R Type: FlexibleArray(Integer) --E 7 ---S 8 of 16 +--S 8 of 17 concat!(f,f) --R --R @@ -50071,7 +52158,7 @@ concat!(f,f) --R Type: FlexibleArray(Integer) --E 8 ---S 9 of 16 +--S 9 of 17 insert!(22,f,1) --R --R @@ -50079,7 +52166,7 @@ insert!(22,f,1) --R Type: FlexibleArray(Integer) --E 9 ---S 10 of 16 +--S 10 of 17 g := f(10..) --R --R @@ -50087,7 +52174,7 @@ g := f(10..) --R Type: FlexibleArray(Integer) --E 10 ---S 11 of 16 +--S 11 of 17 insert!(g,f,1) --R --R @@ -50095,7 +52182,7 @@ insert!(g,f,1) --R Type: FlexibleArray(Integer) --E 11 ---S 12 of 16 +--S 12 of 17 merge!(sort! f, sort! g) --R --R @@ -50103,7 +52190,7 @@ merge!(sort! f, sort! g) --R Type: FlexibleArray(Integer) --E 12 ---S 13 of 16 +--S 13 of 17 removeDuplicates! f --R --R @@ -50111,7 +52198,7 @@ removeDuplicates! f --R Type: FlexibleArray(Integer) --E 13 ---S 14 of 16 +--S 14 of 17 select!(i +-> even? i,f) --R --R @@ -50119,7 +52206,7 @@ select!(i +-> even? i,f) --R Type: FlexibleArray(Integer) --E 14 ---S 15 of 16 +--S 15 of 17 physicalLength f --R --R @@ -50127,13 +52214,105 @@ physicalLength f --R Type: PositiveInteger --E 15 ---S 16 of 16 +--S 16 of 17 shrinkable(false)$FlexibleArray(Integer) --R --R --R (16) true --R Type: Boolean --E 16 + +--S 17 of 17 +)show FlexibleArray + FlexibleArray(S: Type) is a domain constructor + Abbreviation for FlexibleArray is FARRAY + This constructor is exposed in this frame. + Issue )edit bookvol10.3.pamphlet to see algebra source code for FARRAY + +------------------------------- Operations -------------------------------- + concat : List(%) -> % concat : (%,%) -> % + concat : (S,%) -> % concat : (%,S) -> % + concat! : (%,S) -> % concat! : (%,%) -> % + construct : List(S) -> % copy : % -> % + delete : (%,Integer) -> % delete! : (%,Integer) -> % + ?.? : (%,Integer) -> S elt : (%,Integer,S) -> S + empty : () -> % empty? : % -> Boolean + entries : % -> List(S) eq? : (%,%) -> Boolean + flexibleArray : List(S) -> % index? : (Integer,%) -> Boolean + indices : % -> List(Integer) insert : (%,%,Integer) -> % + insert : (S,%,Integer) -> % insert! : (S,%,Integer) -> % + insert! : (%,%,Integer) -> % latex : % -> String if S has SETCAT + map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % + max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET + new : (NonNegativeInteger,S) -> % physicalLength! : (%,Integer) -> % + qelt : (%,Integer) -> S remove! : ((S -> Boolean),%) -> % + reverse : % -> % sample : () -> % + select! : ((S -> Boolean),%) -> % shrinkable : Boolean -> Boolean + sort : % -> % if S has ORDSET sort : (((S,S) -> Boolean),%) -> % + #? : % -> NonNegativeInteger if $ has finiteAggregate + ? Boolean if S has ORDSET + ?<=? : (%,%) -> Boolean if S has ORDSET + ?=? : (%,%) -> Boolean if S has SETCAT + ?>? : (%,%) -> Boolean if S has ORDSET + ?>=? : (%,%) -> Boolean if S has ORDSET + any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate + coerce : % -> OutputForm if S has SETCAT + convert : % -> InputForm if S has KONVERT(INFORM) + copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable + count : (S,%) -> NonNegativeInteger if $ has finiteAggregate and S has SETCAT + count : ((S -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate + delete : (%,UniversalSegment(Integer)) -> % + delete! : (%,UniversalSegment(Integer)) -> % + ?.? : (%,UniversalSegment(Integer)) -> % + entry? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT + eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT + eval : (%,S,S) -> % if S has EVALAB(S) and S has SETCAT + eval : (%,Equation(S)) -> % if S has EVALAB(S) and S has SETCAT + eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT + every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate + fill! : (%,S) -> % if $ has shallowlyMutable + find : ((S -> Boolean),%) -> Union(S,"failed") + first : % -> S if Integer has ORDSET + hash : % -> SingleInteger if S has SETCAT + less? : (%,NonNegativeInteger) -> Boolean + map! : ((S -> S),%) -> % if $ has shallowlyMutable + maxIndex : % -> Integer if Integer has ORDSET + member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT + members : % -> List(S) if $ has finiteAggregate + merge : (%,%) -> % if S has ORDSET + merge : (((S,S) -> Boolean),%,%) -> % + merge! : (((S,S) -> Boolean),%,%) -> % + merge! : (%,%) -> % if S has ORDSET + minIndex : % -> Integer if Integer has ORDSET + more? : (%,NonNegativeInteger) -> Boolean + parts : % -> List(S) if $ has finiteAggregate + physicalLength : % -> NonNegativeInteger + position : (S,%,Integer) -> Integer if S has SETCAT + position : (S,%) -> Integer if S has SETCAT + position : ((S -> Boolean),%) -> Integer + qsetelt! : (%,Integer,S) -> S if $ has shallowlyMutable + reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate + reduce : (((S,S) -> S),%,S) -> S if $ has finiteAggregate + reduce : (((S,S) -> S),%,S,S) -> S if $ has finiteAggregate and S has SETCAT + remove : ((S -> Boolean),%) -> % if $ has finiteAggregate + remove : (S,%) -> % if $ has finiteAggregate and S has SETCAT + remove! : (S,%) -> % if S has SETCAT + removeDuplicates : % -> % if $ has finiteAggregate and S has SETCAT + removeDuplicates! : % -> % if S has SETCAT + reverse! : % -> % if $ has shallowlyMutable + select : ((S -> Boolean),%) -> % if $ has finiteAggregate + setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable + setelt : (%,Integer,S) -> S if $ has shallowlyMutable + size? : (%,NonNegativeInteger) -> Boolean + sort! : % -> % if $ has shallowlyMutable and S has ORDSET + sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable + sorted? : % -> Boolean if S has ORDSET + sorted? : (((S,S) -> Boolean),%) -> Boolean + swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable + ?~=? : (%,%) -> Boolean if S has SETCAT + +--E 17 + )spool )lisp (bye) \end{chunk} @@ -50417,7 +52596,8 @@ parts of Axiom. )set message test on )set message auto off )clear all ---S 1 of 64 + +--S 1 of 65 1.234 --R --R @@ -50425,7 +52605,7 @@ parts of Axiom. --R Type: Float --E 1 ---S 2 of 64 +--S 2 of 65 1.234E2 --R --R @@ -50433,7 +52613,7 @@ parts of Axiom. --R Type: Float --E 2 ---S 3 of 64 +--S 3 of 65 sqrt(1.2 + 2.3 / 3.4 ** 4.5) --R --R @@ -50441,7 +52621,7 @@ sqrt(1.2 + 2.3 / 3.4 ** 4.5) --R Type: Float --E 3 ---S 4 of 64 +--S 4 of 65 i := 3 :: Float --R --R @@ -50449,7 +52629,7 @@ i := 3 :: Float --R Type: Float --E 4 ---S 5 of 64 +--S 5 of 65 i :: Integer --R --R @@ -50457,7 +52637,7 @@ i :: Integer --R Type: Integer --E 5 ---S 6 of 64 +--S 6 of 65 i :: Fraction Integer --R --R @@ -50465,7 +52645,7 @@ i :: Fraction Integer --R Type: Fraction(Integer) --E 6 ---S 7 of 64 +--S 7 of 65 r := 3/7 :: Float --R --R @@ -50473,7 +52653,7 @@ r := 3/7 :: Float --R Type: Float --E 7 ---S 8 of 64 +--S 8 of 65 r :: Fraction Integer --R --R @@ -50483,7 +52663,7 @@ r :: Fraction Integer --R Type: Fraction(Integer) --E 8 ---S 9 of 64 +--S 9 of 65 r :: Integer --R --R @@ -50493,7 +52673,7 @@ r :: Integer --R --E 9 ---S 10 of 64 +--S 10 of 65 truncate 3.6 --R --R @@ -50501,7 +52681,7 @@ truncate 3.6 --R Type: Float --E 10 ---S 11 of 64 +--S 11 of 65 round 3.6 --R --R @@ -50509,7 +52689,7 @@ round 3.6 --R Type: Float --E 11 ---S 12 of 64 +--S 12 of 65 truncate(-3.6) --R --R @@ -50517,7 +52697,7 @@ truncate(-3.6) --R Type: Float --E 12 ---S 13 of 64 +--S 13 of 65 round(-3.6) --R --R @@ -50525,7 +52705,7 @@ round(-3.6) --R Type: Float --E 13 ---S 14 of 64 +--S 14 of 65 fractionPart 3.6 --R --R @@ -50533,7 +52713,7 @@ fractionPart 3.6 --R Type: Float --E 14 ---S 15 of 64 +--S 15 of 65 digits 40 --R --R @@ -50541,7 +52721,7 @@ digits 40 --R Type: PositiveInteger --E 15 ---S 16 of 64 +--S 16 of 65 sqrt 0.2 --R --R @@ -50549,7 +52729,7 @@ sqrt 0.2 --R Type: Float --E 16 ---S 17 of 64 +--S 17 of 65 pi()$Float --R --R @@ -50557,7 +52737,7 @@ pi()$Float --R Type: Float --E 17 ---S 18 of 64 +--S 18 of 65 digits 500 --R --R @@ -50565,7 +52745,7 @@ digits 500 --R Type: PositiveInteger --E 18 ---S 19 of 64 +--S 19 of 65 pi()$Float --R --R @@ -50581,7 +52761,7 @@ pi()$Float --R Type: Float --E 19 ---S 20 of 64 +--S 20 of 65 digits 20 --R --R @@ -50589,7 +52769,7 @@ digits 20 --R Type: PositiveInteger --E 20 ---S 21 of 64 +--S 21 of 65 outputSpacing 0; x := sqrt 0.2 --R --R @@ -50597,7 +52777,7 @@ outputSpacing 0; x := sqrt 0.2 --R Type: Float --E 21 ---S 22 of 64 +--S 22 of 65 outputSpacing 5; x --R --R @@ -50605,7 +52785,7 @@ outputSpacing 5; x --R Type: Float --E 22 ---S 23 of 64 +--S 23 of 65 y := x/10**10 --R --R @@ -50613,7 +52793,7 @@ y := x/10**10 --R Type: Float --E 23 ---S 24 of 64 +--S 24 of 65 outputFloating(); x --R --R @@ -50621,7 +52801,7 @@ outputFloating(); x --R Type: Float --E 24 ---S 25 of 64 +--S 25 of 65 outputFixed(); y --R --R @@ -50629,7 +52809,7 @@ outputFixed(); y --R Type: Float --E 25 ---S 26 of 64 +--S 26 of 65 outputFloating 2; y --R --R @@ -50637,7 +52817,7 @@ outputFloating 2; y --R Type: Float --E 26 ---S 27 of 64 +--S 27 of 65 outputFixed 2; x --R --R @@ -50645,13 +52825,13 @@ outputFixed 2; x --R Type: Float --E 27 ---S 28 of 64 +--S 28 of 65 outputGeneral() --R --R Type: Void --E 28 ---S 29 of 64 +--S 29 of 65 a: Matrix Fraction Integer := matrix [ [1/(i+j+1) for j in 0..9] for i in 0..9] --R --R @@ -50697,7 +52877,7 @@ a: Matrix Fraction Integer := matrix [ [1/(i+j+1) for j in 0..9] for i in 0..9] --R Type: Matrix(Fraction(Integer)) --E 29 ---S 30 of 64 +--S 30 of 65 d:= determinant a --R --R @@ -50707,7 +52887,7 @@ d:= determinant a --R Type: Fraction(Integer) --E 30 ---S 31 of 64 +--S 31 of 65 d :: Float --R --R @@ -50715,7 +52895,7 @@ d :: Float --R Type: Float --E 31 ---S 32 of 64 +--S 32 of 65 b: Matrix DoubleFloat := matrix [ [1/(i+j+1$DoubleFloat) for j in 0..9] for i in 0..9] --R --R @@ -50775,7 +52955,7 @@ b: Matrix DoubleFloat := matrix [ [1/(i+j+1$DoubleFloat) for j in 0..9] for i in --R Type: Matrix(DoubleFloat) --E 32 ---S 33 of 64 +--S 33 of 65 determinant b --R --R @@ -50783,7 +52963,7 @@ determinant b --R Type: DoubleFloat --E 33 ---S 34 of 64 +--S 34 of 65 digits 40 --R --R @@ -50791,7 +52971,7 @@ digits 40 --R Type: PositiveInteger --E 34 ---S 35 of 64 +--S 35 of 65 c: Matrix Float := matrix [ [1/(i+j+1$Float) for j in 0..9] for i in 0..9] --R --R @@ -50885,7 +53065,7 @@ c: Matrix Float := matrix [ [1/(i+j+1$Float) for j in 0..9] for i in 0..9] --R Type: Matrix(Float) --E 35 ---S 36 of 64 +--S 36 of 65 determinant c --R --R @@ -50893,7 +53073,7 @@ determinant c --R Type: Float --E 36 ---S 37 of 64 +--S 37 of 65 digits 20 --R --R @@ -50903,13 +53083,13 @@ digits 20 )clear all ---S 38 of 64 +--S 38 of 65 outputFixed() --R --R Type: Void --E 38 ---S 39 of 64 +--S 39 of 65 a:=3.0 --R --R @@ -50917,7 +53097,7 @@ a:=3.0 --R Type: Float --E 39 ---S 40 of 64 +--S 40 of 65 b:=3.1 --R --R @@ -50925,7 +53105,7 @@ b:=3.1 --R Type: Float --E 40 ---S 41 of 64 +--S 41 of 65 c:=numeric pi() --R --R @@ -50933,7 +53113,7 @@ c:=numeric pi() --R Type: Float --E 41 ---S 42 of 64 +--S 42 of 65 d:=0.0 --R --R @@ -50941,13 +53121,13 @@ d:=0.0 --R Type: Float --E 42 ---S 43 of 64 +--S 43 of 65 outputFixed 2 --R --R Type: Void --E 43 ---S 44 of 64 +--S 44 of 65 a --R --R @@ -50955,7 +53135,7 @@ a --R Type: Float --E 44 ---S 45 of 64 +--S 45 of 65 b --R --R @@ -50963,7 +53143,7 @@ b --R Type: Float --E 45 ---S 46 of 64 +--S 46 of 65 c --R --R @@ -50971,7 +53151,7 @@ c --R Type: Float --E 46 ---S 47 of 64 +--S 47 of 65 d --R --R @@ -50979,13 +53159,13 @@ d --R Type: Float --E 47 ---S 48 of 64 +--S 48 of 65 outputFixed 0 --R --R Type: Void --E 48 ---S 49 of 64 +--S 49 of 65 a --R --R @@ -50993,7 +53173,7 @@ a --R Type: Float --E 49 ---S 50 of 64 +--S 50 of 65 b --R --R @@ -51001,7 +53181,7 @@ b --R Type: Float --E 50 ---S 51 of 64 +--S 51 of 65 c --R --R @@ -51009,7 +53189,7 @@ c --R Type: Float --E 51 ---S 52 of 64 +--S 52 of 65 31.1 --R --R @@ -51017,7 +53197,7 @@ c --R Type: Float --E 52 ---S 53 of 64 +--S 53 of 65 310.1 --R --R @@ -51025,7 +53205,7 @@ c --R Type: Float --E 53 ---S 54 of 64 +--S 54 of 65 d --R --R @@ -51033,68 +53213,178 @@ d --R Type: Float --E 54 ---S 55 of 64 +--S 55 of 65 outputFixed(0) --R Type: Void --E 55 ---S 56 of 64 +--S 56 of 65 1.1 --R --R (19) 1. --R Type: Float --E 56 ---S 57 of 64 +--S 57 of 65 3111.1 --R --R (20) 3111. --R Type: Float --E 57 ---S 58 of 64 +--S 58 of 65 1234567890.1 --R --R (21) 12345 67890. --R Type: Float --E 58 ---S 59 of 64 +--S 59 of 65 outputFixed(12) --R Type: Void --E 59 ---S 60 of 64 +--S 60 of 65 1234567890.1 --R --R (23) 12345 67890.09999 99999 99 --R Type: Float --E 60 ---S 61 of 64 +--S 61 of 65 outputFixed(15) --R Type: Void --E 61 ---S 62 of 64 +--S 62 of 65 1234567890.1 --R --R (25) 12345 67890.09999 99999 98545 --R Type: Float --E 62 ---S 63 of 64 +--S 63 of 65 outputFixed(2) --R Type: Void --E 63 ---S 64 of 64 +--S 64 of 65 1234567890.1 --R --R (27) 12345 67890.10 --R Type: Float --E 64 +--S 65 of 65 +)show Float +--R +--R Float is a domain constructor +--R Abbreviation for Float is FLOAT +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FLOAT +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,%) -> % ?**? : (%,Fraction(Integer)) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (%,Integer) -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean D : % -> % +--R D : (%,NonNegativeInteger) -> % OMwrite : (%,Boolean) -> String +--R OMwrite : % -> String 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % acos : % -> % +--R acosh : % -> % acot : % -> % +--R acoth : % -> % acsc : % -> % +--R acsch : % -> % asec : % -> % +--R asech : % -> % asin : % -> % +--R asinh : % -> % associates? : (%,%) -> Boolean +--R atan : (%,%) -> % atan : % -> % +--R atanh : % -> % base : () -> PositiveInteger +--R bits : () -> PositiveInteger ceiling : % -> % +--R coerce : % -> DoubleFloat coerce : Fraction(Integer) -> % +--R coerce : Integer -> % coerce : Fraction(Integer) -> % +--R coerce : % -> % coerce : Integer -> % +--R coerce : % -> OutputForm convert : DoubleFloat -> % +--R convert : % -> InputForm convert : % -> String +--R convert : % -> Pattern(Float) convert : % -> DoubleFloat +--R convert : % -> Float cos : % -> % +--R cosh : % -> % cot : % -> % +--R coth : % -> % csc : % -> % +--R csch : % -> % differentiate : % -> % +--R digits : () -> PositiveInteger exp : % -> % +--R exp1 : () -> % exponent : % -> Integer +--R factor : % -> Factored(%) float : (Integer,Integer) -> % +--R floor : % -> % fractionPart : % -> % +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % log : % -> % +--R log10 : % -> % log10 : () -> % +--R log2 : % -> % log2 : () -> % +--R mantissa : % -> Integer max : (%,%) -> % +--R min : (%,%) -> % negative? : % -> Boolean +--R norm : % -> % normalize : % -> % +--R nthRoot : (%,Integer) -> % one? : % -> Boolean +--R order : % -> Integer outputFixed : () -> Void +--R outputFloating : () -> Void outputGeneral : () -> Void +--R pi : () -> % positive? : % -> Boolean +--R precision : () -> PositiveInteger prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R relerror : (%,%) -> Integer ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Integer +--R round : % -> % sample : () -> % +--R sec : % -> % sech : % -> % +--R shift : (%,Integer) -> % sign : % -> Integer +--R sin : % -> % sinh : % -> % +--R sizeLess? : (%,%) -> Boolean sqrt : % -> % +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R tan : % -> % tanh : % -> % +--R truncate : % -> % unit? : % -> Boolean +--R unitCanonical : % -> % wholePart : % -> Integer +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R OMwrite : (OpenMathDevice,%,Boolean) -> Void +--R OMwrite : (OpenMathDevice,%) -> Void +--R bits : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R characteristic : () -> NonNegativeInteger +--R decreasePrecision : Integer -> PositiveInteger if $ has arbitraryPrecision +--R differentiate : (%,NonNegativeInteger) -> % +--R digits : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R euclideanSize : % -> NonNegativeInteger +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R float : (Integer,Integer,PositiveInteger) -> % +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R increasePrecision : Integer -> PositiveInteger if $ has arbitraryPrecision +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R max : () -> % if not(has($,arbitraryExponent)) and not(has($,arbitraryPrecision)) +--R min : () -> % if not(has($,arbitraryExponent)) and not(has($,arbitraryPrecision)) +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R outputFixed : NonNegativeInteger -> Void +--R outputFloating : NonNegativeInteger -> Void +--R outputGeneral : NonNegativeInteger -> Void +--R outputSpacing : NonNegativeInteger -> Void +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) +--R precision : PositiveInteger -> PositiveInteger if $ has arbitraryPrecision +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R rationalApproximation : (%,NonNegativeInteger,NonNegativeInteger) -> Fraction(Integer) +--R rationalApproximation : (%,NonNegativeInteger) -> Fraction(Integer) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") +--R retractIfCan : % -> Union(Integer,"failed") +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R +--E 65 + )spool )lisp (bye) \end{chunk} @@ -52552,14 +54842,15 @@ Float(): --R ?=? : (%,%) -> Boolean assign : (Symbol,String) -> % --R block : List(%) -> % call : String -> % --R coerce : % -> OutputForm comment : List(String) -> % ---R comment : String -> % cond : (Switch,%,%) -> % ---R cond : (Switch,%) -> % continue : SingleInteger -> % ---R getCode : % -> SExpression goto : SingleInteger -> % ---R hash : % -> SingleInteger latex : % -> String ---R printCode : % -> Void returns : Expression(Float) -> % ---R returns : () -> % save : () -> % ---R stop : () -> % whileLoop : (Switch,%) -> % ---R ?~=? : (%,%) -> Boolean +--R comment : String -> % common : (Symbol,List(Symbol)) -> % +--R cond : (Switch,%,%) -> % cond : (Switch,%) -> % +--R continue : SingleInteger -> % getCode : % -> SExpression +--R goto : SingleInteger -> % hash : % -> SingleInteger +--R latex : % -> String printCode : % -> Void +--R repeatUntilLoop : (Switch,%) -> % returns : Expression(Integer) -> % +--R returns : Expression(Float) -> % returns : () -> % +--R save : () -> % stop : () -> % +--R whileLoop : (Switch,%) -> % ?~=? : (%,%) -> Boolean --R assign : (Symbol,List(Polynomial(Integer)),Expression(Complex(Float))) -> % --R assign : (Symbol,List(Polynomial(Integer)),Expression(Float)) -> % --R assign : (Symbol,List(Polynomial(Integer)),Expression(Integer)) -> % @@ -52591,14 +54882,11 @@ Float(): --R assign : (Symbol,Expression(MachineFloat)) -> % --R assign : (Symbol,Expression(MachineInteger)) -> % --R code : % -> Union(nullBranch: null,assignmentBranch: Record(var: Symbol,arrayIndex: List(Polynomial(Integer)),rand: Record(ints2Floats?: Boolean,expr: OutputForm)),arrayAssignmentBranch: Record(var: Symbol,rand: OutputForm,ints2Floats?: Boolean),conditionalBranch: Record(switch: Switch,thenClause: %,elseClause: %),returnBranch: Record(empty?: Boolean,value: Record(ints2Floats?: Boolean,expr: OutputForm)),blockBranch: List(%),commentBranch: List(String),callBranch: String,forBranch: Record(range: SegmentBinding(Polynomial(Integer)),span: Polynomial(Integer),body: %),labelBranch: SingleInteger,loopBranch: Record(switch: Switch,body: %),commonBranch: Record(name: Symbol,contents: List(Symbol)),printBranch: List(OutputForm)) ---R common : (Symbol,List(Symbol)) -> % --R forLoop : (SegmentBinding(Polynomial(Integer)),Polynomial(Integer),%) -> % --R forLoop : (SegmentBinding(Polynomial(Integer)),%) -> % --R operation : % -> Union(Null: null,Assignment: assignment,Conditional: conditional,Return: return,Block: block,Comment: comment,Call: call,For: for,While: while,Repeat: repeat,Goto: goto,Continue: continue,ArrayAssignment: arrayAssignment,Save: save,Stop: stop,Common: common,Print: print) --R printStatement : List(OutputForm) -> % ---R repeatUntilLoop : (Switch,%) -> % --R returns : Expression(Complex(Float)) -> % ---R returns : Expression(Integer) -> % --R returns : Expression(MachineComplex) -> % --R returns : Expression(MachineInteger) -> % --R returns : Expression(MachineFloat) -> % @@ -53269,32 +55557,37 @@ FortranCode(): public == private where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for FEXPR --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (%,%) -> % ?