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last edited 8 years ago by test1 |
Edit detail for Cartesian Product revision 2 of 4
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Editor: Bill Page
Time: 2008/05/20 18:51:38 GMT-7 |
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added:
From BillPage Tue May 20 18:49:34 -0700 2008
From: Bill Page
Date: Tue, 20 May 2008 18:49:34 -0700
Subject: break
Message-ID: <20080520184934-0700@axiom-wiki.newsynthesis.org>
\begin{axiom}
for i in 1.. repeat 0
\end{axiom}
++ This domain implements cartesian product\begin{axiom} )show Product \end{axiom}
\begin{spad} )abbrev domain PRODUCT Product Product (A:SetCategory,B:SetCategory) : C == T where C == SetCategory with if A has Finite and B has Finite then Finite if A has Monoid and B has Monoid then Monoid if A has AbelianMonoid and B has AbelianMonoid then AbelianMonoid if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid then CancellationAbelianMonoid if A has Group and B has Group then Group if A has AbelianGroup and B has AbelianGroup then AbelianGroup if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup if A has OrderedSet and B has OrderedSet then OrderedSet
makeprod : (A,B) -> % ++ makeprod(a,b) \undocumented selectfirst : % -> A ++ selectfirst(x) \undocumented selectsecond : % -> B ++ selectsecond(x) \undocumented
T == add
--representations Rep := Record(acomp:A,bcomp:B)
--declarations x,y: % i: NonNegativeInteger p: NonNegativeInteger a: A b: B d: Integer
--define coerce(x):OutputForm == paren [(x.acomp)::OutputForm, (x.bcomp)::OutputForm] x=y == x.acomp = y.acomp => x.bcomp = y.bcomp false makeprod(a:A,b:B) :% == [a,b]
selectfirst(x:%) : A == x.acomp
selectsecond (x:%) : B == x.bcomp
if A has Monoid and B has Monoid then 1 == [1$A,1$B] x * y == [x.acomp y.acomp,x.bcomp y.bcomp] x ** p == [x.acomp p ,x.bcomp p]
if A has Finite and B has Finite then size == size$A * size$B index(n) == [index((((n::Integer-1) quo size$B )+1)::PositiveInteger)$A, index((((n::Integer-1) rem size$B )+1)::PositiveInteger)$B] random() == [random()$A,random()$B] lookup(x) == ((lookup(x.acomp)$A::Integer-1) size$B::Integer + lookup(x.bcomp)$B::Integer)::PositiveInteger hash(x) == hash(x.acomp)$A size$B::SingleInteger + hash(x.bcomp)$B
if A has Group and B has Group then inv(x) == [inv(x.acomp),inv(x.bcomp)]
if A has AbelianMonoid and B has AbelianMonoid then 0 == [0$A,0$B]
x + y == [x.acomp + y.acomp,x.bcomp + y.bcomp]
c:NonNegativeInteger * x == [c x.acomp,cx.bcomp]
if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid then subtractIfCan(x, y) : Union(%,"failed") == (na:= subtractIfCan(x.acomp, y.acomp)) case "failed" => "failed" (nb:= subtractIfCan(x.bcomp, y.bcomp)) case "failed" => "failed" [na::A,nb::B]
if A has AbelianGroup and B has AbelianGroup then - x == [- x.acomp,-x.bcomp] (x - y):% == [x.acomp - y.acomp,x.bcomp - y.bcomp] d * x == [d x.acomp,d x.bcomp]
if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then sup(x,y) == [sup(x.acomp,y.acomp),sup(x.bcomp,y.bcomp)]
if A has OrderedSet and B has OrderedSet then x < y == xa:= x.acomp ; ya:= y.acomp xa < ya => true xb:= x.bcomp ; yb:= y.bcomp xa = ya => (xb < yb) false
- -- coerce(x:%):Symbol ==
- PrintableForm() -- formList([x.acomp::Expression,x.bcomp::Expression])$PrintableForm \end{spad}
\begin{axiom} X:=Product(IntegerMod 3,Set PF 3) size()$X [index(i)$X for i in 1..size()$X::PositiveInteger] reduce(_and,[(lookup(index(i)$X)=i)::Boolean for i in 1..size()$X::PositiveInteger]) lookup(makeprod(2,[2])$X) [random()$X for i in 1..5] \end{axiom}
Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8667117888879070386-25px.axm KilledChecking for foreign routines FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu" spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so" foreign routines found openServer result -2 FriCAS Computer Algebra System Version: FriCAS 1.3.12 built with sbcl 2.2.9.debian Timestamp: Sat 7 Jun 23:54:49 CEST 2025 ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave FriCAS and return to shell. -----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> )show Product
Product(A: SetCategory,B: SetCategory) is a domain constructor Abbreviation for Product is PRODUCT This constructor is not exposed in this frame. ------------------------------- Operations --------------------------------
?=? : (%, %) -> Boolean coerce : % -> OutputForm construct : (A, B) -> % first : % -> A latex : % -> String second : % -> B ?~=? : (%, %) -> Boolean ?? : (Integer, %) -> % if A has ABELGRP and B has ABELGRP ?? : (NonNegativeInteger, %) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?? : (PositiveInteger, %) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?? : (%, %) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID ?+? : (%, %) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?-? : (%, %) -> % if A has ABELGRP and B has ABELGRP -? : % -> % if A has ABELGRP and B has ABELGRP ?/? : (%, %) -> % if A has GROUP and B has GROUP ?<? : (%, %) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?<=? : (%, %) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?>? : (%, %) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?