*? : (%,R) -> % ---R ?*? : (R,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (PositiveInteger,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (%,%) -> % +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?**? : (%,NonNegativeInteger) -> % --R ?+? : (%,%) -> % -? : % -> % --R ?-? : (%,%) -> % ? Boolean --R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean --R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean --R D : (%,Symbol) -> % D : (%,List(Symbol)) -> % --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % abs : % -> % ---R acos : % -> % asin : % -> % ---R atan : % -> % belong? : BasicOperator -> Boolean ---R box : List(%) -> % box : % -> % ---R coerce : % -> Expression(R) coerce : Integer -> % ---R coerce : R -> % coerce : Kernel(%) -> % ---R coerce : % -> OutputForm cos : % -> % ---R cosh : % -> % differentiate : (%,Symbol) -> % ---R distribute : (%,%) -> % distribute : % -> % ---R elt : (BasicOperator,%,%) -> % elt : (BasicOperator,%) -> % ---R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R ?^? : (%,PositiveInteger) -> % ?^? : (%,NonNegativeInteger) -> % +--R abs : % -> % acos : % -> % +--R asin : % -> % atan : % -> % +--R belong? : BasicOperator -> Boolean box : List(%) -> % +--R box : % -> % coerce : % -> Expression(R) +--R coerce : Integer -> % coerce : R -> % +--R coerce : Kernel(%) -> % coerce : % -> OutputForm +--R cos : % -> % cosh : % -> % +--R differentiate : (%,Symbol) -> % distribute : (%,%) -> % +--R distribute : % -> % elt : (BasicOperator,List(%)) -> % +--R elt : (BasicOperator,%,%,%) -> % elt : (BasicOperator,%,%) -> % +--R elt : (BasicOperator,%) -> % eval : (%,Symbol,(% -> %)) -> % +--R eval : (%,List(%),List(%)) -> % eval : (%,%,%) -> % +--R eval : (%,Equation(%)) -> % eval : (%,List(Equation(%))) -> % --R eval : (%,Kernel(%),%) -> % exp : % -> % --R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean --R hash : % -> SingleInteger height : % -> NonNegativeInteger ---R is? : (%,Symbol) -> Boolean kernel : (BasicOperator,%) -> % ---R kernels : % -> List(Kernel(%)) latex : % -> String ---R log : % -> % log10 : % -> % +--R is? : (%,Symbol) -> Boolean is? : (%,BasicOperator) -> Boolean +--R kernel : (BasicOperator,%) -> % kernels : % -> List(Kernel(%)) +--R latex : % -> String log : % -> % +--R log10 : % -> % map : ((% -> %),Kernel(%)) -> % --R max : (%,%) -> % min : (%,%) -> % --R one? : % -> Boolean paren : List(%) -> % --R paren : % -> % pi : () -> % @@ -53304,38 +55597,28 @@ FortranCode(): public == private where --R sin : % -> % sinh : % -> % --R sqrt : % -> % subst : (%,Equation(%)) -> % --R tan : % -> % tanh : % -> % ---R tower : % -> List(Kernel(%)) useNagFunctions : () -> Boolean ---R variables : % -> List(Symbol) zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R tower : % -> List(Kernel(%)) useNagFunctions : Boolean -> Boolean +--R useNagFunctions : () -> Boolean variables : % -> List(Symbol) +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R D : (%,Symbol,NonNegativeInteger) -> % --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R characteristic : () -> NonNegativeInteger --R definingPolynomial : % -> % if $ has RING --R differentiate : (%,List(Symbol)) -> % --R differentiate : (%,Symbol,NonNegativeInteger) -> % --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % ---R elt : (BasicOperator,List(%)) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,%,%,%) -> % --R eval : (%,BasicOperator,(% -> %)) -> % --R eval : (%,BasicOperator,(List(%) -> %)) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R eval : (%,List(Kernel(%)),List(%)) -> % --R even? : % -> Boolean if $ has RETRACT(INT) ---R is? : (%,BasicOperator) -> Boolean --R kernel : (BasicOperator,List(%)) -> % --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R odd? : % -> Boolean if $ has RETRACT(INT) --R operator : BasicOperator -> BasicOperator @@ -53359,7 +55642,6 @@ FortranCode(): public == private where --R subst : (%,List(Kernel(%)),List(%)) -> % --R subst : (%,List(Equation(%))) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") ---R useNagFunctions : Boolean -> Boolean --R --E 1 @@ -54112,6 +56394,7 @@ FortranProgram(name,returnType,arguments,symbols): Exports == Implement where --S 1 of 1 )show FortranScalarType +--R --R FortranScalarType is a domain constructor --R Abbreviation for FortranScalarType is FST --R This constructor is exposed in this frame. @@ -54330,6 +56613,7 @@ FortranScalarType() : exports == implementation where --S 1 of 1 )show FortranTemplate +--R --R FortranTemplate is a domain constructor --R Abbreviation for FortranTemplate is FTEM --R This constructor is exposed in this frame. @@ -54339,12 +56623,12 @@ FortranScalarType() : exports == implementation where --R ?=? : (%,%) -> Boolean close! : % -> % --R coerce : % -> OutputForm flush : % -> Void --R fortranCarriageReturn : () -> Void fortranLiteral : String -> Void ---R hash : % -> SingleInteger iomode : % -> String ---R latex : % -> String name : % -> FileName ---R open : (FileName,String) -> % open : FileName -> % ---R read! : % -> String reopen! : (%,String) -> % ---R write! : (%,String) -> String ?~=? : (%,%) -> Boolean ---R fortranLiteralLine : String -> Void +--R fortranLiteralLine : String -> Void hash : % -> SingleInteger +--R iomode : % -> String latex : % -> String +--R name : % -> FileName open : (FileName,String) -> % +--R open : FileName -> % read! : % -> String +--R reopen! : (%,String) -> % write! : (%,String) -> String +--R ?~=? : (%,%) -> Boolean --R processTemplate : FileName -> FileName --R processTemplate : (FileName,FileName) -> FileName --R @@ -54808,20 +57092,19 @@ FourierComponent(E:OrderedSet): --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : FourierComponent(E) -> % ---R coerce : R -> % coerce : Integer -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String makeCos : (E,R) -> % ---R makeSin : (E,R) -> % one? : % -> Boolean ---R recip : % -> Union(%,"failed") sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : FourierComponent(E) -> % coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R makeCos : (E,R) -> % makeSin : (E,R) -> % +--R one? : % -> Boolean recip : % -> Union(%,"failed") +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -54957,7 +57240,8 @@ FourierSeries(R:Join(CommutativeRing,Algebra(Fraction Integer)), )set message test on )set message auto off )clear all ---S 1 of 12 + +--S 1 of 13 a := 11/12 --R --R @@ -54967,7 +57251,7 @@ a := 11/12 --R Type: Fraction(Integer) --E 1 ---S 2 of 12 +--S 2 of 13 b := 23/24 --R --R @@ -54977,7 +57261,7 @@ b := 23/24 --R Type: Fraction(Integer) --E 2 ---S 3 of 12 +--S 3 of 13 3 - a*b**2 + a + b/a --R --R @@ -54987,7 +57271,7 @@ b := 23/24 --R Type: Fraction(Integer) --E 3 ---S 4 of 12 +--S 4 of 13 numer(a) --R --R @@ -54995,7 +57279,7 @@ numer(a) --R Type: PositiveInteger --E 4 ---S 5 of 12 +--S 5 of 13 denom(b) --R --R @@ -55003,7 +57287,7 @@ denom(b) --R Type: PositiveInteger --E 5 ---S 6 of 12 +--S 6 of 13 r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) --R --R @@ -55015,7 +57299,7 @@ r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) --R Type: Fraction(Polynomial(Integer)) --E 6 ---S 7 of 12 +--S 7 of 13 factor(r) --R --R @@ -55027,7 +57311,7 @@ factor(r) --R Type: Factored(Fraction(Polynomial(Integer))) --E 7 ---S 8 of 12 +--S 8 of 13 map(factor,r) --R --R @@ -55039,7 +57323,7 @@ map(factor,r) --R Type: Fraction(Factored(Polynomial(Integer))) --E 8 ---S 9 of 12 +--S 9 of 13 continuedFraction(7/12) --R --R @@ -55049,7 +57333,7 @@ continuedFraction(7/12) --R Type: ContinuedFraction(Integer) --E 9 ---S 10 of 12 +--S 10 of 13 partialFraction(7,12) --R --R @@ -55060,7 +57344,7 @@ partialFraction(7,12) --R Type: PartialFraction(Integer) --E 10 ---S 11 of 12 +--S 11 of 13 g := 2/3 + 4/5*%i --R --R @@ -55070,7 +57354,7 @@ g := 2/3 + 4/5*%i --R Type: Complex(Fraction(Integer)) --E 11 ---S 12 of 12 +--S 12 of 13 g :: FRAC COMPLEX INT --R --R @@ -55080,6 +57364,122 @@ g :: FRAC COMPLEX INT --R Type: Fraction(Complex(Integer)) --E 12 +--S 13 of 13 +)show Fraction +--R +--R Fraction(S: IntegralDomain) is a domain constructor +--R Abbreviation for Fraction is FRAC +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FRAC +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,S) -> % ?*? : (S,%) -> % +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (S,S) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean D : (%,(S -> S)) -> % +--R D : % -> % if S has DIFRING 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R abs : % -> % if S has OINTDOM associates? : (%,%) -> Boolean +--R ceiling : % -> S if S has INS coerce : S -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R convert : % -> Float if S has REAL denom : % -> S +--R denominator : % -> % differentiate : (%,(S -> S)) -> % +--R factor : % -> Factored(%) floor : % -> S if S has INS +--R gcd : List(%) -> % gcd : (%,%) -> % +--R hash : % -> SingleInteger init : () -> % if S has STEP +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R map : ((S -> S),%) -> % max : (%,%) -> % if S has ORDSET +--R min : (%,%) -> % if S has ORDSET numer : % -> S +--R numerator : % -> % one? : % -> Boolean +--R prime? : % -> Boolean ?quo? : (%,%) -> % +--R random : () -> % if S has INS recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % retract : % -> S +--R sample : () -> % sizeLess? : (%,%) -> Boolean +--R squareFree : % -> Factored(%) squareFreePart : % -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R wholePart : % -> S if S has EUCDOM zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R ? Boolean if S has ORDSET +--R ?<=? : (%,%) -> Boolean if S has ORDSET +--R ?>? : (%,%) -> Boolean if S has ORDSET +--R ?>=? : (%,%) -> Boolean if S has ORDSET +--R D : (%,(S -> S),NonNegativeInteger) -> % +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if S has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if S has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if S has PDRING(SYMBOL) +--R D : (%,Symbol) -> % if S has PDRING(SYMBOL) +--R D : (%,NonNegativeInteger) -> % if S has DIFRING +--R OMwrite : (OpenMathDevice,%,Boolean) -> Void if S has INS and S has OM +--R OMwrite : (OpenMathDevice,%) -> Void if S has INS and S has OM +--R OMwrite : (%,Boolean) -> String if S has INS and S has OM +--R OMwrite : % -> String if S has INS and S has OM +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and S has PFECAT or S has CHARNZ +--R coerce : Symbol -> % if S has RETRACT(SYMBOL) +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and S has PFECAT +--R convert : % -> DoubleFloat if S has REAL +--R convert : % -> InputForm if S has KONVERT(INFORM) +--R convert : % -> Pattern(Float) if S has KONVERT(PATTERN(FLOAT)) +--R convert : % -> Pattern(Integer) if S has KONVERT(PATTERN(INT)) +--R differentiate : (%,(S -> S),NonNegativeInteger) -> % +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,Symbol) -> % if S has PDRING(SYMBOL) +--R differentiate : (%,NonNegativeInteger) -> % if S has DIFRING +--R differentiate : % -> % if S has DIFRING +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R ?.? : (%,S) -> % if S has ELTAB(S,S) +--R euclideanSize : % -> NonNegativeInteger +--R eval : (%,Symbol,S) -> % if S has IEVALAB(SYMBOL,S) +--R eval : (%,List(Symbol),List(S)) -> % if S has IEVALAB(SYMBOL,S) +--R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) +--R eval : (%,Equation(S)) -> % if S has EVALAB(S) +--R eval : (%,S,S) -> % if S has EVALAB(S) +--R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if S has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if S has PFECAT +--R fractionPart : % -> % if S has EUCDOM +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R negative? : % -> Boolean if S has OINTDOM +--R nextItem : % -> Union(%,"failed") if S has STEP +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if S has PATMAB(FLOAT) +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if S has PATMAB(INT) +--R positive? : % -> Boolean if S has OINTDOM +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R reducedSystem : Matrix(%) -> Matrix(S) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(S),vec: Vector(S)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if S has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if S has LINEXP(INT) +--R retract : % -> Integer if S has RETRACT(INT) +--R retract : % -> Fraction(Integer) if S has RETRACT(INT) +--R retract : % -> Symbol if S has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(Integer,"failed") if S has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if S has RETRACT(INT) +--R retractIfCan : % -> Union(Symbol,"failed") if S has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(S,"failed") +--R sign : % -> Integer if S has OINTDOM +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if S has PFECAT +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if S has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R +--E 13 + )spool )lisp (bye) \end{chunk} @@ -55645,9 +58045,10 @@ Fraction(S: IntegralDomain): QuotientFieldCategory S with --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?**? : (%,Integer) -> % ---R ?**? : (%,PositiveInteger) -> % ?/? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R basis : % -> Vector(A) coerce : % -> OutputForm --R commutator : (%,%) -> % conjugate : (%,%) -> % --R denom : % -> R hash : % -> SingleInteger @@ -55656,8 +58057,6 @@ Fraction(S: IntegralDomain): QuotientFieldCategory S with --R norm : % -> F numer : % -> Vector(A) --R one? : % -> Boolean recip : % -> Union(%,"failed") --R sample : () -> % ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R randomLC : (NonNegativeInteger,Vector(A)) -> A --R --E 1 @@ -55907,16 +58306,15 @@ FractionalIdeal(R, F, UP, A): Exports == Implementation where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for FRMOD --R --R------------------------------- Operations -------------------------------- ---R ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?=? : (%,%) -> Boolean 1 : () -> % +--R ?*? : (%,%) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?=? : (%,%) -> Boolean +--R 1 : () -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % basis : % -> Vector(A) --R coerce : % -> OutputForm hash : % -> SingleInteger --R latex : % -> String module : Vector(A) -> % --R norm : % -> F one? : % -> Boolean --R recip : % -> Union(%,"failed") sample : () -> % --R ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R module : FractionalIdeal(R,F,UP,A) -> % if A has RETRACT(F) --R --E 1 @@ -56095,26 +58493,24 @@ FramedModule(R, F, UP, A, ibasis): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,S) -> % ?*? : (%,Integer) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (S,%) -> % ?+? : (%,%) -> % ---R ?-? : (%,%) -> % -? : % -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (S,%) -> Integer coerce : S -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String mapGen : ((S -> S),%) -> % ---R nthCoef : (%,Integer) -> Integer nthFactor : (%,Integer) -> S ---R retract : % -> S sample : () -> % ---R size : % -> NonNegativeInteger zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (S,%) -> Integer +--R coerce : S -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R mapGen : ((S -> S),%) -> % max : (%,%) -> % if S has ORDSET +--R min : (%,%) -> % if S has ORDSET nthCoef : (%,Integer) -> Integer +--R nthFactor : (%,Integer) -> S retract : % -> S +--R sample : () -> % size : % -> NonNegativeInteger +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET --R ?>? : (%,%) -> Boolean if S has ORDSET --R ?>=? : (%,%) -> Boolean if S has ORDSET --R highCommonTerms : (%,%) -> % if Integer has OAMON --R mapCoef : ((Integer -> Integer),%) -> % ---R max : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R retractIfCan : % -> Union(S,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") --R terms : % -> List(Record(gen: S,exp: Integer)) @@ -56253,16 +58649,15 @@ FreeAbelianGroup(S:SetCategory): Exports == Implementation where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for FAMONOID --R --R------------------------------- Operations -------------------------------- ---R ?*? : (NonNegativeInteger,S) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (S,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : S -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R mapGen : ((S -> S),%) -> % nthFactor : (%,Integer) -> S ---R retract : % -> S sample : () -> % ---R size : % -> NonNegativeInteger zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,S) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : S -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String mapGen : ((S -> S),%) -> % +--R nthFactor : (%,Integer) -> S retract : % -> S +--R sample : () -> % size : % -> NonNegativeInteger +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R coefficient : (S,%) -> NonNegativeInteger --R highCommonTerms : (%,%) -> % if NonNegativeInteger has OAMON --R mapCoef : ((NonNegativeInteger -> NonNegativeInteger),%) -> % @@ -56369,9 +58764,10 @@ FreeAbelianMonoid(S: SetCategory): --R------------------------------- Operations -------------------------------- --R ?*? : (%,S) -> % ?*? : (S,%) -> % --R ?*? : (%,%) -> % ?**? : (S,Integer) -> % ---R ?**? : (%,Integer) -> % ?**? : (%,PositiveInteger) -> % ---R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % ?^? : (%,Integer) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R ?^? : (%,Integer) -> % ?^? : (%,NonNegativeInteger) -> % --R ?^? : (%,PositiveInteger) -> % coerce : S -> % --R coerce : % -> OutputForm commutator : (%,%) -> % --R conjugate : (%,%) -> % hash : % -> SingleInteger @@ -56381,8 +58777,6 @@ FreeAbelianMonoid(S: SetCategory): --R recip : % -> Union(%,"failed") retract : % -> S --R sample : () -> % size : % -> NonNegativeInteger --R ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R factors : % -> List(Record(gen: S,exp: Integer)) --R mapExpon : ((Integer -> Integer),%) -> % --R retractIfCan : % -> Union(S,"failed") @@ -56528,6 +58922,7 @@ FreeGroup(S: SetCategory): Join(Group, RetractableTo S) with --S 1 of 1 )show FreeModule +--R --R FreeModule(R: Ring,S: OrderedSet) is a domain constructor --R Abbreviation for FreeModule is FM --R This constructor is not exposed in this frame. @@ -56535,16 +58930,16 @@ FreeGroup(S: SetCategory): Join(Group, RetractableTo S) with --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R leadingCoefficient : % -> R leadingSupport : % -> S ---R map : ((R -> R),%) -> % monomial : (R,S) -> % ---R reductum : % -> % sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String leadingCoefficient : % -> R +--R leadingSupport : % -> S map : ((R -> R),%) -> % +--R monomial : (R,S) -> % reductum : % -> % +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -56684,19 +59079,19 @@ FreeModule(R:Ring,S:OrderedSet): --R------------------------------- Operations -------------------------------- --R ?*? : (S,R) -> % ?*? : (R,S) -> % --R ?*? : (%,R) -> % ?*? : (R,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coefficient : (%,S) -> R ---R coefficients : % -> List(R) coerce : S -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> R ---R leadingMonomial : % -> S map : ((R -> R),%) -> % ---R monom : (S,R) -> % monomial? : % -> Boolean ---R monomials : % -> List(%) reductum : % -> % ---R retract : % -> S sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coefficient : (%,S) -> R coefficients : % -> List(R) +--R coerce : S -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingCoefficient : % -> R leadingMonomial : % -> S +--R map : ((R -> R),%) -> % monom : (S,R) -> % +--R monomial? : % -> Boolean monomials : % -> List(%) +--R reductum : % -> % retract : % -> S +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R leadingTerm : % -> Record(k: S,c: R) --R listOfTerms : % -> List(Record(k: S,c: R)) --R numberOfMonomials : % -> NonNegativeInteger @@ -56878,33 +59273,29 @@ FreeModule1(R:Ring,S:OrderedSet): FMcat == FMdef where --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,S) -> % ?*? : (S,%) -> % ---R ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (%,%) -> % ?**? : (S,NonNegativeInteger) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R ?^? : (%,PositiveInteger) -> % coerce : S -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R hclf : (%,%) -> % hcrf : (%,%) -> % ---R latex : % -> String mapGen : ((S -> S),%) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R coerce : S -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger hclf : (%,%) -> % +--R hcrf : (%,%) -> % latex : % -> String +--R lquo : (%,%) -> Union(%,"failed") mapGen : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET --R nthFactor : (%,Integer) -> S one? : % -> Boolean --R recip : % -> Union(%,"failed") retract : % -> S ---R sample : () -> % size : % -> NonNegativeInteger ---R ?~=? : (%,%) -> Boolean ---R ?**? : (S,NonNegativeInteger) -> % ---R ?**? : (%,NonNegativeInteger) -> % +--R rquo : (%,%) -> Union(%,"failed") sample : () -> % +--R size : % -> NonNegativeInteger ?~=? : (%,%) -> Boolean --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET --R ?>? : (%,%) -> Boolean if S has ORDSET --R ?>=? : (%,%) -> Boolean if S has ORDSET ---R ?^? : (%,NonNegativeInteger) -> % --R divide : (%,%) -> Union(Record(lm: %,rm: %),"failed") --R factors : % -> List(Record(gen: S,exp: NonNegativeInteger)) ---R lquo : (%,%) -> Union(%,"failed") --R mapExpon : ((NonNegativeInteger -> NonNegativeInteger),%) -> % ---R max : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R nthExpon : (%,Integer) -> NonNegativeInteger --R overlap : (%,%) -> Record(lm: %,mm: %,rm: %) --R retractIfCan : % -> Union(S,"failed") ---R rquo : (%,%) -> Union(%,"failed") --R --E 1 @@ -57174,6 +59565,7 @@ FreeMonoid(S: SetCategory): FMcategory == FMdefinition where --S 1 of 1 )show FreeNilpotentLie +--R --R FreeNilpotentLie(n: NonNegativeInteger,class: NonNegativeInteger,R: CommutativeRing) is a domain constructor --R Abbreviation for FreeNilpotentLie is FNLA --R This constructor is exposed in this frame. @@ -57182,18 +59574,17 @@ FreeMonoid(S: SetCategory): FMcategory == FMdefinition where --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % antiCommutator : (%,%) -> % ---R associator : (%,%,%) -> % coerce : % -> OutputForm ---R commutator : (%,%) -> % deepExpand : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R sample : () -> % shallowExpand : % -> OutputForm ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R dimension : () -> NonNegativeInteger ---R generator : NonNegativeInteger -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R antiCommutator : (%,%) -> % associator : (%,%,%) -> % +--R coerce : % -> OutputForm commutator : (%,%) -> % +--R deepExpand : % -> OutputForm dimension : () -> NonNegativeInteger +--R generator : NonNegativeInteger -> % hash : % -> SingleInteger +--R latex : % -> String sample : () -> % +--R shallowExpand : % -> OutputForm zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R leftPower : (%,PositiveInteger) -> % --R plenaryPower : (%,PositiveInteger) -> % --R rightPower : (%,PositiveInteger) -> % @@ -57381,7 +59772,8 @@ FreeNilpotentLie(n:NNI,class:NNI,R: CommutativeRing): Export == Implement where )set message test on )set message auto off )clear all ---S 1 of 16 + +--S 1 of 17 Fx := FRAC UP(x, FRAC INT) --R --R @@ -57389,7 +59781,7 @@ Fx := FRAC UP(x, FRAC INT) --R Type: Domain --E 1 ---S 2 of 16 +--S 2 of 17 f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) --R --R @@ -57400,7 +59792,7 @@ f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 2 ---S 3 of 16 +--S 3 of 17 g := fullPartialFraction f --R --R @@ -57412,7 +59804,7 @@ g := fullPartialFraction f --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 3 ---S 4 of 16 +--S 4 of 17 g :: Fx --R --R @@ -57423,7 +59815,7 @@ g :: Fx --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 4 ---S 5 of 16 +--S 5 of 17 g5 := D(g, 5) --R --R @@ -57435,7 +59827,7 @@ g5 := D(g, 5) --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 5 ---S 6 of 16 +--S 6 of 17 f5 := D(f, 5) --R --R @@ -57457,7 +59849,7 @@ f5 := D(f, 5) --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 6 ---S 7 of 16 +--S 7 of 17 g5::Fx - f5 --R --R @@ -57465,7 +59857,7 @@ g5::Fx - f5 --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 7 ---S 8 of 16 +--S 8 of 17 f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) --R --R @@ -57477,7 +59869,7 @@ f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 8 ---S 9 of 16 +--S 9 of 17 g := fullPartialFraction f --R --R @@ -57500,7 +59892,7 @@ g := fullPartialFraction f --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 9 ---S 10 of 16 +--S 10 of 17 g :: Fx - f --R --R @@ -57508,7 +59900,7 @@ g :: Fx - f --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 10 ---S 11 of 16 +--S 11 of 17 f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) --R --R @@ -57520,7 +59912,7 @@ f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 11 ---S 12 of 16 +--S 12 of 17 g := fullPartialFraction f --R --R @@ -57534,7 +59926,7 @@ g := fullPartialFraction f --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 12 ---S 13 of 16 +--S 13 of 17 g :: Fx - f --R --R @@ -57542,7 +59934,7 @@ g :: Fx - f --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 13 ---S 14 of 16 +--S 14 of 17 f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1) --R --R @@ -57560,7 +59952,7 @@ f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 14 ---S 15 of 16 +--S 15 of 17 g := fullPartialFraction f --R --R @@ -57619,13 +60011,36 @@ g := fullPartialFraction f --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 15 ---S 16 of 16 +--S 16 of 17 g :: Fx - f --R --R --R (16) 0 --R Type: Fraction(UnivariatePolynomial(x,Fraction(Integer))) --E 16 + +--S 17 of 17 +)show FullPartialFractionExpansion +--R +--R FullPartialFractionExpansion(F: Join(Field,CharacteristicZero),UP: UnivariatePolynomialCategory(F)) is a domain constructor +--R Abbreviation for FullPartialFractionExpansion is FPARFRAC +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for FPARFRAC +--R +--R------------------------------- Operations -------------------------------- +--R ?+? : (UP,%) -> % ?=? : (%,%) -> Boolean +--R D : (%,NonNegativeInteger) -> % D : % -> % +--R coerce : % -> OutputForm convert : % -> Fraction(UP) +--R differentiate : % -> % hash : % -> SingleInteger +--R latex : % -> String polyPart : % -> UP +--R ?~=? : (%,%) -> Boolean +--R construct : List(Record(exponent: NonNegativeInteger,center: UP,num: UP)) -> % +--R differentiate : (%,NonNegativeInteger) -> % +--R fracPart : % -> List(Record(exponent: NonNegativeInteger,center: UP,num: UP)) +--R fullPartialFraction : Fraction(UP) -> % +--R +--E 17 + )spool )lisp (bye) \end{chunk} @@ -58141,13 +60556,14 @@ FunctionCalled(f:Symbol): SetCategory with )set message test on )set message auto off )clear all ---S 1 of 10 + +--S 1 of 11 (d1,d2,d3) : DMP([z,y,x],FRAC INT) --R --R Type: Void --E 1 ---S 2 of 10 +--S 2 of 11 d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R --R @@ -58156,7 +60572,7 @@ d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 2 ---S 3 of 10 +--S 3 of 11 d2 := 2*z*y**2 + 4*x + 1 --R --R @@ -58165,7 +60581,7 @@ d2 := 2*z*y**2 + 4*x + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 3 ---S 4 of 10 +--S 4 of 11 d3 := 2*z*x**2 - 2*y**2 - x --R --R @@ -58174,7 +60590,7 @@ d3 := 2*z*x**2 - 2*y**2 - x --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 4 ---S 5 of 10 +--S 5 of 11 groebner [d1,d2,d3] --R --R @@ -58191,13 +60607,13 @@ groebner [d1,d2,d3] --R Type: List(DistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 5 ---S 6 of 10 +--S 6 of 11 (n1,n2,n3) : HDMP([z,y,x],FRAC INT) --R --R Type: Void --E 6 ---S 7 of 10 +--S 7 of 11 n1 := d1 --R --R @@ -58206,7 +60622,7 @@ n1 := d1 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 7 ---S 8 of 10 +--S 8 of 11 n2 := d2 --R --R @@ -58215,7 +60631,7 @@ n2 := d2 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 8 ---S 9 of 10 +--S 9 of 11 n3 := d3 --R --R @@ -58224,7 +60640,7 @@ n3 := d3 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 9 ---S 10 of 10 +--S 10 of 11 groebner [n1,n2,n3] --R --R @@ -58240,6 +60656,131 @@ groebner [n1,n2,n3] --R 4 2 --RType: List(HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 10 + +--S 11 of 11 +)show GeneralDistributedMultivariatePolynomial +--R +--R GeneralDistributedMultivariatePolynomial(vl: List(Symbol),R: Ring,E: DirectProductCategory(#(vl),NonNegativeInteger)) is a domain constructor +--R Abbreviation for GeneralDistributedMultivariatePolynomial is GDMP +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for GDMP +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficient : (%,E) -> R +--R coefficients : % -> List(R) coerce : % -> % if R has INTDOM +--R coerce : R -> % coerce : Integer -> % +--R coerce : % -> OutputForm content : % -> R if R has GCDDOM +--R degree : % -> E eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R +--R ground? : % -> Boolean hash : % -> SingleInteger +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leadingCoefficient : % -> R +--R leadingMonomial : % -> % map : ((R -> R),%) -> % +--R mapExponents : ((E -> E),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET minimumDegree : % -> E +--R monomial : (R,E) -> % monomial? : % -> Boolean +--R monomials : % -> List(%) one? : % -> Boolean +--R pomopo! : (%,R,E,%) -> % primitiveMonomials : % -> List(%) +--R recip : % -> Union(%,"failed") reductum : % -> % +--R reorder : (%,List(Integer)) -> % retract : % -> R +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) +--R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) +--R ? Boolean if R has ORDSET +--R ?<=? : (%,%) -> Boolean if R has ORDSET +--R ?>? : (%,%) -> Boolean if R has ORDSET +--R ?>=? : (%,%) -> Boolean if R has ORDSET +--R D : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R D : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R D : (%,List(OrderedVariableList(vl))) -> % +--R D : (%,OrderedVariableList(vl)) -> % +--R associates? : (%,%) -> Boolean if R has INTDOM +--R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and R has PFECAT or R has CHARNZ +--R coefficient : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R coefficient : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R coerce : Fraction(Integer) -> % if R has ALGEBRA(FRAC(INT)) or R has RETRACT(FRAC(INT)) +--R coerce : OrderedVariableList(vl) -> % +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and R has PFECAT +--R content : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R convert : % -> InputForm if OrderedVariableList(vl) has KONVERT(INFORM) and R has KONVERT(INFORM) +--R convert : % -> Pattern(Integer) if OrderedVariableList(vl) has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) +--R convert : % -> Pattern(Float) if OrderedVariableList(vl) has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) +--R degree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R degree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R differentiate : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R differentiate : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R differentiate : (%,List(OrderedVariableList(vl))) -> % +--R differentiate : (%,OrderedVariableList(vl)) -> % +--R discriminant : (%,OrderedVariableList(vl)) -> % if R has COMRING +--R eval : (%,List(OrderedVariableList(vl)),List(%)) -> % +--R eval : (%,OrderedVariableList(vl),%) -> % +--R eval : (%,List(OrderedVariableList(vl)),List(R)) -> % +--R eval : (%,OrderedVariableList(vl),R) -> % +--R exquo : (%,%) -> Union(%,"failed") if R has INTDOM +--R exquo : (%,R) -> Union(%,"failed") if R has INTDOM +--R factor : % -> Factored(%) if R has PFECAT +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM +--R isExpt : % -> Union(Record(var: OrderedVariableList(vl),exponent: NonNegativeInteger),"failed") +--R isPlus : % -> Union(List(%),"failed") +--R isTimes : % -> Union(List(%),"failed") +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM +--R mainVariable : % -> Union(OrderedVariableList(vl),"failed") +--R minimumDegree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R minimumDegree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R monicDivide : (%,%,OrderedVariableList(vl)) -> Record(quotient: %,remainder: %) +--R monomial : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R monomial : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R multivariate : (SparseUnivariatePolynomial(%),OrderedVariableList(vl)) -> % +--R multivariate : (SparseUnivariatePolynomial(R),OrderedVariableList(vl)) -> % +--R numberOfMonomials : % -> NonNegativeInteger +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if OrderedVariableList(vl) has PATMAB(INT) and R has PATMAB(INT) +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if OrderedVariableList(vl) has PATMAB(FLOAT) and R has PATMAB(FLOAT) +--R prime? : % -> Boolean if R has PFECAT +--R primitivePart : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R primitivePart : % -> % if R has GCDDOM +--R reducedSystem : Matrix(%) -> Matrix(R) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) +--R resultant : (%,%,OrderedVariableList(vl)) -> % if R has COMRING +--R retract : % -> OrderedVariableList(vl) +--R retract : % -> Integer if R has RETRACT(INT) +--R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(OrderedVariableList(vl),"failed") +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(R,"failed") +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has PFECAT +--R squareFree : % -> Factored(%) if R has GCDDOM +--R squareFreePart : % -> % if R has GCDDOM +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R totalDegree : (%,List(OrderedVariableList(vl))) -> NonNegativeInteger +--R totalDegree : % -> NonNegativeInteger +--R unit? : % -> Boolean if R has INTDOM +--R unitCanonical : % -> % if R has INTDOM +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has INTDOM +--R univariate : % -> SparseUnivariatePolynomial(R) +--R univariate : (%,OrderedVariableList(vl)) -> SparseUnivariatePolynomial(%) +--R variables : % -> List(OrderedVariableList(vl)) +--R +--E 11 + )spool )lisp (bye) \end{chunk} @@ -58706,17 +61247,17 @@ GeneralDistributedMultivariatePolynomial(vl,R,E): public == private where --R------------------------------- Operations -------------------------------- --R ?*? : (R,%) -> % ?*? : (%,R) -> % --R ?*? : (%,P) -> % ?*? : (P,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % build : (R,IS,E) -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> R ---R leadingExponent : % -> E leadingIndex : % -> IS ---R multMonom : (R,E,%) -> % reductum : % -> % ---R sample : () -> % unitVector : IS -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R build : (R,IS,E) -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingCoefficient : % -> R leadingExponent : % -> E +--R leadingIndex : % -> IS multMonom : (R,E,%) -> % +--R reductum : % -> % sample : () -> % +--R unitVector : IS -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R leadingMonomial : % -> ModuleMonomial(IS,E,ff) --R monomial : (R,ModuleMonomial(IS,E,ff)) -> % --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -58870,15 +61411,16 @@ GeneralModulePolynomial(vl, R, IS, E, ff, P): public == private where --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % alternative? : () -> Boolean ---R antiAssociative? : () -> Boolean antiCommutative? : () -> Boolean ---R antiCommutator : (%,%) -> % associative? : () -> Boolean ---R associator : (%,%,%) -> % basis : () -> Vector(%) ---R coerce : % -> OutputForm commutative? : () -> Boolean ---R commutator : (%,%) -> % flexible? : () -> Boolean +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R alternative? : () -> Boolean antiAssociative? : () -> Boolean +--R antiCommutative? : () -> Boolean antiCommutator : (%,%) -> % +--R associative? : () -> Boolean associator : (%,%,%) -> % +--R basis : () -> Vector(%) coerce : % -> OutputForm +--R commutative? : () -> Boolean commutator : (%,%) -> % +--R flexible? : () -> Boolean generic : (Symbol,Vector(%)) -> % --R generic : Vector(%) -> % generic : Vector(Symbol) -> % --R generic : Symbol -> % generic : () -> % --R hash : % -> SingleInteger jacobiIdentity? : () -> Boolean @@ -58892,7 +61434,6 @@ GeneralModulePolynomial(vl, R, IS, E, ff, P): public == private where --R ?*? : (SquareMatrix(n,Fraction(Polynomial(R))),%) -> % --R ?*? : (Fraction(Polynomial(R)),%) -> % --R ?*? : (%,Fraction(Polynomial(R))) -> % ---R ?*? : (NonNegativeInteger,%) -> % --R apply : (Matrix(Fraction(Polynomial(R))),%) -> % --R associatorDependence : () -> List(Vector(Fraction(Polynomial(R)))) if Fraction(Polynomial(R)) has INTDOM --R coerce : Vector(Fraction(Polynomial(R))) -> % @@ -58908,7 +61449,6 @@ GeneralModulePolynomial(vl, R, IS, E, ff, P): public == private where --R coordinates : (%,Vector(%)) -> Vector(Fraction(Polynomial(R))) --R ?.? : (%,Integer) -> Fraction(Polynomial(R)) --R generic : (Vector(Symbol),Vector(%)) -> % ---R generic : (Symbol,Vector(%)) -> % --R genericLeftDiscriminant : () -> Fraction(Polynomial(R)) if R has INTDOM --R genericLeftMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(R))) if R has INTDOM --R genericLeftNorm : % -> Fraction(Polynomial(R)) if R has INTDOM @@ -59611,33 +62151,119 @@ GeneralPolynomialSet(R,E,VarSet,P) : Exports == Implementation where )set message auto off )set break resume )clear all ---S 1 of 7 + +--S 1 of 8 patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ; --E 1 ---S 2 of 7 +--S 2 of 8 patrons."Smith" := 10500 --E 2 ---S 3 of 7 +--S 3 of 8 patrons."Jones" := 22000 --E 3 ---S 4 of 7 +--S 4 of 8 patrons."Jones" --E 4 ---S 5 of 7 +--S 5 of 8 patrons."Stingy" --E 5 ---S 6 of 7 +--S 6 of 8 reduce(+, entries patrons) --E 6 ---S 7 of 7 +--S 7 of 8 )system rm -r kaf*.sdata --E 7 + +--S 8 of 8 +)show GeneralSparseTable +--R +--R GeneralSparseTable(Key: SetCategory,Entry: SetCategory,Tbl: TableAggregate(Key,Entry),dent: Entry) is a domain constructor +--R Abbreviation for GeneralSparseTable is GSTBL +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for GSTBL +--R +--R------------------------------- Operations -------------------------------- +--R copy : % -> % dictionary : () -> % +--R elt : (%,Key,Entry) -> Entry ?.? : (%,Key) -> Entry +--R empty : () -> % empty? : % -> Boolean +--R entries : % -> List(Entry) eq? : (%,%) -> Boolean +--R index? : (Key,%) -> Boolean indices : % -> List(Key) +--R key? : (Key,%) -> Boolean keys : % -> List(Key) +--R map : ((Entry -> Entry),%) -> % qelt : (%,Key) -> Entry +--R sample : () -> % setelt : (%,Key,Entry) -> Entry +--R table : () -> % +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R ?=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R any? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R any? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R bag : List(Record(key: Key,entry: Entry)) -> % +--R coerce : % -> OutputForm if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R construct : List(Record(key: Key,entry: Entry)) -> % +--R convert : % -> InputForm if Record(key: Key,entry: Entry) has KONVERT(INFORM) +--R count : ((Entry -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Entry,%) -> NonNegativeInteger if $ has finiteAggregate and Entry has SETCAT +--R count : (Record(key: Key,entry: Entry),%) -> NonNegativeInteger if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R count : ((Record(key: Key,entry: Entry) -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R dictionary : List(Record(key: Key,entry: Entry)) -> % +--R entry? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R eval : (%,List(Equation(Entry))) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Equation(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,Entry,Entry) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Entry),List(Entry)) -> % if Entry has EVALAB(Entry) and Entry has SETCAT +--R eval : (%,List(Record(key: Key,entry: Entry)),List(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,Equation(Record(key: Key,entry: Entry))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R eval : (%,List(Equation(Record(key: Key,entry: Entry)))) -> % if Record(key: Key,entry: Entry) has EVALAB(Record(key: Key,entry: Entry)) and Record(key: Key,entry: Entry) has SETCAT +--R every? : ((Entry -> Boolean),%) -> Boolean if $ has finiteAggregate +--R every? : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R extract! : % -> Record(key: Key,entry: Entry) +--R fill! : (%,Entry) -> % if $ has shallowlyMutable +--R find : ((Record(key: Key,entry: Entry) -> Boolean),%) -> Union(Record(key: Key,entry: Entry),"failed") +--R first : % -> Entry if Key has ORDSET +--R hash : % -> SingleInteger if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R insert! : (Record(key: Key,entry: Entry),%) -> % +--R inspect : % -> Record(key: Key,entry: Entry) +--R latex : % -> String if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R less? : (%,NonNegativeInteger) -> Boolean +--R map : (((Entry,Entry) -> Entry),%,%) -> % +--R map : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % +--R map! : ((Entry -> Entry),%) -> % if $ has shallowlyMutable +--R map! : ((Record(key: Key,entry: Entry) -> Record(key: Key,entry: Entry)),%) -> % if $ has shallowlyMutable +--R maxIndex : % -> Key if Key has ORDSET +--R member? : (Entry,%) -> Boolean if $ has finiteAggregate and Entry has SETCAT +--R member? : (Record(key: Key,entry: Entry),%) -> Boolean if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R members : % -> List(Entry) if $ has finiteAggregate +--R members : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R minIndex : % -> Key if Key has ORDSET +--R more? : (%,NonNegativeInteger) -> Boolean +--R parts : % -> List(Entry) if $ has finiteAggregate +--R parts : % -> List(Record(key: Key,entry: Entry)) if $ has finiteAggregate +--R qsetelt! : (%,Key,Entry) -> Entry if $ has shallowlyMutable +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate +--R reduce : (((Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry)),%,Record(key: Key,entry: Entry),Record(key: Key,entry: Entry)) -> Record(key: Key,entry: Entry) if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R remove! : (Key,%) -> Union(Entry,"failed") +--R remove! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R remove! : (Record(key: Key,entry: Entry),%) -> % if $ has finiteAggregate +--R removeDuplicates : % -> % if $ has finiteAggregate and Record(key: Key,entry: Entry) has SETCAT +--R search : (Key,%) -> Union(Entry,"failed") +--R select : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R select! : ((Record(key: Key,entry: Entry) -> Boolean),%) -> % if $ has finiteAggregate +--R size? : (%,NonNegativeInteger) -> Boolean +--R swap! : (%,Key,Key) -> Void if $ has shallowlyMutable +--R table : List(Record(key: Key,entry: Entry)) -> % +--R ?~=? : (%,%) -> Boolean if Record(key: Key,entry: Entry) has SETCAT or Entry has SETCAT +--R +--E 8 + )spool )lisp (bye) \end{chunk} @@ -59831,7 +62457,8 @@ GeneralSparseTable(Key, Entry, Tbl, dent): TableAggregate(Key, Entry) == Impl --R normalized? : (P,%) -> Boolean reduceByQuasiMonic : (P,%) -> P --R removeZero : (P,%) -> P rest : % -> Union(%,"failed") --R retract : List(P) -> % sample : () -> % ---R stronglyReduce : (P,%) -> P stronglyReduced? : % -> Boolean +--R select : (%,V) -> Union(P,"failed") stronglyReduce : (P,%) -> P +--R stronglyReduced? : % -> Boolean stronglyReduced? : (P,%) -> Boolean --R trivialIdeal? : % -> Boolean variables : % -> List(V) --R zeroSetSplit : List(P) -> List(%) ?~=? : (%,%) -> Boolean --R #? : % -> NonNegativeInteger if $ has finiteAggregate @@ -59876,11 +62503,9 @@ GeneralSparseTable(Key, Entry, Tbl, dent): TableAggregate(Key, Entry) == Impl --R roughEqualIdeals? : (%,%) -> Boolean if R has INTDOM --R roughSubIdeal? : (%,%) -> Boolean if R has INTDOM --R roughUnitIdeal? : % -> Boolean if R has INTDOM ---R select : (%,V) -> Union(P,"failed") --R select : ((P -> Boolean),%) -> % if $ has finiteAggregate --R size? : (%,NonNegativeInteger) -> Boolean --R sort : (%,V) -> Record(under: %,floor: %,upper: %) ---R stronglyReduced? : (P,%) -> Boolean --R triangular? : % -> Boolean if R has INTDOM --R zeroSetSplitIntoTriangularSystems : List(P) -> List(Record(close: %,open: List(P))) --R @@ -60127,14 +62752,17 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coerce : % -> % if Coef has INTDOM --R coerce : Variable(var) -> % coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % ---R degree : % -> Fraction(Integer) hash : % -> SingleInteger +--R degree : % -> Fraction(Integer) ?.? : (%,Fraction(Integer)) -> Coef +--R hash : % -> SingleInteger inv : % -> % if Coef has FIELD --R latex : % -> String leadingCoefficient : % -> Coef --R leadingMonomial : % -> % map : ((Coef -> Coef),%) -> % --R monomial? : % -> Boolean one? : % -> Boolean @@ -60144,11 +62772,9 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % --R ?**? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,%) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?**? : (%,Integer) -> % if Coef has FIELD ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,%) -> % if Coef has FIELD --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Fraction(Integer),Coef) -> Coef @@ -60158,7 +62784,6 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has *: (Fraction(Integer),Coef) -> Coef and Coef has PDRING(SYMBOL) --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has *: (Fraction(Integer),Coef) -> Coef and Coef has PDRING(SYMBOL) --R ?^? : (%,Integer) -> % if Coef has FIELD ---R ?^? : (%,NonNegativeInteger) -> % --R acos : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acosh : % -> % if Coef has ALGEBRA(FRAC(INT)) --R acot : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -60176,7 +62801,6 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coefficient : (%,Fraction(Integer)) -> Coef ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R coerce : UnivariatePuiseuxSeries(Coef,var,cen) -> % --R coerce : Coef -> % if Coef has COMRING @@ -60195,7 +62819,6 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has *: (Fraction(Integer),Coef) -> Coef and Coef has PDRING(SYMBOL) --R divide : (%,%) -> Record(quotient: %,remainder: %) if Coef has FIELD --R ?.? : (%,%) -> % if Fraction(Integer) has SGROUP ---R ?.? : (%,Fraction(Integer)) -> Coef --R euclideanSize : % -> NonNegativeInteger if Coef has FIELD --R eval : (%,Coef) -> Stream(Coef) if Coef has **: (Coef,Fraction(Integer)) -> Coef --R exp : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -60211,7 +62834,6 @@ GeneralTriangularSet(R,E,V,P) : Exports == Implementation where --R integrate : (%,Variable(var)) -> % if Coef has ALGEBRA(FRAC(INT)) --R integrate : (%,Symbol) -> % if Coef has integrate: (Coef,Symbol) -> Coef and Coef has variables: Coef -> List(Symbol) and Coef has ALGEBRA(FRAC(INT)) or Coef has ACFS(INT) and Coef has ALGEBRA(FRAC(INT)) and Coef has PRIMCAT and Coef has TRANFUN --R integrate : % -> % if Coef has ALGEBRA(FRAC(INT)) ---R inv : % -> % if Coef has FIELD --R lcm : (%,%) -> % if Coef has FIELD --R lcm : List(%) -> % if Coef has FIELD --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if Coef has FIELD @@ -60506,8 +63128,8 @@ GeneralUnivariatePowerSeries(Coef,var,cen): Exports == Implementation where --R ?=? : (%,%) -> Boolean coerce : % -> OutputForm --R graphImage : () -> % hash : % -> SingleInteger --R key : % -> Integer latex : % -> String ---R makeGraphImage : % -> % units : % -> List(Float) ---R ?~=? : (%,%) -> Boolean +--R makeGraphImage : % -> % ranges : % -> List(Segment(Float)) +--R units : % -> List(Float) ?~=? : (%,%) -> Boolean --R appendPoint : (%,Point(DoubleFloat)) -> Void --R coerce : List(List(Point(DoubleFloat))) -> % --R component : (%,Point(DoubleFloat),Palette,Palette,PositiveInteger) -> Void @@ -60521,7 +63143,6 @@ GeneralUnivariatePowerSeries(Coef,var,cen): Exports == Implementation where --R pointLists : % -> List(List(Point(DoubleFloat))) --R putColorInfo : (List(List(Point(DoubleFloat))),List(Palette)) -> List(List(Point(DoubleFloat))) --R ranges : (%,List(Segment(Float))) -> List(Segment(Float)) ---R ranges : % -> List(Segment(Float)) --R units : (%,List(Float)) -> List(Float) --R --E 1 @@ -61258,14 +63879,14 @@ GuessOption(): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R ?=? : (%,%) -> Boolean coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R one : List(GuessOption) -> Boolean ?~=? : (%,%) -> Boolean +--R debug : List(GuessOption) -> Boolean hash : % -> SingleInteger +--R latex : % -> String one : List(GuessOption) -> Boolean +--R ?~=? : (%,%) -> Boolean --R Somos : List(GuessOption) -> Union(PositiveInteger,Boolean) --R allDegrees : List(GuessOption) -> Boolean --R check : List(GuessOption) -> Union(skip,MonteCarlo,deterministic) --R checkExtraValues : List(GuessOption) -> Boolean --R checkOptions : List(GuessOption) -> Void ---R debug : List(GuessOption) -> Boolean --R displayAsGF : List(GuessOption) -> Boolean --R functionName : List(GuessOption) -> Symbol --R homogeneous : List(GuessOption) -> Union(PositiveInteger,Boolean) @@ -62192,9 +64813,10 @@ coerce a --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean extract! : % -> S --R heap : List(S) -> % insert! : (S,%) -> % ---R inspect : % -> S map : ((S -> S),%) -> % ---R max : % -> S merge : (%,%) -> % ---R merge! : (%,%) -> % sample : () -> % +--R inspect : % -> S latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % max : % -> S +--R merge : (%,%) -> % merge! : (%,%) -> % +--R sample : () -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if S has SETCAT --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -62207,7 +64829,6 @@ coerce a --R eval : (%,List(Equation(S))) -> % if S has EVALAB(S) and S has SETCAT --R every? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT @@ -62709,7 +65330,8 @@ Heap(S:OrderedSet): Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 7 + +--S 1 of 8 r := hex(22/7) --R --R @@ -62718,7 +65340,7 @@ r := hex(22/7) --R Type: HexadecimalExpansion --E 1 ---S 2 of 7 +--S 2 of 8 r + hex(6/7) --R --R @@ -62726,7 +65348,7 @@ r + hex(6/7) --R Type: HexadecimalExpansion --E 2 ---S 3 of 7 +--S 3 of 8 [hex(1/i) for i in 350..354] --R --R @@ -62738,7 +65360,7 @@ r + hex(6/7) --R Type: List(HexadecimalExpansion) --E 3 ---S 4 of 7 +--S 4 of 8 hex(1/1007) --R --R @@ -62750,7 +65372,7 @@ hex(1/1007) --R Type: HexadecimalExpansion --E 4 ---S 5 of 7 +--S 5 of 8 p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9) --R --R @@ -62759,7 +65381,7 @@ p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9) --R Type: Polynomial(HexadecimalExpansion) --E 5 ---S 6 of 7 +--S 6 of 8 q := D(p, x) --R --R @@ -62768,7 +65390,7 @@ q := D(p, x) --R Type: Polynomial(HexadecimalExpansion) --E 6 ---S 7 of 7 +--S 7 of 8 g := gcd(p, q) --R --R @@ -62776,6 +65398,126 @@ g := gcd(p, q) --R (7) x + 1.5 --R Type: Polynomial(HexadecimalExpansion) --E 7 + +--S 8 of 8 +)show HexadecimalExpansion +--R +--R HexadecimalExpansion is a domain constructor +--R Abbreviation for HexadecimalExpansion is HEXADEC +--R This constructor is exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for HEXADEC +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,Integer) -> % ?*? : (Integer,%) -> % +--R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?/? : (Integer,Integer) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean D : (%,(Integer -> Integer)) -> % +--R D : % -> % if Integer has DIFRING 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> RadixExpansion(16) +--R coerce : % -> Fraction(Integer) coerce : Integer -> % +--R coerce : Fraction(Integer) -> % coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R denom : % -> Integer denominator : % -> % +--R factor : % -> Factored(%) gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R hex : Fraction(Integer) -> % init : () -> % if Integer has STEP +--R inv : % -> % latex : % -> String +--R lcm : List(%) -> % lcm : (%,%) -> % +--R numer : % -> Integer numerator : % -> % +--R one? : % -> Boolean prime? : % -> Boolean +--R ?quo? : (%,%) -> % random : () -> % if Integer has INS +--R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % +--R retract : % -> Integer sample : () -> % +--R sizeLess? : (%,%) -> Boolean squareFree : % -> Factored(%) +--R squareFreePart : % -> % toint : String -> Integer +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean +--R ? Boolean if Integer has ORDSET +--R ?<=? : (%,%) -> Boolean if Integer has ORDSET +--R ?>? : (%,%) -> Boolean if Integer has ORDSET +--R ?>=? : (%,%) -> Boolean if Integer has ORDSET +--R D : (%,(Integer -> Integer),NonNegativeInteger) -> % +--R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) +--R D : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) +--R D : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) +--R D : (%,Symbol) -> % if Integer has PDRING(SYMBOL) +--R D : (%,NonNegativeInteger) -> % if Integer has DIFRING +--R abs : % -> % if Integer has OINTDOM +--R ceiling : % -> Integer if Integer has INS +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and Integer has PFECAT or Integer has CHARNZ +--R coerce : Symbol -> % if Integer has RETRACT(SYMBOL) +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and Integer has PFECAT +--R convert : % -> DoubleFloat if Integer has REAL +--R convert : % -> Float if Integer has REAL +--R convert : % -> InputForm if Integer has KONVERT(INFORM) +--R convert : % -> Pattern(Float) if Integer has KONVERT(PATTERN(FLOAT)) +--R convert : % -> Pattern(Integer) if Integer has KONVERT(PATTERN(INT)) +--R differentiate : (%,(Integer -> Integer)) -> % +--R differentiate : (%,(Integer -> Integer),NonNegativeInteger) -> % +--R differentiate : (%,List(Symbol),List(NonNegativeInteger)) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,List(Symbol)) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,Symbol) -> % if Integer has PDRING(SYMBOL) +--R differentiate : (%,NonNegativeInteger) -> % if Integer has DIFRING +--R differentiate : % -> % if Integer has DIFRING +--R divide : (%,%) -> Record(quotient: %,remainder: %) +--R ?.? : (%,Integer) -> % if Integer has ELTAB(INT,INT) +--R euclideanSize : % -> NonNegativeInteger +--R eval : (%,Symbol,Integer) -> % if Integer has IEVALAB(SYMBOL,INT) +--R eval : (%,List(Symbol),List(Integer)) -> % if Integer has IEVALAB(SYMBOL,INT) +--R eval : (%,List(Equation(Integer))) -> % if Integer has EVALAB(INT) +--R eval : (%,Equation(Integer)) -> % if Integer has EVALAB(INT) +--R eval : (%,Integer,Integer) -> % if Integer has EVALAB(INT) +--R eval : (%,List(Integer),List(Integer)) -> % if Integer has EVALAB(INT) +--R expressIdealMember : (List(%),%) -> Union(List(%),"failed") +--R exquo : (%,%) -> Union(%,"failed") +--R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") +--R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R floor : % -> Integer if Integer has INS +--R fractionPart : % -> Fraction(Integer) +--R fractionPart : % -> % if Integer has EUCDOM +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) +--R map : ((Integer -> Integer),%) -> % +--R max : (%,%) -> % if Integer has ORDSET +--R min : (%,%) -> % if Integer has ORDSET +--R multiEuclidean : (List(%),%) -> Union(List(%),"failed") +--R negative? : % -> Boolean if Integer has OINTDOM +--R nextItem : % -> Union(%,"failed") if Integer has STEP +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if Integer has PATMAB(FLOAT) +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if Integer has PATMAB(INT) +--R positive? : % -> Boolean if Integer has OINTDOM +--R principalIdeal : List(%) -> Record(coef: List(%),generator: %) +--R reducedSystem : Matrix(%) -> Matrix(Integer) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if Integer has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if Integer has LINEXP(INT) +--R retract : % -> Integer if Integer has RETRACT(INT) +--R retract : % -> Fraction(Integer) if Integer has RETRACT(INT) +--R retract : % -> Symbol if Integer has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(Integer,"failed") if Integer has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if Integer has RETRACT(INT) +--R retractIfCan : % -> Union(Symbol,"failed") if Integer has RETRACT(SYMBOL) +--R retractIfCan : % -> Union(Integer,"failed") +--R sign : % -> Integer if Integer has OINTDOM +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if Integer has PFECAT +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Integer has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) +--R wholePart : % -> Integer if Integer has EUCDOM +--R +--E 8 + )spool )lisp (bye) \end{chunk} @@ -63170,6 +65912,7 @@ information. --S 1 of 9 )show HTMLFormat +--R --R HTMLFormat is a domain constructor --R Abbreviation for HTMLFormat is HTMLFORM --R This constructor is exposed in this frame. @@ -64242,33 +66985,35 @@ HTMLFormat(): public == private where --S 1 of 1 )show HomogeneousDirectProduct +--R --R HomogeneousDirectProduct(dim: NonNegativeInteger,S: OrderedAbelianMonoidSup) is a domain constructor --R Abbreviation for HomogeneousDirectProduct is HDP --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for HDP --R --R------------------------------- Operations -------------------------------- ---R -? : % -> % if S has RING 1 : () -> % if S has MONOID ---R 0 : () -> % if S has CABMON coerce : % -> Vector(S) +--R ?*? : (S,%) -> % if S has MONOID ?*? : (%,S) -> % if S has MONOID +--R ?*? : (%,%) -> % if S has MONOID ?+? : (%,%) -> % if S has ABELSG +--R -? : % -> % if S has RING ?-? : (%,%) -> % if S has RING +--R ?/? : (%,S) -> % if S has FIELD 1 : () -> % if S has MONOID +--R 0 : () -> % if S has CABMON abs : % -> % if S has ORDRING +--R coerce : S -> % if S has SETCAT coerce : % -> Vector(S) --R copy : % -> % directProduct : Vector(S) -> % ---R ?.? : (%,Integer) -> S elt : (%,Integer,S) -> S ---R empty : () -> % empty? : % -> Boolean ---R entries : % -> List(S) eq? : (%,%) -> Boolean ---R index? : (Integer,%) -> Boolean indices : % -> List(Integer) ---R map : ((S -> S),%) -> % qelt : (%,Integer) -> S ---R sample : () -> % +--R dot : (%,%) -> S if S has RING ?.? : (%,Integer) -> S +--R elt : (%,Integer,S) -> S empty : () -> % +--R empty? : % -> Boolean entries : % -> List(S) +--R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean +--R indices : % -> List(Integer) latex : % -> String if S has SETCAT +--R map : ((S -> S),%) -> % one? : % -> Boolean if S has MONOID +--R qelt : (%,Integer) -> S random : () -> % if S has FINITE +--R retract : % -> S if S has SETCAT sample : () -> % +--R sup : (%,%) -> % if S has OAMONS --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (PositiveInteger,%) -> % if S has ABELSG --R ?*? : (NonNegativeInteger,%) -> % if S has CABMON ---R ?*? : (S,%) -> % if S has MONOID ---R ?*? : (%,S) -> % if S has MONOID ---R ?*? : (%,%) -> % if S has MONOID --R ?*? : (Integer,%) -> % if S has RING --R ?**? : (%,PositiveInteger) -> % if S has MONOID --R ?**? : (%,NonNegativeInteger) -> % if S has MONOID ---R ?+? : (%,%) -> % if S has ABELSG ---R ?-? : (%,%) -> % if S has RING ---R ?/? : (%,S) -> % if S has FIELD --R ? Boolean if S has OAMONS or S has ORDRING --R ?<=? : (%,%) -> Boolean if S has OAMONS or S has ORDRING --R ?=? : (%,%) -> Boolean if S has SETCAT @@ -64284,10 +67029,8 @@ HTMLFormat(): public == private where --R D : % -> % if S has DIFRING and S has RING --R ?^? : (%,PositiveInteger) -> % if S has MONOID --R ?^? : (%,NonNegativeInteger) -> % if S has MONOID ---R abs : % -> % if S has ORDRING --R any? : ((S -> Boolean),%) -> Boolean if $ has finiteAggregate --R characteristic : () -> NonNegativeInteger if S has RING ---R coerce : S -> % if S has SETCAT --R coerce : Fraction(Integer) -> % if S has RETRACT(FRAC(INT)) and S has SETCAT --R coerce : Integer -> % if S has RETRACT(INT) and S has SETCAT or S has RING --R coerce : % -> OutputForm if S has SETCAT @@ -64302,7 +67045,6 @@ HTMLFormat(): public == private where --R differentiate : (%,NonNegativeInteger) -> % if S has DIFRING and S has RING --R differentiate : % -> % if S has DIFRING and S has RING --R dimension : () -> CardinalNumber if S has FIELD ---R dot : (%,%) -> S if S has RING --R entry? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R enumerate : () -> List(%) if S has FINITE --R eval : (%,List(S),List(S)) -> % if S has EVALAB(S) and S has SETCAT @@ -64314,7 +67056,6 @@ HTMLFormat(): public == private where --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT --R index : PositiveInteger -> % if S has FINITE ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R lookup : % -> PositiveInteger if S has FINITE --R map! : ((S -> S),%) -> % if $ has shallowlyMutable @@ -64326,17 +67067,14 @@ HTMLFormat(): public == private where --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R negative? : % -> Boolean if S has ORDRING ---R one? : % -> Boolean if S has MONOID --R parts : % -> List(S) if $ has finiteAggregate --R positive? : % -> Boolean if S has ORDRING --R qsetelt! : (%,Integer,S) -> S if $ has shallowlyMutable ---R random : () -> % if S has FINITE --R recip : % -> Union(%,"failed") if S has MONOID --R reducedSystem : Matrix(%) -> Matrix(S) if S has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(S),vec: Vector(S)) if S has RING --R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if S has LINEXP(INT) and S has RING --R reducedSystem : Matrix(%) -> Matrix(Integer) if S has LINEXP(INT) and S has RING ---R retract : % -> S if S has SETCAT --R retract : % -> Fraction(Integer) if S has RETRACT(FRAC(INT)) and S has SETCAT --R retract : % -> Integer if S has RETRACT(INT) and S has SETCAT --R retractIfCan : % -> Union(S,"failed") if S has SETCAT @@ -64347,7 +67085,6 @@ HTMLFormat(): public == private where --R size : () -> NonNegativeInteger if S has FINITE --R size? : (%,NonNegativeInteger) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") if S has CABMON ---R sup : (%,%) -> % if S has OAMONS --R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable --R unitVector : PositiveInteger -> % if S has RING --R zero? : % -> Boolean if S has CABMON @@ -64507,13 +67244,14 @@ HomogeneousDirectProduct(dim,S) : T == C where )set message test on )set message auto off )clear all ---S 1 of 10 + +--S 1 of 11 (d1,d2,d3) : DMP([z,y,x],FRAC INT) --R --R Type: Void --E 1 ---S 2 of 10 +--S 2 of 11 d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R --R @@ -64522,7 +67260,7 @@ d1 := -4*z + 4*y**2*x + 16*x**2 + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 2 ---S 3 of 10 +--S 3 of 11 d2 := 2*z*y**2 + 4*x + 1 --R --R @@ -64531,7 +67269,7 @@ d2 := 2*z*y**2 + 4*x + 1 --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 3 ---S 4 of 10 +--S 4 of 11 d3 := 2*z*x**2 - 2*y**2 - x --R --R @@ -64540,7 +67278,7 @@ d3 := 2*z*x**2 - 2*y**2 - x --R Type: DistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 4 ---S 5 of 10 +--S 5 of 11 groebner [d1,d2,d3] --R --R @@ -64557,13 +67295,13 @@ groebner [d1,d2,d3] --R Type: List(DistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 5 ---S 6 of 10 +--S 6 of 11 (n1,n2,n3) : HDMP([z,y,x],FRAC INT) --R --R Type: Void --E 6 ---S 7 of 10 +--S 7 of 11 n1 := d1 --R --R @@ -64572,7 +67310,7 @@ n1 := d1 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 7 ---S 8 of 10 +--S 8 of 11 n2 := d2 --R --R @@ -64581,7 +67319,7 @@ n2 := d2 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 8 ---S 9 of 10 +--S 9 of 11 n3 := d3 --R --R @@ -64590,7 +67328,7 @@ n3 := d3 --RType: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer)) --E 9 ---S 10 of 10 +--S 10 of 11 groebner [n1,n2,n3] --R --R @@ -64606,6 +67344,134 @@ groebner [n1,n2,n3] --R 4 2 --RType: List(HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction(Integer))) --E 10 + +--S 11 of 11 +)show HomogeneousDistributedMultivariatePolynomial +--R +--R HomogeneousDistributedMultivariatePolynomial(vl: List(Symbol),R: Ring) is a domain constructor +--R Abbreviation for HomogeneousDistributedMultivariatePolynomial is HDMP +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for HDMP +--R +--R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % ?*? : (R,%) -> % +--R ?*? : (%,%) -> % ?*? : (Integer,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % +--R ?+? : (%,%) -> % ?-? : (%,%) -> % +--R -? : % -> % ?/? : (%,R) -> % if R has FIELD +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficients : % -> List(R) +--R coerce : % -> % if R has INTDOM coerce : R -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R content : % -> R if R has GCDDOM eval : (%,List(%),List(%)) -> % +--R eval : (%,%,%) -> % eval : (%,Equation(%)) -> % +--R eval : (%,List(Equation(%))) -> % gcd : (%,%) -> % if R has GCDDOM +--R gcd : List(%) -> % if R has GCDDOM ground : % -> R +--R ground? : % -> Boolean hash : % -> SingleInteger +--R latex : % -> String lcm : (%,%) -> % if R has GCDDOM +--R lcm : List(%) -> % if R has GCDDOM leadingCoefficient : % -> R +--R leadingMonomial : % -> % map : ((R -> R),%) -> % +--R max : (%,%) -> % if R has ORDSET min : (%,%) -> % if R has ORDSET +--R monomial? : % -> Boolean monomials : % -> List(%) +--R one? : % -> Boolean primitiveMonomials : % -> List(%) +--R recip : % -> Union(%,"failed") reductum : % -> % +--R reorder : (%,List(Integer)) -> % retract : % -> R +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean +--R ?*? : (Fraction(Integer),%) -> % if R has ALGEBRA(FRAC(INT)) +--R ?*? : (%,Fraction(Integer)) -> % if R has ALGEBRA(FRAC(INT)) +--R ? Boolean if R has ORDSET +--R ?<=? : (%,%) -> Boolean if R has ORDSET +--R ?>? : (%,%) -> Boolean if R has ORDSET +--R ?>=? : (%,%) -> Boolean if R has ORDSET +--R D : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R D : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R D : (%,List(OrderedVariableList(vl))) -> % +--R D : (%,OrderedVariableList(vl)) -> % +--R associates? : (%,%) -> Boolean if R has INTDOM +--R binomThmExpt : (%,%,NonNegativeInteger) -> % if R has COMRING +--R characteristic : () -> NonNegativeInteger +--R charthRoot : % -> Union(%,"failed") if $ has CHARNZ and R has PFECAT or R has CHARNZ +--R coefficient : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R coefficient : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R coefficient : (%,HomogeneousDirectProduct(#(vl),NonNegativeInteger)) -> R +--R coerce : Fraction(Integer) -> % if R has ALGEBRA(FRAC(INT)) or R has RETRACT(FRAC(INT)) +--R coerce : OrderedVariableList(vl) -> % +--R conditionP : Matrix(%) -> Union(Vector(%),"failed") if $ has CHARNZ and R has PFECAT +--R content : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R convert : % -> InputForm if OrderedVariableList(vl) has KONVERT(INFORM) and R has KONVERT(INFORM) +--R convert : % -> Pattern(Integer) if OrderedVariableList(vl) has KONVERT(PATTERN(INT)) and R has KONVERT(PATTERN(INT)) +--R convert : % -> Pattern(Float) if OrderedVariableList(vl) has KONVERT(PATTERN(FLOAT)) and R has KONVERT(PATTERN(FLOAT)) +--R degree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R degree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R degree : % -> HomogeneousDirectProduct(#(vl),NonNegativeInteger) +--R differentiate : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R differentiate : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R differentiate : (%,List(OrderedVariableList(vl))) -> % +--R differentiate : (%,OrderedVariableList(vl)) -> % +--R discriminant : (%,OrderedVariableList(vl)) -> % if R has COMRING +--R eval : (%,List(OrderedVariableList(vl)),List(%)) -> % +--R eval : (%,OrderedVariableList(vl),%) -> % +--R eval : (%,List(OrderedVariableList(vl)),List(R)) -> % +--R eval : (%,OrderedVariableList(vl),R) -> % +--R exquo : (%,%) -> Union(%,"failed") if R has INTDOM +--R exquo : (%,R) -> Union(%,"failed") if R has INTDOM +--R factor : % -> Factored(%) if R has PFECAT +--R factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if R has GCDDOM +--R isExpt : % -> Union(Record(var: OrderedVariableList(vl),exponent: NonNegativeInteger),"failed") +--R isPlus : % -> Union(List(%),"failed") +--R isTimes : % -> Union(List(%),"failed") +--R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) if R has GCDDOM +--R mainVariable : % -> Union(OrderedVariableList(vl),"failed") +--R mapExponents : ((HomogeneousDirectProduct(#(vl),NonNegativeInteger) -> HomogeneousDirectProduct(#(vl),NonNegativeInteger)),%) -> % +--R minimumDegree : (%,List(OrderedVariableList(vl))) -> List(NonNegativeInteger) +--R minimumDegree : (%,OrderedVariableList(vl)) -> NonNegativeInteger +--R minimumDegree : % -> HomogeneousDirectProduct(#(vl),NonNegativeInteger) +--R monicDivide : (%,%,OrderedVariableList(vl)) -> Record(quotient: %,remainder: %) +--R monomial : (%,List(OrderedVariableList(vl)),List(NonNegativeInteger)) -> % +--R monomial : (%,OrderedVariableList(vl),NonNegativeInteger) -> % +--R monomial : (R,HomogeneousDirectProduct(#(vl),NonNegativeInteger)) -> % +--R multivariate : (SparseUnivariatePolynomial(%),OrderedVariableList(vl)) -> % +--R multivariate : (SparseUnivariatePolynomial(R),OrderedVariableList(vl)) -> % +--R numberOfMonomials : % -> NonNegativeInteger +--R patternMatch : (%,Pattern(Integer),PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if OrderedVariableList(vl) has PATMAB(INT) and R has PATMAB(INT) +--R patternMatch : (%,Pattern(Float),PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if OrderedVariableList(vl) has PATMAB(FLOAT) and R has PATMAB(FLOAT) +--R pomopo! : (%,R,HomogeneousDirectProduct(#(vl),NonNegativeInteger),%) -> % +--R prime? : % -> Boolean if R has PFECAT +--R primitivePart : (%,OrderedVariableList(vl)) -> % if R has GCDDOM +--R primitivePart : % -> % if R has GCDDOM +--R reducedSystem : Matrix(%) -> Matrix(R) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(R),vec: Vector(R)) +--R reducedSystem : (Matrix(%),Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if R has LINEXP(INT) +--R reducedSystem : Matrix(%) -> Matrix(Integer) if R has LINEXP(INT) +--R resultant : (%,%,OrderedVariableList(vl)) -> % if R has COMRING +--R retract : % -> OrderedVariableList(vl) +--R retract : % -> Integer if R has RETRACT(INT) +--R retract : % -> Fraction(Integer) if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(OrderedVariableList(vl),"failed") +--R retractIfCan : % -> Union(Integer,"failed") if R has RETRACT(INT) +--R retractIfCan : % -> Union(Fraction(Integer),"failed") if R has RETRACT(FRAC(INT)) +--R retractIfCan : % -> Union(R,"failed") +--R solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)),SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)),"failed") if R has PFECAT +--R squareFree : % -> Factored(%) if R has GCDDOM +--R squareFreePart : % -> % if R has GCDDOM +--R squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if R has PFECAT +--R subtractIfCan : (%,%) -> Union(%,"failed") +--R totalDegree : (%,List(OrderedVariableList(vl))) -> NonNegativeInteger +--R totalDegree : % -> NonNegativeInteger +--R unit? : % -> Boolean if R has INTDOM +--R unitCanonical : % -> % if R has INTDOM +--R unitNormal : % -> Record(unit: %,canonical: %,associate: %) if R has INTDOM +--R univariate : % -> SparseUnivariatePolynomial(R) +--R univariate : (%,OrderedVariableList(vl)) -> SparseUnivariatePolynomial(%) +--R variables : % -> List(OrderedVariableList(vl)) +--R +--E 11 + )spool )lisp (bye) \end{chunk} @@ -64858,20 +67724,19 @@ HomogeneousDistributedMultivariatePolynomial(vl,R): public == private where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for HELLFDIV --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R divisor : (R,UP,UP,UP,F) -> % divisor : (F,F,Integer) -> % ---R divisor : (F,F) -> % divisor : R -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm divisor : (R,UP,UP,UP,F) -> % +--R divisor : (F,F,Integer) -> % divisor : (F,F) -> % +--R divisor : R -> % generator : % -> Union(R,"failed") --R hash : % -> SingleInteger latex : % -> String --R principal? : % -> Boolean reduce : % -> % --R sample : () -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % --R decompose : % -> Record(id: FractionalIdeal(UP,Fraction(UP),UPUP,R),principalPart: R) --R divisor : FractionalIdeal(UP,Fraction(UP),UPUP,R) -> % ---R generator : % -> Union(R,"failed") --R ideal : % -> FractionalIdeal(UP,Fraction(UP),UPUP,R) --R subtractIfCan : (%,%) -> Union(%,"failed") --R @@ -65112,6 +67977,7 @@ HyperellipticFiniteDivisor(F, UP, UPUP, R): Exports == Implementation where --S 1 of 1 )show InfClsPt +--R --R InfClsPt(K: Field,symb: List(Symbol),BLMET: BlowUpMethodCategory) is a domain constructor --R Abbreviation for InfClsPt is ICP --R This constructor is exposed in this frame. @@ -65120,16 +67986,17 @@ HyperellipticFiniteDivisor(F, UP, UPUP, R): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?=? : (%,%) -> Boolean actualExtensionV : % -> K --R chartV : % -> BLMET coerce : % -> OutputForm ---R degree : % -> PositiveInteger fullOut : % -> OutputForm ---R fullOutput : () -> Boolean fullOutput : Boolean -> Boolean ---R hash : % -> SingleInteger latex : % -> String ---R localPointV : % -> AffinePlane(K) multV : % -> NonNegativeInteger ---R pointV : % -> ProjectivePlane(K) setchart! : (%,BLMET) -> BLMET ---R symbNameV : % -> Symbol ?~=? : (%,%) -> Boolean +--R degree : % -> PositiveInteger excpDivV : % -> Divisor(Places(K)) +--R fullOut : % -> OutputForm fullOutput : () -> Boolean +--R fullOutput : Boolean -> Boolean hash : % -> SingleInteger +--R latex : % -> String localPointV : % -> AffinePlane(K) +--R multV : % -> NonNegativeInteger pointV : % -> ProjectivePlane(K) +--R setchart! : (%,BLMET) -> BLMET setsymbName! : (%,Symbol) -> Symbol +--R subMultV : % -> NonNegativeInteger symbNameV : % -> Symbol +--R ?~=? : (%,%) -> Boolean --R create : (ProjectivePlane(K),DistributedMultivariatePolynomial(symb,K)) -> % --R create : (ProjectivePlane(K),DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K),AffinePlane(K),NonNegativeInteger,BLMET,NonNegativeInteger,Divisor(Places(K)),K,Symbol) -> % --R curveV : % -> DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K) ---R excpDivV : % -> Divisor(Places(K)) --R localParamV : % -> List(NeitherSparseOrDensePowerSeries(K)) --R setcurve! : (%,DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K)) -> DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K) --R setexcpDiv! : (%,Divisor(Places(K))) -> Divisor(Places(K)) @@ -65138,8 +68005,6 @@ HyperellipticFiniteDivisor(F, UP, UPUP, R): Exports == Implementation where --R setmult! : (%,NonNegativeInteger) -> NonNegativeInteger --R setpoint! : (%,ProjectivePlane(K)) -> ProjectivePlane(K) --R setsubmult! : (%,NonNegativeInteger) -> NonNegativeInteger ---R setsymbName! : (%,Symbol) -> Symbol ---R subMultV : % -> NonNegativeInteger --R --E 1 @@ -65251,6 +68116,7 @@ InfClsPt(K,symb,BLMET):Exports == Implementation where --S 1 of 1 )show IndexCard +--R --R IndexCard is a domain constructor --R Abbreviation for IndexCard is ICARD --R This constructor is exposed in this frame. @@ -65384,7 +68250,8 @@ IndexCard() : Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 13 + +--S 1 of 14 a:IBITS(32):=new(32,false) --R --R @@ -65392,7 +68259,7 @@ a:IBITS(32):=new(32,false) --R Type: IndexedBits(32) --E 1 ---S 2 of 13 +--S 2 of 14 b:IBITS(32):=new(32,true) --R --R @@ -65400,7 +68267,7 @@ b:IBITS(32):=new(32,true) --R Type: IndexedBits(32) --E 2 ---S 3 of 13 +--S 3 of 14 elt(a,3) --R --R @@ -65408,7 +68275,7 @@ elt(a,3) --R Type: Boolean --E 3 ---S 4 of 13 +--S 4 of 14 setelt(a,3,true) --R --R @@ -65416,7 +68283,7 @@ setelt(a,3,true) --R Type: Boolean --E 4 ---S 5 of 13 +--S 5 of 14 a --R --R @@ -65424,7 +68291,7 @@ a --R Type: IndexedBits(32) --E 5 ---S 6 of 13 +--S 6 of 14 #a --R --R @@ -65432,7 +68299,7 @@ a --R Type: PositiveInteger --E 6 ---S 7 of 13 +--S 7 of 14 (a=a)$IBITS(32) --R --R @@ -65440,7 +68307,7 @@ a --R Type: Boolean --E 7 ---S 8 of 13 +--S 8 of 14 (a=b)$IBITS(32) --R --R @@ -65448,7 +68315,7 @@ a --R Type: Boolean --E 8 ---S 9 of 13 +--S 9 of 14 (a ~= b) --R --R @@ -65456,7 +68323,7 @@ a --R Type: Boolean --E 9 ---S 10 of 13 +--S 10 of 14 Or(a,b) --R --R @@ -65464,7 +68331,7 @@ Or(a,b) --R Type: IndexedBits(32) --E 10 ---S 11 of 13 +--S 11 of 14 And(a,b) --R --R @@ -65472,7 +68339,7 @@ And(a,b) --R Type: IndexedBits(32) --E 11 ---S 12 of 13 +--S 12 of 14 Not(a) --R --R @@ -65480,13 +68347,101 @@ Not(a) --R Type: IndexedBits(32) --E 12 ---S 13 of 13 +--S 13 of 14 c:=copy a --R --R --R (13) "00000000000000000000000000000100" --R Type: IndexedBits(32) --E 13 + +--S 14 of 14 +)show IndexedBits +--R +--R IndexedBits(mn: Integer) is a domain constructor +--R Abbreviation for IndexedBits is IBITS +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.pamphlet to see algebra source code for IBITS +--R +--R------------------------------- Operations -------------------------------- +--R ?/\? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R And : (%,%) -> % Not : % -> % +--R Or : (%,%) -> % ?\/? : (%,%) -> % +--R ^? : % -> % ?and? : (%,%) -> % +--R coerce : % -> OutputForm concat : (%,Boolean) -> % +--R concat : (Boolean,%) -> % concat : (%,%) -> % +--R concat : List(%) -> % construct : List(Boolean) -> % +--R copy : % -> % delete : (%,Integer) -> % +--R ?.? : (%,Integer) -> Boolean empty : () -> % +--R empty? : % -> Boolean entries : % -> List(Boolean) +--R eq? : (%,%) -> Boolean hash : % -> SingleInteger +--R index? : (Integer,%) -> Boolean indices : % -> List(Integer) +--R insert : (Boolean,%,Integer) -> % insert : (%,%,Integer) -> % +--R latex : % -> String max : (%,%) -> % +--R min : (%,%) -> % nand : (%,%) -> % +--R nor : (%,%) -> % not? : % -> % +--R ?or? : (%,%) -> % qelt : (%,Integer) -> Boolean +--R reverse : % -> % sample : () -> % +--R xor : (%,%) -> % ~? : % -> % +--R ?~=? : (%,%) -> Boolean +--R #? : % -> NonNegativeInteger if $ has finiteAggregate +--R any? : ((Boolean -> Boolean),%) -> Boolean if $ has finiteAggregate +--R convert : % -> InputForm if Boolean has KONVERT(INFORM) +--R copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable +--R count : ((Boolean -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate +--R count : (Boolean,%) -> NonNegativeInteger if $ has finiteAggregate and Boolean has SETCAT +--R delete : (%,UniversalSegment(Integer)) -> % +--R elt : (%,Integer,Boolean) -> Boolean +--R ?.? : (%,UniversalSegment(Integer)) -> % +--R entry? : (Boolean,%) -> Boolean if $ has finiteAggregate and Boolean has SETCAT +--R eval : (%,List(Equation(Boolean))) -> % if Boolean has EVALAB(BOOLEAN) and Boolean has SETCAT +--R eval : (%,Equation(Boolean)) -> % if Boolean has EVALAB(BOOLEAN) and Boolean has SETCAT +--R eval : (%,Boolean,Boolean) -> % if Boolean has EVALAB(BOOLEAN) and Boolean has SETCAT +--R eval : (%,List(Boolean),List(Boolean)) -> % if Boolean has EVALAB(BOOLEAN) and Boolean has SETCAT +--R every? : ((Boolean -> Boolean),%) -> Boolean if $ has finiteAggregate +--R fill! : (%,Boolean) -> % if $ has shallowlyMutable +--R find : ((Boolean -> Boolean),%) -> Union(Boolean,"failed") +--R first : % -> Boolean if Integer has ORDSET +--R less? : (%,NonNegativeInteger) -> Boolean +--R map : ((Boolean -> Boolean),%) -> % +--R map : (((Boolean,Boolean) -> Boolean),%,%) -> % +--R map! : ((Boolean -> Boolean),%) -> % if $ has shallowlyMutable +--R maxIndex : % -> Integer if Integer has ORDSET +--R member? : (Boolean,%) -> Boolean if $ has finiteAggregate and Boolean has SETCAT +--R members : % -> List(Boolean) if $ has finiteAggregate +--R merge : (((Boolean,Boolean) -> Boolean),%,%) -> % +--R merge : (%,%) -> % if Boolean has ORDSET +--R minIndex : % -> Integer if Integer has ORDSET +--R more? : (%,NonNegativeInteger) -> Boolean +--R new : (NonNegativeInteger,Boolean) -> % +--R parts : % -> List(Boolean) if $ has finiteAggregate +--R position : ((Boolean -> Boolean),%) -> Integer +--R position : (Boolean,%) -> Integer if Boolean has SETCAT +--R position : (Boolean,%,Integer) -> Integer if Boolean has SETCAT +--R qsetelt! : (%,Integer,Boolean) -> Boolean if $ has shallowlyMutable +--R reduce : (((Boolean,Boolean) -> Boolean),%,Boolean,Boolean) -> Boolean if $ has finiteAggregate and Boolean has SETCAT +--R reduce : (((Boolean,Boolean) -> Boolean),%,Boolean) -> Boolean if $ has finiteAggregate +--R reduce : (((Boolean,Boolean) -> Boolean),%) -> Boolean if $ has finiteAggregate +--R remove : (Boolean,%) -> % if $ has finiteAggregate and Boolean has SETCAT +--R remove : ((Boolean -> Boolean),%) -> % if $ has finiteAggregate +--R removeDuplicates : % -> % if $ has finiteAggregate and Boolean has SETCAT +--R reverse! : % -> % if $ has shallowlyMutable +--R select : ((Boolean -> Boolean),%) -> % if $ has finiteAggregate +--R setelt : (%,Integer,Boolean) -> Boolean if $ has shallowlyMutable +--R setelt : (%,UniversalSegment(Integer),Boolean) -> Boolean if $ has shallowlyMutable +--R size? : (%,NonNegativeInteger) -> Boolean +--R sort : (((Boolean,Boolean) -> Boolean),%) -> % +--R sort : % -> % if Boolean has ORDSET +--R sort! : (((Boolean,Boolean) -> Boolean),%) -> % if $ has shallowlyMutable +--R sort! : % -> % if $ has shallowlyMutable and Boolean has ORDSET +--R sorted? : (((Boolean,Boolean) -> Boolean),%) -> Boolean +--R sorted? : % -> Boolean if Boolean has ORDSET +--R swap! : (%,Integer,Integer) -> Void if $ has shallowlyMutable +--R +--E 14 + )spool )lisp (bye) \end{chunk} @@ -65701,22 +68656,23 @@ IndexedBits(mn:Integer): BitAggregate() with --S 1 of 1 )show IndexedDirectProductAbelianGroup +--R --R IndexedDirectProductAbelianGroup(A: AbelianGroup,S: OrderedSet) is a domain constructor --R Abbreviation for IndexedDirectProductAbelianGroup is IDPAG --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IDPAG --R --R------------------------------- Operations -------------------------------- ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (%,%) -> % ?-? : (%,%) -> % ---R -? : % -> % ?=? : (%,%) -> Boolean ---R 0 : () -> % coerce : % -> OutputForm ---R hash : % -> SingleInteger latex : % -> String ---R leadingCoefficient : % -> A leadingSupport : % -> S ---R map : ((A -> A),%) -> % monomial : (A,S) -> % ---R reductum : % -> % sample : () -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % +--R ?-? : (%,%) -> % -? : % -> % +--R ?=? : (%,%) -> Boolean 0 : () -> % +--R coerce : % -> OutputForm hash : % -> SingleInteger +--R latex : % -> String leadingCoefficient : % -> A +--R leadingSupport : % -> S map : ((A -> A),%) -> % +--R monomial : (A,S) -> % reductum : % -> % +--R sample : () -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -65859,21 +68815,21 @@ IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedSet): --S 1 of 1 )show IndexedDirectProductAbelianMonoid +--R --R IndexedDirectProductAbelianMonoid(A: AbelianMonoid,S: OrderedSet) is a domain constructor --R Abbreviation for IndexedDirectProductAbelianMonoid is IDPAM --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IDPAM --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> A ---R leadingSupport : % -> S map : ((A -> A),%) -> % ---R monomial : (A,S) -> % reductum : % -> % ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingCoefficient : % -> A leadingSupport : % -> S +--R map : ((A -> A),%) -> % monomial : (A,S) -> % +--R reductum : % -> % sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R --E 1 @@ -66023,6 +68979,7 @@ IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedSet): --S 1 of 1 )show IndexedDirectProductObject +--R --R IndexedDirectProductObject(A: SetCategory,S: OrderedSet) is a domain constructor --R Abbreviation for IndexedDirectProductObject is IDPO --R This constructor is not exposed in this frame. @@ -66140,24 +69097,24 @@ IndexedDirectProductObject(A:SetCategory,S:OrderedSet): _ --S 1 of 1 )show IndexedDirectProductOrderedAbelianMonoid +--R --R IndexedDirectProductOrderedAbelianMonoid(A: OrderedAbelianMonoid,S: OrderedSet) is a domain constructor --R Abbreviation for IndexedDirectProductOrderedAbelianMonoid is IDPOAM --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IDPOAM --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> A ---R leadingSupport : % -> S map : ((A -> A),%) -> % ---R max : (%,%) -> % min : (%,%) -> % ---R monomial : (A,S) -> % reductum : % -> % ---R sample : () -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingCoefficient : % -> A leadingSupport : % -> S +--R map : ((A -> A),%) -> % max : (%,%) -> % +--R min : (%,%) -> % monomial : (A,S) -> % +--R reductum : % -> % sample : () -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R --E 1 @@ -66256,24 +69213,25 @@ IndexedDirectProductOrderedAbelianMonoid(A:OrderedAbelianMonoid,S:OrderedSet): --S 1 of 1 )show IndexedDirectProductOrderedAbelianMonoidSup +--R --R IndexedDirectProductOrderedAbelianMonoidSup(A: OrderedAbelianMonoidSup,S: OrderedSet) is a domain constructor --R Abbreviation for IndexedDirectProductOrderedAbelianMonoidSup is IDPOAMS --R This constructor is not exposed in this frame. --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IDPOAMS --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingCoefficient : % -> A ---R leadingSupport : % -> S map : ((A -> A),%) -> % ---R max : (%,%) -> % min : (%,%) -> % ---R monomial : (A,S) -> % reductum : % -> % ---R sample : () -> % sup : (%,%) -> % ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingCoefficient : % -> A leadingSupport : % -> S +--R map : ((A -> A),%) -> % max : (%,%) -> % +--R min : (%,%) -> % monomial : (A,S) -> % +--R reductum : % -> % sample : () -> % +--R sup : (%,%) -> % zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R subtractIfCan : (%,%) -> Union(%,"failed") --R --E 1 @@ -66398,17 +69356,16 @@ IndexedDirectProductOrderedAbelianMonoidSup(A:OrderedAbelianMonoidSup,S:OrderedS --R Issue )edit bookvol10.3.pamphlet to see algebra source code for INDE --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?+? : (%,%) -> % ---R ? Boolean ?<=? : (%,%) -> Boolean ---R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean ---R ?>=? : (%,%) -> Boolean 0 : () -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String leadingSupport : % -> Varset ---R max : (%,%) -> % min : (%,%) -> % ---R reductum : % -> % sample : () -> % ---R sup : (%,%) -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?+? : (%,%) -> % ? Boolean +--R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean +--R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean +--R 0 : () -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R leadingSupport : % -> Varset max : (%,%) -> % +--R min : (%,%) -> % reductum : % -> % +--R sample : () -> % sup : (%,%) -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R leadingCoefficient : % -> NonNegativeInteger --R map : ((NonNegativeInteger -> NonNegativeInteger),%) -> % --R monomial : (NonNegativeInteger,Varset) -> % @@ -66529,10 +69486,14 @@ IndexedExponents(Varset:OrderedSet): C == T where --R flexibleArray : List(S) -> % index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % --R insert : (S,%,Integer) -> % insert! : (S,%,Integer) -> % ---R insert! : (%,%,Integer) -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % ---R qelt : (%,Integer) -> S reverse : % -> % ---R sample : () -> % shrinkable : Boolean -> Boolean +--R insert! : (%,%,Integer) -> % latex : % -> String if S has SETCAT +--R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET +--R new : (NonNegativeInteger,S) -> % physicalLength! : (%,Integer) -> % +--R qelt : (%,Integer) -> S remove! : ((S -> Boolean),%) -> % +--R reverse : % -> % sample : () -> % +--R select! : ((S -> Boolean),%) -> % shrinkable : Boolean -> Boolean +--R sort : % -> % if S has ORDSET sort : (((S,S) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -66558,10 +69519,8 @@ IndexedExponents(Varset:OrderedSet): C == T where --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R members : % -> List(S) if $ has finiteAggregate @@ -66569,12 +69528,10 @@ IndexedExponents(Varset:OrderedSet): C == T where --R merge : (((S,S) -> Boolean),%,%) -> % --R merge! : (((S,S) -> Boolean),%,%) -> % --R merge! : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(S) if $ has finiteAggregate --R physicalLength : % -> NonNegativeInteger ---R physicalLength! : (%,Integer) -> % --R position : (S,%,Integer) -> Integer if S has SETCAT --R position : (S,%) -> Integer if S has SETCAT --R position : ((S -> Boolean),%) -> Integer @@ -66584,18 +69541,14 @@ IndexedExponents(Varset:OrderedSet): C == T where --R reduce : (((S,S) -> S),%,S,S) -> S if $ has finiteAggregate and S has SETCAT --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate --R remove : (S,%) -> % if $ has finiteAggregate and S has SETCAT ---R remove! : ((S -> Boolean),%) -> % --R remove! : (S,%) -> % if S has SETCAT --R removeDuplicates : % -> % if $ has finiteAggregate and S has SETCAT --R removeDuplicates! : % -> % if S has SETCAT --R reverse! : % -> % if $ has shallowlyMutable --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R select! : ((S -> Boolean),%) -> % --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if S has ORDSET ---R sort : (((S,S) -> Boolean),%) -> % --R sort! : % -> % if $ has shallowlyMutable and S has ORDSET --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if S has ORDSET @@ -67000,15 +69953,20 @@ IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where --R first : % -> S index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (S,%,Integer) -> % --R insert : (%,%,Integer) -> % insert! : (S,%,Integer) -> % ---R insert! : (%,%,Integer) -> % last : % -> S +--R insert! : (%,%,Integer) -> % last : (%,NonNegativeInteger) -> % +--R last : % -> S latex : % -> String if S has SETCAT --R leaf? : % -> Boolean leaves : % -> List(S) --R list : S -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % +--R map : ((S -> S),%) -> % max : (%,%) -> % if S has ORDSET +--R min : (%,%) -> % if S has ORDSET new : (NonNegativeInteger,S) -> % --R nodes : % -> List(%) possiblyInfinite? : % -> Boolean ---R qelt : (%,Integer) -> S rest : % -> % +--R qelt : (%,Integer) -> S remove! : ((S -> Boolean),%) -> % +--R rest : (%,NonNegativeInteger) -> % rest : % -> % --R reverse : % -> % sample : () -> % ---R second : % -> S tail : % -> % ---R third : % -> S value : % -> S +--R second : % -> S select! : ((S -> Boolean),%) -> % +--R sort : (((S,S) -> Boolean),%) -> % sort : % -> % if S has ORDSET +--R tail : % -> % third : % -> S +--R value : % -> S --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -67037,11 +69995,8 @@ IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : (%,NonNegativeInteger) -> % --R hash : % -> SingleInteger if S has SETCAT ---R last : (%,NonNegativeInteger) -> % ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R members : % -> List(S) if $ has finiteAggregate @@ -67049,7 +70004,6 @@ IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where --R merge : (%,%) -> % if S has ORDSET --R merge! : (((S,S) -> Boolean),%,%) -> % --R merge! : (%,%) -> % if S has ORDSET ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R node? : (%,%) -> Boolean if S has SETCAT @@ -67063,14 +70017,11 @@ IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where --R reduce : (((S,S) -> S),%) -> S if $ has finiteAggregate --R remove : (S,%) -> % if $ has finiteAggregate and S has SETCAT --R remove : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R remove! : ((S -> Boolean),%) -> % --R remove! : (S,%) -> % if S has SETCAT --R removeDuplicates : % -> % if $ has finiteAggregate and S has SETCAT --R removeDuplicates! : % -> % if S has SETCAT ---R rest : (%,NonNegativeInteger) -> % --R reverse! : % -> % if $ has shallowlyMutable --R select : ((S -> Boolean),%) -> % if $ has finiteAggregate ---R select! : ((S -> Boolean),%) -> % --R setchildren! : (%,List(%)) -> % if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable @@ -67083,8 +70034,6 @@ IndexedFlexibleArray(S:Type, mn: Integer): Exports == Implementation where --R setrest! : (%,%) -> % if $ has shallowlyMutable --R setvalue! : (%,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : (((S,S) -> Boolean),%) -> % ---R sort : % -> % if S has ORDSET --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sort! : % -> % if $ has shallowlyMutable and S has ORDSET --R sorted? : (((S,S) -> Boolean),%) -> Boolean @@ -67427,6 +70376,7 @@ IndexedList(S:Type, mn:Integer): Exports == Implementation where --S 1 of 1 )show IndexedMatrix +--R --R IndexedMatrix(R: Ring,mnRow: Integer,mnCol: Integer) is a domain constructor --R Abbreviation for IndexedMatrix is IMATRIX --R This constructor is not exposed in this frame. @@ -67435,30 +70385,30 @@ IndexedList(S:Type, mn:Integer): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (Integer,%) -> % ?*? : (%,R) -> % --R ?*? : (R,%) -> % ?*? : (%,%) -> % ---R ?+? : (%,%) -> % -? : % -> % ---R ?-? : (%,%) -> % antisymmetric? : % -> Boolean +--R ?**? : (%,NonNegativeInteger) -> % ?+? : (%,%) -> % +--R -? : % -> % ?-? : (%,%) -> % +--R ?/? : (%,R) -> % if R has FIELD antisymmetric? : % -> Boolean --R copy : % -> % diagonal? : % -> Boolean --R diagonalMatrix : List(%) -> % diagonalMatrix : List(R) -> % --R elt : (%,Integer,Integer,R) -> R elt : (%,Integer,Integer) -> R --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean fill! : (%,R) -> % ---R horizConcat : (%,%) -> % listOfLists : % -> List(List(R)) ---R map : (((R,R) -> R),%,%,R) -> % map : (((R,R) -> R),%,%) -> % ---R map : ((R -> R),%) -> % map! : ((R -> R),%) -> % ---R matrix : List(List(R)) -> % maxColIndex : % -> Integer ---R maxRowIndex : % -> Integer minColIndex : % -> Integer ---R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger ---R nrows : % -> NonNegativeInteger parts : % -> List(R) +--R horizConcat : (%,%) -> % latex : % -> String if R has SETCAT +--R listOfLists : % -> List(List(R)) map : (((R,R) -> R),%,%,R) -> % +--R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R map! : ((R -> R),%) -> % matrix : List(List(R)) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R parts : % -> List(R) pfaffian : % -> R if R has COMRING --R qelt : (%,Integer,Integer) -> R sample : () -> % ---R square? : % -> Boolean squareTop : % -> % ---R symmetric? : % -> Boolean transpose : % -> % ---R vertConcat : (%,%) -> % +--R setelt : (%,Integer,Integer,R) -> R square? : % -> Boolean +--R squareTop : % -> % symmetric? : % -> Boolean +--R transpose : % -> % vertConcat : (%,%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?*? : (IndexedVector(R,mnCol),%) -> IndexedVector(R,mnCol) --R ?*? : (%,IndexedVector(R,mnRow)) -> IndexedVector(R,mnRow) --R ?**? : (%,Integer) -> % if R has FIELD ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,R) -> % if R has FIELD --R ?=? : (%,%) -> Boolean if R has SETCAT --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R coerce : IndexedVector(R,mnRow) -> % @@ -67477,7 +70427,6 @@ IndexedList(S:Type, mn:Integer): Exports == Implementation where --R exquo : (%,R) -> Union(%,"failed") if R has INTDOM --R hash : % -> SingleInteger if R has SETCAT --R inverse : % -> Union(%,"failed") if R has FIELD ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R matrix : (NonNegativeInteger,NonNegativeInteger,((Integer,Integer) -> R)) -> % --R member? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT @@ -67487,7 +70436,6 @@ IndexedList(S:Type, mn:Integer): Exports == Implementation where --R new : (NonNegativeInteger,NonNegativeInteger,R) -> % --R nullSpace : % -> List(IndexedVector(R,mnRow)) if R has INTDOM --R nullity : % -> NonNegativeInteger if R has INTDOM ---R pfaffian : % -> R if R has COMRING --R qsetelt! : (%,Integer,Integer,R) -> R --R rank : % -> NonNegativeInteger if R has INTDOM --R row : (%,Integer) -> IndexedVector(R,mnCol) @@ -67496,7 +70444,6 @@ IndexedList(S:Type, mn:Integer): Exports == Implementation where --R setColumn! : (%,Integer,IndexedVector(R,mnRow)) -> % --R setRow! : (%,Integer,IndexedVector(R,mnCol)) -> % --R setelt : (%,List(Integer),List(Integer),%) -> % ---R setelt : (%,Integer,Integer,R) -> R --R setsubMatrix! : (%,Integer,Integer,%) -> % --R size? : (%,NonNegativeInteger) -> Boolean --R subMatrix : (%,Integer,Integer,Integer,Integer) -> % @@ -67707,10 +70654,12 @@ IndexedMatrix(R,mnRow,mnCol): Exports == Implementation where --R empty? : % -> Boolean entries : % -> List(S) --R eq? : (%,%) -> Boolean index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R insert : (S,%,Integer) -> % map : (((S,S) -> S),%,%) -> % ---R map : ((S -> S),%) -> % new : (NonNegativeInteger,S) -> % ---R qelt : (%,Integer) -> S reverse : % -> % ---R sample : () -> % +--R insert : (S,%,Integer) -> % latex : % -> String if S has SETCAT +--R map : (((S,S) -> S),%,%) -> % map : ((S -> S),%) -> % +--R max : (%,%) -> % if S has ORDSET min : (%,%) -> % if S has ORDSET +--R new : (NonNegativeInteger,S) -> % qelt : (%,Integer) -> S +--R reverse : % -> % sample : () -> % +--R sort : % -> % if S has ORDSET sort : (((S,S) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if S has ORDSET --R ?<=? : (%,%) -> Boolean if S has ORDSET @@ -67735,16 +70684,13 @@ IndexedMatrix(R,mnRow,mnCol): Exports == Implementation where --R find : ((S -> Boolean),%) -> Union(S,"failed") --R first : % -> S if Integer has ORDSET --R hash : % -> SingleInteger if S has SETCAT ---R latex : % -> String if S has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R map! : ((S -> S),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if S has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (S,%) -> Boolean if $ has finiteAggregate and S has SETCAT --R members : % -> List(S) if $ has finiteAggregate --R merge : (%,%) -> % if S has ORDSET --R merge : (((S,S) -> Boolean),%,%) -> % ---R min : (%,%) -> % if S has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R parts : % -> List(S) if $ has finiteAggregate @@ -67763,8 +70709,6 @@ IndexedMatrix(R,mnRow,mnCol): Exports == Implementation where --R setelt : (%,UniversalSegment(Integer),S) -> S if $ has shallowlyMutable --R setelt : (%,Integer,S) -> S if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if S has ORDSET ---R sort : (((S,S) -> Boolean),%) -> % --R sort! : % -> % if $ has shallowlyMutable and S has ORDSET --R sort! : (((S,S) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if S has ORDSET @@ -68004,13 +70948,14 @@ IndexedOneDimensionalArray(S:Type, mn:Integer): --R entries : % -> List(Character) eq? : (%,%) -> Boolean --R hash : % -> Integer index? : (Integer,%) -> Boolean --R indices : % -> List(Integer) insert : (%,%,Integer) -> % ---R leftTrim : (%,Character) -> % lowerCase : % -> % ---R lowerCase! : % -> % prefix? : (%,%) -> Boolean ---R qelt : (%,Integer) -> Character reverse : % -> % ---R rightTrim : (%,Character) -> % sample : () -> % ---R split : (%,Character) -> List(%) suffix? : (%,%) -> Boolean ---R trim : (%,CharacterClass) -> % trim : (%,Character) -> % ---R upperCase : % -> % upperCase! : % -> % +--R leftTrim : (%,CharacterClass) -> % leftTrim : (%,Character) -> % +--R lowerCase : % -> % lowerCase! : % -> % +--R prefix? : (%,%) -> Boolean qelt : (%,Integer) -> Character +--R reverse : % -> % rightTrim : (%,Character) -> % +--R sample : () -> % split : (%,Character) -> List(%) +--R suffix? : (%,%) -> Boolean trim : (%,CharacterClass) -> % +--R trim : (%,Character) -> % upperCase : % -> % +--R upperCase! : % -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ? Boolean if Character has ORDSET --R ?<=? : (%,%) -> Boolean if Character has ORDSET @@ -68038,7 +70983,6 @@ IndexedOneDimensionalArray(S:Type, mn:Integer): --R hash : % -> SingleInteger if Character has SETCAT --R insert : (Character,%,Integer) -> % --R latex : % -> String if Character has SETCAT ---R leftTrim : (%,CharacterClass) -> % --R less? : (%,NonNegativeInteger) -> Boolean --R map : (((Character,Character) -> Character),%,%) -> % --R map : ((Character -> Character),%) -> % @@ -68422,13 +71366,14 @@ first column in an array and vice versa. --R copy : % -> % elt : (%,Integer,Integer,R) -> R --R elt : (%,Integer,Integer) -> R empty : () -> % --R empty? : % -> Boolean eq? : (%,%) -> Boolean ---R fill! : (%,R) -> % map : (((R,R) -> R),%,%,R) -> % ---R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % ---R map! : ((R -> R),%) -> % maxColIndex : % -> Integer ---R maxRowIndex : % -> Integer minColIndex : % -> Integer ---R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger ---R nrows : % -> NonNegativeInteger parts : % -> List(R) ---R qelt : (%,Integer,Integer) -> R sample : () -> % +--R fill! : (%,R) -> % latex : % -> String if R has SETCAT +--R map : (((R,R) -> R),%,%,R) -> % map : (((R,R) -> R),%,%) -> % +--R map : ((R -> R),%) -> % map! : ((R -> R),%) -> % +--R maxColIndex : % -> Integer maxRowIndex : % -> Integer +--R minColIndex : % -> Integer minRowIndex : % -> Integer +--R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger +--R parts : % -> List(R) qelt : (%,Integer,Integer) -> R +--R sample : () -> % setelt : (%,Integer,Integer,R) -> R --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if R has SETCAT --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -68442,7 +71387,6 @@ first column in an array and vice versa. --R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) and R has SETCAT --R every? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R member? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R members : % -> List(R) if $ has finiteAggregate @@ -68452,7 +71396,6 @@ first column in an array and vice versa. --R row : (%,Integer) -> IndexedOneDimensionalArray(R,mnCol) --R setColumn! : (%,Integer,IndexedOneDimensionalArray(R,mnRow)) -> % --R setRow! : (%,Integer,IndexedOneDimensionalArray(R,mnCol)) -> % ---R setelt : (%,Integer,Integer,R) -> R --R size? : (%,NonNegativeInteger) -> Boolean --R ?~=? : (%,%) -> Boolean if R has SETCAT --R @@ -68569,24 +71512,26 @@ IndexedTwoDimensionalArray(R,mnRow,mnCol):Exports == Implementation where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IVECTOR --R --R------------------------------- Operations -------------------------------- +--R ?*? : (%,R) -> % if R has MONOID ?*? : (R,%) -> % if R has MONOID +--R ?+? : (%,%) -> % if R has ABELSG ?-? : (%,%) -> % if R has ABELGRP --R -? : % -> % if R has ABELGRP concat : List(%) -> % --R concat : (%,%) -> % concat : (R,%) -> % --R concat : (%,R) -> % construct : List(R) -> % ---R copy : % -> % delete : (%,Integer) -> % +--R copy : % -> % cross : (%,%) -> % if R has RING +--R delete : (%,Integer) -> % dot : (%,%) -> R if R has RING --R ?.? : (%,Integer) -> R elt : (%,Integer,R) -> R --R empty : () -> % empty? : % -> Boolean --R entries : % -> List(R) eq? : (%,%) -> Boolean --R index? : (Integer,%) -> Boolean indices : % -> List(Integer) --R insert : (%,%,Integer) -> % insert : (R,%,Integer) -> % ---R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % ---R new : (NonNegativeInteger,R) -> % qelt : (%,Integer) -> R ---R reverse : % -> % sample : () -> % +--R latex : % -> String if R has SETCAT map : (((R,R) -> R),%,%) -> % +--R map : ((R -> R),%) -> % max : (%,%) -> % if R has ORDSET +--R min : (%,%) -> % if R has ORDSET new : (NonNegativeInteger,R) -> % +--R qelt : (%,Integer) -> R reverse : % -> % +--R sample : () -> % sort : % -> % if R has ORDSET +--R sort : (((R,R) -> Boolean),%) -> % --R #? : % -> NonNegativeInteger if $ has finiteAggregate ---R ?*? : (%,R) -> % if R has MONOID ---R ?*? : (R,%) -> % if R has MONOID --R ?*? : (Integer,%) -> % if R has ABELGRP ---R ?+? : (%,%) -> % if R has ABELSG ---R ?-? : (%,%) -> % if R has ABELGRP --R ? Boolean if R has ORDSET --R ?<=? : (%,%) -> Boolean if R has ORDSET --R ?=? : (%,%) -> Boolean if R has SETCAT @@ -68598,9 +71543,7 @@ IndexedTwoDimensionalArray(R,mnRow,mnCol):Exports == Implementation where --R copyInto! : (%,%,Integer) -> % if $ has shallowlyMutable --R count : (R,%) -> NonNegativeInteger if $ has finiteAggregate and R has SETCAT --R count : ((R -> Boolean),%) -> NonNegativeInteger if $ has finiteAggregate ---R cross : (%,%) -> % if R has RING --R delete : (%,UniversalSegment(Integer)) -> % ---R dot : (%,%) -> R if R has RING --R ?.? : (%,UniversalSegment(Integer)) -> % --R entry? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R eval : (%,List(R),List(R)) -> % if R has EVALAB(R) and R has SETCAT @@ -68612,18 +71555,15 @@ IndexedTwoDimensionalArray(R,mnRow,mnCol):Exports == Implementation where --R find : ((R -> Boolean),%) -> Union(R,"failed") --R first : % -> R if Integer has ORDSET --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R length : % -> R if R has RADCAT and R has RING --R less? : (%,NonNegativeInteger) -> Boolean --R magnitude : % -> R if R has RADCAT and R has RING --R map! : ((R -> R),%) -> % if $ has shallowlyMutable ---R max : (%,%) -> % if R has ORDSET --R maxIndex : % -> Integer if Integer has ORDSET --R member? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R members : % -> List(R) if $ has finiteAggregate --R merge : (%,%) -> % if R has ORDSET --R merge : (((R,R) -> Boolean),%,%) -> % ---R min : (%,%) -> % if R has ORDSET --R minIndex : % -> Integer if Integer has ORDSET --R more? : (%,NonNegativeInteger) -> Boolean --R outerProduct : (%,%) -> Matrix(R) if R has RING @@ -68643,8 +71583,6 @@ IndexedTwoDimensionalArray(R,mnRow,mnCol):Exports == Implementation where --R setelt : (%,UniversalSegment(Integer),R) -> R if $ has shallowlyMutable --R setelt : (%,Integer,R) -> R if $ has shallowlyMutable --R size? : (%,NonNegativeInteger) -> Boolean ---R sort : % -> % if R has ORDSET ---R sort : (((R,R) -> Boolean),%) -> % --R sort! : % -> % if $ has shallowlyMutable and R has ORDSET --R sort! : (((R,R) -> Boolean),%) -> % if $ has shallowlyMutable --R sorted? : % -> Boolean if R has ORDSET @@ -68895,6 +71833,7 @@ InfiniteTuple(S:Type): Exports == Implementation where --R localParamV : % -> List(PCS) localPointV : % -> AffinePlane(K) --R multV : % -> NonNegativeInteger pointV : % -> ProjPt --R setchart! : (%,BLMET) -> BLMET setpoint! : (%,ProjPt) -> ProjPt +--R setsymbName! : (%,Symbol) -> Symbol subMultV : % -> NonNegativeInteger --R symbNameV : % -> Symbol ?~=? : (%,%) -> Boolean --R create : (ProjPt,DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K),AffinePlane(K),NonNegativeInteger,BLMET,NonNegativeInteger,DIVISOR,K,Symbol) -> % --R curveV : % -> DistributedMultivariatePolynomial([construct,QUOTEX,QUOTEY],K) @@ -68904,10 +71843,9 @@ InfiniteTuple(S:Type): Exports == Implementation where --R setlocalPoint! : (%,AffinePlane(K)) -> AffinePlane(K) --R setmult! : (%,NonNegativeInteger) -> NonNegativeInteger --R setsubmult! : (%,NonNegativeInteger) -> NonNegativeInteger ---R setsymbName! : (%,Symbol) -> Symbol ---R subMultV : % -> NonNegativeInteger --R --E 1 + )spool )lisp (bye) @@ -69162,7 +72100,8 @@ InfinitlyClosePoint(K,symb,PolyRing,E,ProjPt,PCS,Plc,DIVISOR,BLMET):Exports == I --R fullOut : % -> OutputForm fullOutput : () -> Boolean --R fullOutput : Boolean -> Boolean hash : % -> SingleInteger --R latex : % -> String multV : % -> NonNegativeInteger ---R setchart! : (%,BLMET) -> BLMET symbNameV : % -> Symbol +--R setchart! : (%,BLMET) -> BLMET setsymbName! : (%,Symbol) -> Symbol +--R subMultV : % -> NonNegativeInteger symbNameV : % -> Symbol --R ?~=? : (%,%) -> Boolean --R actualExtensionV : % -> PseudoAlgebraicClosureOfFiniteField(K) --R create : (ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField(K),DistributedMultivariatePolynomial(symb,PseudoAlgebraicClosureOfFiniteField(K))) -> % @@ -69179,8 +72118,6 @@ InfinitlyClosePoint(K,symb,PolyRing,E,ProjPt,PCS,Plc,DIVISOR,BLMET):Exports == I --R setmult! : (%,NonNegativeInteger) -> NonNegativeInteger --R setpoint! : (%,ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField(K)) -> ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField(K) --R setsubmult! : (%,NonNegativeInteger) -> NonNegativeInteger ---R setsymbName! : (%,Symbol) -> Symbol ---R subMultV : % -> NonNegativeInteger --R --E 1 @@ -69300,17 +72237,19 @@ InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField(K,symb,BLMET):_ --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IAN --R --R------------------------------- Operations -------------------------------- ---R ?*? : (PositiveInteger,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (%,%) -> % ?*? : (%,Fraction(Integer)) -> % ---R ?*? : (Fraction(Integer),%) -> % ?**? : (%,PositiveInteger) -> % ---R ?**? : (%,Integer) -> % ?+? : (%,%) -> % ---R -? : % -> % ?-? : (%,%) -> % ---R ?/? : (%,%) -> % ? Boolean ---R ?<=? : (%,%) -> Boolean ?=? : (%,%) -> Boolean ---R ?>? : (%,%) -> Boolean ?>=? : (%,%) -> Boolean ---R D : % -> % D : (%,NonNegativeInteger) -> % ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % ?^? : (%,Integer) -> % +--R ?*? : (PositiveInteger,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (%,%) -> % +--R ?*? : (%,Fraction(Integer)) -> % ?*? : (Fraction(Integer),%) -> % +--R ?**? : (%,PositiveInteger) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,Fraction(Integer)) -> % +--R ?+? : (%,%) -> % -? : % -> % +--R ?-? : (%,%) -> % ?/? : (%,%) -> % +--R ? Boolean ?<=? : (%,%) -> Boolean +--R ?=? : (%,%) -> Boolean ?>? : (%,%) -> Boolean +--R ?>=? : (%,%) -> Boolean D : % -> % +--R D : (%,NonNegativeInteger) -> % 1 : () -> % +--R 0 : () -> % ?^? : (%,PositiveInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,Integer) -> % --R associates? : (%,%) -> Boolean belong? : BasicOperator -> Boolean --R box : List(%) -> % box : % -> % --R coerce : Integer -> % coerce : % -> % @@ -69318,55 +72257,51 @@ InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField(K,symb,BLMET):_ --R coerce : % -> OutputForm convert : % -> Complex(Float) --R convert : % -> DoubleFloat convert : % -> Float --R differentiate : % -> % distribute : (%,%) -> % ---R distribute : % -> % elt : (BasicOperator,%,%) -> % ---R elt : (BasicOperator,%) -> % eval : (%,%,%) -> % ---R eval : (%,Equation(%)) -> % eval : (%,Kernel(%),%) -> % ---R factor : % -> Factored(%) freeOf? : (%,Symbol) -> Boolean ---R freeOf? : (%,%) -> Boolean gcd : (%,%) -> % ---R gcd : List(%) -> % hash : % -> SingleInteger ---R height : % -> NonNegativeInteger inv : % -> % ---R is? : (%,Symbol) -> Boolean kernel : (BasicOperator,%) -> % +--R distribute : % -> % elt : (BasicOperator,List(%)) -> % +--R elt : (BasicOperator,%,%,%) -> % elt : (BasicOperator,%,%) -> % +--R elt : (BasicOperator,%) -> % eval : (%,Symbol,(% -> %)) -> % +--R eval : (%,List(%),List(%)) -> % eval : (%,%,%) -> % +--R eval : (%,Equation(%)) -> % eval : (%,List(Equation(%))) -> % +--R eval : (%,Kernel(%),%) -> % factor : % -> Factored(%) +--R freeOf? : (%,Symbol) -> Boolean freeOf? : (%,%) -> Boolean +--R gcd : (%,%) -> % gcd : List(%) -> % +--R hash : % -> SingleInteger height : % -> NonNegativeInteger +--R inv : % -> % is? : (%,Symbol) -> Boolean +--R is? : (%,BasicOperator) -> Boolean kernel : (BasicOperator,%) -> % --R kernels : % -> List(Kernel(%)) latex : % -> String --R lcm : (%,%) -> % lcm : List(%) -> % ---R max : (%,%) -> % min : (%,%) -> % ---R norm : (%,List(Kernel(%))) -> % norm : (%,Kernel(%)) -> % ---R nthRoot : (%,Integer) -> % one? : % -> Boolean ---R paren : List(%) -> % paren : % -> % ---R prime? : % -> Boolean ?quo? : (%,%) -> % ---R recip : % -> Union(%,"failed") reduce : % -> % ---R ?rem? : (%,%) -> % retract : % -> Fraction(Integer) ---R retract : % -> Integer retract : % -> Kernel(%) ---R rootOf : Polynomial(%) -> % sample : () -> % +--R map : ((% -> %),Kernel(%)) -> % max : (%,%) -> % +--R min : (%,%) -> % norm : (%,List(Kernel(%))) -> % +--R norm : (%,Kernel(%)) -> % nthRoot : (%,Integer) -> % +--R one? : % -> Boolean paren : List(%) -> % +--R paren : % -> % prime? : % -> Boolean +--R ?quo? : (%,%) -> % recip : % -> Union(%,"failed") +--R reduce : % -> % ?rem? : (%,%) -> % +--R retract : % -> Fraction(Integer) retract : % -> Integer +--R retract : % -> Kernel(%) rootOf : Polynomial(%) -> % +--R rootsOf : Polynomial(%) -> List(%) sample : () -> % --R sizeLess? : (%,%) -> Boolean sqrt : % -> % --R squareFree : % -> Factored(%) squareFreePart : % -> % --R subst : (%,Equation(%)) -> % tower : % -> List(Kernel(%)) --R trueEqual : (%,%) -> Boolean unit? : % -> Boolean --R unitCanonical : % -> % zero? : % -> Boolean ---R zeroOf : Polynomial(%) -> % ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?**? : (%,Fraction(Integer)) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R zeroOf : Polynomial(%) -> % zerosOf : Polynomial(%) -> List(%) +--R ?~=? : (%,%) -> Boolean --R characteristic : () -> NonNegativeInteger --R coerce : SparseMultivariatePolynomial(Integer,Kernel(%)) -> % --R definingPolynomial : % -> % if $ has RING --R denom : % -> SparseMultivariatePolynomial(Integer,Kernel(%)) --R differentiate : (%,NonNegativeInteger) -> % --R divide : (%,%) -> Record(quotient: %,remainder: %) ---R elt : (BasicOperator,List(%)) -> % --R elt : (BasicOperator,%,%,%,%) -> % ---R elt : (BasicOperator,%,%,%) -> % --R euclideanSize : % -> NonNegativeInteger --R eval : (%,BasicOperator,(% -> %)) -> % --R eval : (%,BasicOperator,(List(%) -> %)) -> % --R eval : (%,List(BasicOperator),List((List(%) -> %))) -> % --R eval : (%,List(BasicOperator),List((% -> %))) -> % ---R eval : (%,Symbol,(% -> %)) -> % --R eval : (%,Symbol,(List(%) -> %)) -> % --R eval : (%,List(Symbol),List((List(%) -> %))) -> % --R eval : (%,List(Symbol),List((% -> %))) -> % ---R eval : (%,List(%),List(%)) -> % ---R eval : (%,List(Equation(%))) -> % --R eval : (%,List(Kernel(%)),List(%)) -> % --R even? : % -> Boolean if $ has RETRACT(INT) --R expressIdealMember : (List(%),%) -> Union(List(%),"failed") @@ -69374,11 +72309,9 @@ InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField(K,symb,BLMET):_ --R extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %) --R extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed") --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) ---R is? : (%,BasicOperator) -> Boolean --R kernel : (BasicOperator,List(%)) -> % --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R mainKernel : % -> Union(Kernel(%),"failed") ---R map : ((% -> %),Kernel(%)) -> % --R minPoly : Kernel(%) -> SparseUnivariatePolynomial(%) if $ has RING --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") --R norm : (SparseUnivariatePolynomial(%),List(Kernel(%))) -> SparseUnivariatePolynomial(%) @@ -69397,7 +72330,6 @@ InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField(K,symb,BLMET):_ --R retractIfCan : % -> Union(Kernel(%),"failed") --R rootOf : SparseUnivariatePolynomial(%) -> % --R rootOf : (SparseUnivariatePolynomial(%),Symbol) -> % ---R rootsOf : Polynomial(%) -> List(%) --R rootsOf : SparseUnivariatePolynomial(%) -> List(%) --R rootsOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R subst : (%,List(Kernel(%)),List(%)) -> % @@ -69406,7 +72338,6 @@ InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField(K,symb,BLMET):_ --R unitNormal : % -> Record(unit: %,canonical: %,associate: %) --R zeroOf : SparseUnivariatePolynomial(%) -> % --R zeroOf : (SparseUnivariatePolynomial(%),Symbol) -> % ---R zerosOf : Polynomial(%) -> List(%) --R zerosOf : SparseUnivariatePolynomial(%) -> List(%) --R zerosOf : (SparseUnivariatePolynomial(%),Symbol) -> List(%) --R @@ -69695,14 +72626,17 @@ InnerAlgebraicNumber(): Exports == Implementation where --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IFF --R --R------------------------------- Operations -------------------------------- +--R ?*? : (InnerPrimeField(p),%) -> % ?*? : (%,InnerPrimeField(p)) -> % --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % ---R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R ?/? : (%,InnerPrimeField(p)) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R algebraic? : % -> Boolean associates? : (%,%) -> Boolean --R basis : () -> Vector(%) coerce : InnerPrimeField(p) -> % --R coerce : Fraction(Integer) -> % coerce : % -> % @@ -69721,16 +72655,10 @@ InnerAlgebraicNumber(): Exports == Implementation where --R trace : % -> InnerPrimeField(p) transcendent? : % -> Boolean --R unit? : % -> Boolean unitCanonical : % -> % --R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (InnerPrimeField(p),%) -> % ---R ?*? : (%,InnerPrimeField(p)) -> % ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?/? : (%,InnerPrimeField(p)) -> % --R D : (%,NonNegativeInteger) -> % if InnerPrimeField(p) has FINITE --R D : % -> % if InnerPrimeField(p) has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if InnerPrimeField(p) has FINITE --R Frobenius : % -> % if InnerPrimeField(p) has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if InnerPrimeField(p) has CHARNZ or InnerPrimeField(p) has FINITE @@ -69951,17 +72879,17 @@ InnerFiniteField(p:PositiveInteger, n:PositiveInteger) == --R Issue )edit bookvol10.3.pamphlet to see algebra source code for IFAMON --R --R------------------------------- Operations -------------------------------- ---R ?*? : (E,S) -> % ?*? : (PositiveInteger,%) -> % ---R ?+? : (S,%) -> % ?+? : (%,%) -> % ---R ?=? : (%,%) -> Boolean 0 : () -> % ---R coefficient : (S,%) -> E coerce : S -> % ---R coerce : % -> OutputForm hash : % -> SingleInteger ---R latex : % -> String mapCoef : ((E -> E),%) -> % ---R mapGen : ((S -> S),%) -> % nthCoef : (%,Integer) -> E ---R nthFactor : (%,Integer) -> S retract : % -> S ---R sample : () -> % size : % -> NonNegativeInteger ---R zero? : % -> Boolean ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (E,S) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?+? : (S,%) -> % +--R ?+? : (%,%) -> % ?=? : (%,%) -> Boolean +--R 0 : () -> % coefficient : (S,%) -> E +--R coerce : S -> % coerce : % -> OutputForm +--R hash : % -> SingleInteger latex : % -> String +--R mapCoef : ((E -> E),%) -> % mapGen : ((S -> S),%) -> % +--R nthCoef : (%,Integer) -> E nthFactor : (%,Integer) -> S +--R retract : % -> S sample : () -> % +--R size : % -> NonNegativeInteger zero? : % -> Boolean +--R ?~=? : (%,%) -> Boolean --R highCommonTerms : (%,%) -> % if E has OAMON --R retractIfCan : % -> Union(S,"failed") --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -70088,14 +73016,15 @@ This is an internal type which provides an implementation of --R elt : (%,Integer,Integer,R) -> R elt : (%,Integer,Integer) -> R --R empty : () -> % empty? : % -> Boolean --R eq? : (%,%) -> Boolean fill! : (%,R) -> % ---R map : (((R,R) -> R),%,%,R) -> % map : (((R,R) -> R),%,%) -> % ---R map : ((R -> R),%) -> % map! : ((R -> R),%) -> % ---R maxColIndex : % -> Integer maxRowIndex : % -> Integer ---R minColIndex : % -> Integer minRowIndex : % -> Integer ---R ncols : % -> NonNegativeInteger nrows : % -> NonNegativeInteger ---R parts : % -> List(R) qelt : (%,Integer,Integer) -> R ---R row : (%,Integer) -> Row sample : () -> % ---R setRow! : (%,Integer,Row) -> % +--R latex : % -> String if R has SETCAT map : (((R,R) -> R),%,%,R) -> % +--R map : (((R,R) -> R),%,%) -> % map : ((R -> R),%) -> % +--R map! : ((R -> R),%) -> % maxColIndex : % -> Integer +--R maxRowIndex : % -> Integer minColIndex : % -> Integer +--R minRowIndex : % -> Integer ncols : % -> NonNegativeInteger +--R nrows : % -> NonNegativeInteger parts : % -> List(R) +--R qelt : (%,Integer,Integer) -> R row : (%,Integer) -> Row +--R sample : () -> % setColumn! : (%,Integer,Col) -> % +--R setRow! : (%,Integer,Row) -> % setelt : (%,Integer,Integer,R) -> R --R #? : % -> NonNegativeInteger if $ has finiteAggregate --R ?=? : (%,%) -> Boolean if R has SETCAT --R any? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate @@ -70108,15 +73037,12 @@ This is an internal type which provides an implementation of --R eval : (%,List(Equation(R))) -> % if R has EVALAB(R) and R has SETCAT --R every? : ((R -> Boolean),%) -> Boolean if $ has finiteAggregate --R hash : % -> SingleInteger if R has SETCAT ---R latex : % -> String if R has SETCAT --R less? : (%,NonNegativeInteger) -> Boolean --R member? : (R,%) -> Boolean if $ has finiteAggregate and R has SETCAT --R members : % -> List(R) if $ has finiteAggregate --R more? : (%,NonNegativeInteger) -> Boolean --R new : (NonNegativeInteger,NonNegativeInteger,R) -> % --R qsetelt! : (%,Integer,Integer,R) -> R ---R setColumn! : (%,Integer,Col) -> % ---R setelt : (%,Integer,Integer,R) -> R --R size? : (%,NonNegativeInteger) -> Boolean --R ?~=? : (%,%) -> Boolean if R has SETCAT --R @@ -70301,28 +73227,26 @@ InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col):_ --R --R------------------------------- Operations -------------------------------- --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % associates? : (%,%) -> Boolean ---R coerce : % -> % coerce : Integer -> % ---R coerce : % -> OutputForm complete : % -> % ---R digits : % -> Stream(Integer) extend : (%,Integer) -> % ---R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R moduloP : % -> Integer modulus : () -> Integer ---R one? : % -> Boolean order : % -> NonNegativeInteger ---R ?quo? : (%,%) -> % quotientByP : % -> % ---R recip : % -> Union(%,"failed") ?rem? : (%,%) -> % ---R sample : () -> % sizeLess? : (%,%) -> Boolean ---R sqrt : (%,Integer) -> % unit? : % -> Boolean ---R unitCanonical : % -> % zero? : % -> Boolean ---R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R associates? : (%,%) -> Boolean coerce : % -> % +--R coerce : Integer -> % coerce : % -> OutputForm +--R complete : % -> % digits : % -> Stream(Integer) +--R extend : (%,Integer) -> % gcd : List(%) -> % +--R gcd : (%,%) -> % hash : % -> SingleInteger +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % moduloP : % -> Integer +--R modulus : () -> Integer one? : % -> Boolean +--R order : % -> NonNegativeInteger ?quo? : (%,%) -> % +--R quotientByP : % -> % recip : % -> Union(%,"failed") +--R ?rem? : (%,%) -> % sample : () -> % +--R sizeLess? : (%,%) -> Boolean sqrt : (%,Integer) -> % +--R unit? : % -> Boolean unitCanonical : % -> % +--R zero? : % -> Boolean ?~=? : (%,%) -> Boolean --R approximate : (%,Integer) -> Integer --R characteristic : () -> NonNegativeInteger --R divide : (%,%) -> Record(quotient: %,remainder: %) @@ -70689,13 +73613,15 @@ InnerPAdicInteger(p,unBalanced?): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (Fraction(Integer),%) -> % ?*? : (%,Fraction(Integer)) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,Integer) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean --R D : % -> % D : (%,NonNegativeInteger) -> % ---R 1 : () -> % 0 : () -> % ---R ?^? : (%,Integer) -> % ?^? : (%,PositiveInteger) -> % +--R Frobenius : % -> % if $ has FINITE 1 : () -> % +--R 0 : () -> % ?^? : (%,Integer) -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % --R algebraic? : % -> Boolean associates? : (%,%) -> Boolean --R basis : () -> Vector(%) charthRoot : % -> % --R coerce : Fraction(Integer) -> % coerce : % -> % @@ -70705,11 +73631,12 @@ InnerPAdicInteger(p,unBalanced?): Exports == Implementation where --R differentiate : % -> % dimension : () -> CardinalNumber --R enumerate : () -> List(%) factor : % -> Factored(%) --R gcd : List(%) -> % gcd : (%,%) -> % ---R hash : % -> SingleInteger inGroundField? : % -> Boolean ---R index : PositiveInteger -> % init : () -> % ---R inv : % -> % latex : % -> String ---R lcm : List(%) -> % lcm : (%,%) -> % ---R lookup : % -> PositiveInteger norm : % -> % +--R generator : () -> % if $ has FINITE hash : % -> SingleInteger +--R inGroundField? : % -> Boolean index : PositiveInteger -> % +--R init : () -> % inv : % -> % +--R latex : % -> String lcm : List(%) -> % +--R lcm : (%,%) -> % lookup : % -> PositiveInteger +--R nextItem : % -> Union(%,"failed") norm : % -> % --R one? : % -> Boolean order : % -> PositiveInteger --R prime? : % -> Boolean primeFrobenius : % -> % --R primitive? : % -> Boolean primitiveElement : () -> % @@ -70722,11 +73649,7 @@ InnerPAdicInteger(p,unBalanced?): Exports == Implementation where --R transcendent? : % -> Boolean unit? : % -> Boolean --R unitCanonical : % -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R Frobenius : % -> % if $ has FINITE --R Frobenius : (%,NonNegativeInteger) -> % if $ has FINITE ---R ?^? : (%,NonNegativeInteger) -> % --R basis : PositiveInteger -> Vector(%) --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") @@ -70748,7 +73671,6 @@ InnerPAdicInteger(p,unBalanced?): Exports == Implementation where --R extensionDegree : () -> PositiveInteger --R factorsOfCyclicGroupSize : () -> List(Record(factor: Integer,exponent: Integer)) --R gcdPolynomial : (SparseUnivariatePolynomial(%),SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) ---R generator : () -> % if $ has FINITE --R lcmCoef : (%,%) -> Record(llcmres: %,coeff1: %,coeff2: %) --R linearAssociatedExp : (%,SparseUnivariatePolynomial(%)) -> % if $ has FINITE --R linearAssociatedLog : % -> SparseUnivariatePolynomial(%) if $ has FINITE @@ -70757,7 +73679,6 @@ InnerPAdicInteger(p,unBalanced?): Exports == Implementation where --R minimalPolynomial : % -> SparseUnivariatePolynomial(%) --R minimalPolynomial : (%,PositiveInteger) -> SparseUnivariatePolynomial(%) if $ has FINITE --R multiEuclidean : (List(%),%) -> Union(List(%),"failed") ---R nextItem : % -> Union(%,"failed") --R norm : (%,PositiveInteger) -> % if $ has FINITE --R normal? : % -> Boolean if $ has FINITE --R normalElement : () -> % if $ has FINITE @@ -71089,12 +74010,14 @@ InnerPrimeField(p:PositiveInteger): Exports == Implementation where --R------------------------------- Operations -------------------------------- --R ?*? : (Coef,%) -> % ?*? : (%,Coef) -> % --R ?*? : (%,%) -> % ?*? : (Integer,%) -> % ---R ?*? : (PositiveInteger,%) -> % ?**? : (%,PositiveInteger) -> % +--R ?*? : (NonNegativeInteger,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?**? : (%,NonNegativeInteger) -> % ?**? : (%,PositiveInteger) -> % --R ?+? : (%,%) -> % ?-? : (%,%) -> % --R -? : % -> % ?=? : (%,%) -> Boolean --R 1 : () -> % 0 : () -> % ---R ?^? : (%,PositiveInteger) -> % center : % -> Coef ---R coefficient : (%,Integer) -> Coef coerce : Integer -> % +--R ?^? : (%,NonNegativeInteger) -> % ?^? : (%,PositiveInteger) -> % +--R center : % -> Coef coefficient : (%,Integer) -> Coef +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm complete : % -> % --R degree : % -> Integer ?.? : (%,Integer) -> Coef --R extend : (%,Integer) -> % hash : % -> SingleInteger @@ -71110,8 +74033,6 @@ InnerPrimeField(p:PositiveInteger): Exports == Implementation where --R ?~=? : (%,%) -> Boolean --R ?*? : (%,Fraction(Integer)) -> % if Coef has ALGEBRA(FRAC(INT)) --R ?*? : (Fraction(Integer),%) -> % if Coef has ALGEBRA(FRAC(INT)) ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % --R ?/? : (%,Coef) -> % if Coef has FIELD --R D : % -> % if Coef has *: (Integer,Coef) -> Coef --R D : (%,NonNegativeInteger) -> % if Coef has *: (Integer,Coef) -> Coef @@ -71119,7 +74040,6 @@ InnerPrimeField(p:PositiveInteger): Exports == Implementation where --R D : (%,List(Symbol)) -> % if Coef has *: (Integer,Coef) -> Coef and Coef has PDRING(SYMBOL) --R D : (%,Symbol,NonNegativeInteger) -> % if Coef has *: (Integer,Coef) -> Coef and Coef has PDRING(SYMBOL) --R D : (%,List(Symbol),List(NonNegativeInteger)) -> % if Coef has *: (Integer,Coef) -> Coef and Coef has PDRING(SYMBOL) ---R ?^? : (%,NonNegativeInteger) -> % --R approximate : (%,Integer) -> Coef if Coef has **: (Coef,Integer) -> Coef and Coef has coerce: Symbol -> Coef --R associates? : (%,%) -> Boolean if Coef has INTDOM --R cAcos : % -> % if Coef has ALGEBRA(FRAC(INT)) @@ -71153,7 +74073,6 @@ InnerPrimeField(p:PositiveInteger): Exports == Implementation where --R characteristic : () -> NonNegativeInteger --R charthRoot : % -> Union(%,"failed") if Coef has CHARNZ --R coerce : Coef -> % if Coef has COMRING ---R coerce : % -> % if Coef has INTDOM --R coerce : Fraction(Integer) -> % if Coef has ALGEBRA(FRAC(INT)) --R differentiate : % -> % if Coef has *: (Integer,Coef) -> Coef --R differentiate : (%,NonNegativeInteger) -> % if Coef has *: (Integer,Coef) -> Coef @@ -72395,8 +75314,8 @@ InnerSparseUnivariatePowerSeries(Coef): Exports == Implementation where --S 1 of 1 )show InnerTable --R ---R InnerTable(Key: SetCategory,Entry: SetCategory,addDom) where ---R addDom: TableAggregate(Key,Entry) with +--R InnerTable(Key: SetCategory,Entry: SetCategory,addDom)where +--R addDom: TableAggregate(Key,Entry)with --R finiteAggregate is a domain constructor --R Abbreviation for InnerTable is INTABL --R This constructor is not exposed in this frame. @@ -72609,24 +75528,22 @@ InnerTable(Key: SetCategory, Entry: SetCategory, addDom):Exports == Implementati --R------------------------------- Operations -------------------------------- --R ?*? : (%,Integer) -> % ?*? : (%,Coef) -> % --R ?*? : (Coef,%) -> % ?*? : (%,%) -> % ---R ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % +--R ?*? : (Integer,%) -> % ?*? : (NonNegativeInteger,%) -> % +--R ?*? : (PositiveInteger,%) -> % ?**? : (%,NonNegativeInteger) -> % --R ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % --R ?-? : (%,%) -> % -? : % -> % --R ?=? : (%,%) -> Boolean 1 : () -> % ---R 0 : () -> % ?^? : (%,PositiveInteger) -> % ---R coefficients : % -> Stream(Coef) coerce : Integer -> % +--R 0 : () -> % ?^? : (%,NonNegativeInteger) -> % +--R ?^? : (%,PositiveInteger) -> % coefficients : % -> Stream(Coef) +--R coerce : % -> % if Coef has INTDOM coerce : Integer -> % --R coerce : % -> OutputForm hash : % -> SingleInteger --R latex : % -> String one? : % -> Boolean --R order : % -> NonNegativeInteger pole? : % -> Boolean --R recip : % -> Union(%,"failed") sample : () -> % --R series : Stream(Coef) -> % zero? : % -> Boolean --R ?~=? : (%,%) -> Boolean ---R ?*? : (NonNegativeInteger,%) -> % ---R ?**? : (%,NonNegativeInteger) -> % ---R ?^? : (%,NonNegativeInteger) -> % --R associates? : (%,%) -> Boolean if Coef has INTDOM --R characteristic : () -> NonNegativeInteger ---R coerce : % -> % if Coef has INTDOM --R exquo : (%,%) -> Union(%,"failed") if Coef has INTDOM --R order : (%,NonNegativeInteger) -> NonNegativeInteger --R subtractIfCan : (%,%) -> Union(%,"failed") @@ -72850,29 +75767,29 @@ InnerTaylorSeries(Coef): Exports == Implementation where --R --R------------------------------- Operations -------------------------------- --R #? : % -> Integer ?*? : (%,%) -> % ---R ?**? : (%,Integer) -> % ?+? : (%,%) -> % ---R ?/? : (%,%) -> % ?=? : (%,%) -> Boolean ---R 1 : () -> % 0 : () -> % ---R atom? : % -> Boolean binary : (%,List(%)) -> % ---R car : % -> % cdr : % -> % ---R coerce : % -> OutputForm convert : SExpression -> % ---R convert : % -> SExpression convert : OutputForm -> % ---R convert : DoubleFloat -> % convert : Integer -> % ---R convert : Symbol -> % convert : String -> % ---R convert : List(%) -> % declare : List(%) -> Symbol ---R destruct : % -> List(%) ?.? : (%,List(Integer)) -> % ---R ?.? : (%,Integer) -> % eq : (%,%) -> Boolean ---R expr : % -> OutputForm flatten : % -> % ---R float : % -> DoubleFloat float? : % -> Boolean ---R hash : % -> SingleInteger integer : % -> Integer ---R integer? : % -> Boolean interpret : % -> Any ---R lambda : (%,List(Symbol)) -> % latex : % -> String ---R list? : % -> Boolean null? : % -> Boolean ---R pair? : % -> Boolean parse : String -> % ---R string : % -> String string? : % -> Boolean ---R symbol : % -> Symbol symbol? : % -> Boolean ---R unparse : % -> String ?~=? : (%,%) -> Boolean ---R ?**? : (%,NonNegativeInteger) -> % +--R ?**? : (%,Integer) -> % ?**? : (%,NonNegativeInteger) -> % +--R ?+? : (%,%) -> % ?/? : (%,%) -> % +--R ?=? : (%,%) -> Boolean 1 : () -> % +--R 0 : () -> % atom? : % -> Boolean +--R binary : (%,List(%)) -> % car : % -> % +--R cdr : % -> % coerce : % -> OutputForm +--R convert : SExpression -> % convert : % -> SExpression +--R convert : OutputForm -> % convert : DoubleFloat -> % +--R convert : Integer -> % convert : Symbol -> % +--R convert : String -> % convert : List(%) -> % +--R declare : List(%) -> Symbol destruct : % -> List(%) +--R ?.? : (%,List(Integer)) -> % ?.? : (%,Integer) -> % +--R eq : (%,%) -> Boolean expr : % -> OutputForm +--R flatten : % -> % float : % -> DoubleFloat +--R float? : % -> Boolean hash : % -> SingleInteger +--R integer : % -> Integer integer? : % -> Boolean +--R interpret : % -> Any lambda : (%,List(Symbol)) -> % +--R latex : % -> String list? : % -> Boolean +--R null? : % -> Boolean pair? : % -> Boolean +--R parse : String -> % string : % -> String +--R string? : % -> Boolean symbol : % -> Symbol +--R symbol? : % -> Boolean unparse : % -> String +--R ?~=? : (%,%) -> Boolean --R compile : (Symbol,List(%)) -> Symbol --R function : (%,List(Symbol),Symbol) -> % --R @@ -73148,7 +76065,8 @@ Common Lisp and is not defined in Common Lisp. )set message test on )set message auto off )clear all ---S 1 of 42 + +--S 1 of 43 2**(5678 - 4856 + 2 * 17) --R --R @@ -73160,7 +76078,7 @@ Common Lisp and is not defined in Common Lisp. --R Type: PositiveInteger --E 1 ---S 2 of 42 +--S 2 of 43 x := -101 --R --R @@ -73168,7 +76086,7 @@ x := -101 --R Type: Integer --E 2 ---S 3 of 42 +--S 3 of 43 abs(x) --R --R @@ -73176,7 +76094,7 @@ abs(x) --R Type: PositiveInteger --E 3 ---S 4 of 42 +--S 4 of 43 sign(x) --R --R @@ -73184,7 +76102,7 @@ sign(x) --R Type: Integer --E 4 ---S 5 of 42 +--S 5 of 43 x < 0 --R --R @@ -73192,7 +76110,7 @@ x < 0 --R Type: Boolean --E 5 ---S 6 of 42 +--S 6 of 43 x <= -1 --R --R @@ -73200,7 +76118,7 @@ x <= -1 --R Type: Boolean --E 6 ---S 7 of 42 +--S 7 of 43 negative?(x) --R --R @@ -73208,7 +76126,7 @@ negative?(x) --R Type: Boolean --E 7 ---S 8 of 42 +--S 8 of 43 x > 0 --R --R @@ -73216,7 +76134,7 @@ x > 0 --R Type: Boolean --E 8 ---S 9 of 42 +--S 9 of 43 x >= 1 --R --R @@ -73224,7 +76142,7 @@ x >= 1 --R Type: Boolean --E 9 ---S 10 of 42 +--S 10 of 43 positive?(x) --R --R @@ -73232,7 +76150,7 @@ positive?(x) --R Type: Boolean --E 10 ---S 11 of 42 +--S 11 of 43 zero?(x) --R --R @@ -73240,7 +76158,7 @@ zero?(x) --R Type: Boolean --E 11 ---S 12 of 42 +--S 12 of 43 one?(x) --R --R @@ -73248,7 +76166,7 @@ one?(x) --R Type: Boolean --E 12 ---S 13 of 42 +--S 13 of 43 (x = -101)@Boolean --R --R @@ -73256,7 +76174,7 @@ one?(x) --R Type: Boolean --E 13 ---S 14 of 42 +--S 14 of 43 odd?(x) --R --R @@ -73264,7 +76182,7 @@ odd?(x) --R Type: Boolean --E 14 ---S 15 of 42 +--S 15 of 43 even?(x) --R --R @@ -73272,7 +76190,7 @@ even?(x) --R Type: Boolean --E 15 ---S 16 of 42 +--S 16 of 43 gcd(56788,43688) --R --R @@ -73280,7 +76198,7 @@ gcd(56788,43688) --R Type: PositiveInteger --E 16 ---S 17 of 42 +--S 17 of 43 lcm(56788,43688) --R --R @@ -73288,7 +76206,7 @@ lcm(56788,43688) --R Type: PositiveInteger --E 17 ---S 18 of 42 +--S 18 of 43 max(678,567) --R --R @@ -73296,7 +76214,7 @@ max(678,567) --R Type: PositiveInteger --E 18 ---S 19 of 42 +--S 19 of 43 min(678,567) --R --R @@ -73304,7 +76222,7 @@ min(678,567) --R Type: PositiveInteger --E 19 ---S 20 of 42 +--S 20 of 43 reduce(max,[2,45,-89,78,100,-45]) --R --R @@ -73312,7 +76230,7 @@ reduce(max,[2,45,-89,78,100,-45]) --R Type: PositiveInteger --E 20 ---S 21 of 42 +--S 21 of 43 reduce(min,[2,45,-89,78,100,-45]) --R --R @@ -73320,7 +76238,7 @@ reduce(min,[2,45,-89,78,100,-45]) --R Type: Integer --E 21 ---S 22 of 42 +--S 22 of 43 reduce(gcd,[2,45,-89,78,100,-45]) --R --R @@ -73328,7 +76246,7 @@ reduce(gcd,[2,45,-89,78,100,-45]) --R Type: PositiveInteger --E 22 ---S 23 of 42 +--S 23 of 43 reduce(lcm,[2,45,-89,78,100,-45]) --R --R @@ -73336,7 +76254,7 @@ reduce(lcm,[2,45,-89,78,100,-45]) --R Type: PositiveInteger --E 23 ---S 24 of 42 +--S 24 of 43 13 / 4 --R --R @@ -73346,7 +76264,7 @@ reduce(lcm,[2,45,-89,78,100,-45]) --R Type: Fraction(Integer) --E 24 ---S 25 of 42 +--S 25 of 43 13 quo 4 --R --