>=? : (%, %) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET 1 : () -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID 0 : () -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?^? : (%, Integer) -> % if A has GROUP and B has GROUP ?^? : (%, NonNegativeInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID ?^? : (%, PositiveInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID commutator : (%, %) -> % if A has GROUP and B has GROUP conjugate : (%, %) -> % if A has GROUP and B has GROUP convert : % -> InputForm if A has FINITE and B has FINITE enumerate : () -> List(%) if A has FINITE and B has FINITE hash : % -> SingleInteger if A has FINITE and B has FINITE or A has HASHABL and B has HASHABL hashUpdate! : (HashState, %) -> HashState if A has FINITE and B has FINITE or A has HASHABL and B has HASHABL index : PositiveInteger -> % if A has FINITE and B has FINITE inf : (%, %) -> % if A has OAMONS and B has OAMONS inv : % -> % if A has GROUP and B has GROUP leftPower : (%, NonNegativeInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID leftPower : (%, PositiveInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID leftRecip : % -> Union(%,"failed") if A has GROUP and B has GROUP or A has MONOID and B has MONOID lookup : % -> PositiveInteger if A has FINITE and B has FINITE max : (%, %) -> % if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET min : (%, %) -> % if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET one? : % -> Boolean if A has GROUP and B has GROUP or A has MONOID and B has MONOID opposite? : (%, %) -> Boolean if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS random : () -> % if A has FINITE and B has FINITE recip : % -> Union(%,"failed") if A has GROUP and B has GROUP or A has MONOID and B has MONOID rightPower : (%, NonNegativeInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID rightPower : (%, PositiveInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID rightRecip : % -> Union(%,"failed") if A has GROUP and B has GROUP or A has MONOID and B has MONOID sample : () -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has GROUP and B has GROUP or A has MONOID and B has MONOID or A has OAMONS and B has OAMONS size : () -> NonNegativeInteger if A has FINITE and B has FINITE smaller? : (%, %) -> Boolean if A has COMPAR and B has COMPAR or A has FINITE and B has FINITE or A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET subtractIfCan : (%, %) -> Union(%,"failed") if A has ABELGRP and B has ABELGRP or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS sup : (%, %) -> % if A has OAMONS and B has OAMONS zero? : % -> Boolean if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS
(1) -> <spad> )abbrev domain PRODUCT Product Product (A:SetCategory,B:SetCategory) : C == T where C == SetCategory with if A has Finite and B has Finite then Finite if A has Monoid and B has Monoid then Monoid if A has AbelianMonoid and B has AbelianMonoid then AbelianMonoid if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid then CancellationAbelianMonoid if A has Group and B has Group then Group if A has AbelianGroup and B has AbelianGroup then AbelianGroup if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup if A has OrderedSet and B has OrderedSet then OrderedSet
makeprod : (A,B) -> % ++ makeprod(a,b) \undocumented selectfirst : % -> A ++ selectfirst(x) \undocumented selectsecond : % -> B ++ selectsecond(x) \undocumented
T == add
--representations Rep := Record(acomp:A,bcomp:B)
--declarations x,y: % i: NonNegativeInteger p: NonNegativeInteger a: A b: B d: Integer
--define coerce(x):OutputForm == paren [(x.acomp)::OutputForm, (x.bcomp)::OutputForm] x=y == x.acomp = y.acomp => x.bcomp = y.bcomp false makeprod(a:A,b:B) :% == [a,b]
selectfirst(x:%) : A == x.acomp
selectsecond (x:%) : B == x.bcomp
if A has Monoid and B has Monoid then 1 == [1$A,1$B] x * y == [x.acomp y.acomp,x.bcomp y.bcomp] x ** p == [x.acomp p ,x.bcomp p]
if A has Finite and B has Finite then size == size$A * size$B index(n) == [index((((n::Integer-1) quo size$B )+1)::PositiveInteger)$A, index((((n::Integer-1) rem size$B )+1)::PositiveInteger)$B] random() == [random()$A,random()$B] lookup(x) == ((lookup(x.acomp)$A::Integer-1) size$B::Integer + lookup(x.bcomp)$B::Integer)::PositiveInteger hash(x) == hash(x.acomp)$A size$B::SingleInteger + hash(x.bcomp)$B
if A has Group and B has Group then inv(x) == [inv(x.acomp),inv(x.bcomp)]
if A has AbelianMonoid and B has AbelianMonoid then 0 == [0$A,0$B]
x + y == [x.acomp + y.acomp,x.bcomp + y.bcomp]
c:NonNegativeInteger * x == [c x.acomp,cx.bcomp]
if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid then subtractIfCan(x, y) : Union(%,"failed") == (na:= subtractIfCan(x.acomp, y.acomp)) case "failed" => "failed" (nb:= subtractIfCan(x.bcomp, y.bcomp)) case "failed" => "failed" [na::A,nb::B]
if A has AbelianGroup and B has AbelianGroup then - x == [- x.acomp,-x.bcomp] (x - y):% == [x.acomp - y.acomp,x.bcomp - y.bcomp] d * x == [d x.acomp,d x.bcomp]
if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then sup(x,y) == [sup(x.acomp,y.acomp),sup(x.bcomp,y.bcomp)]
if A has OrderedSet and B has OrderedSet then x < y == xa:= x.acomp ; ya:= y.acomp xa < ya => true xb:= x.bcomp ; yb:= y.bcomp xa = ya => (xb < yb) false
- -- coerce(x:%):Symbol ==
- PrintableForm()
-- formList([x.acomp::Expression,x.bcomp::Expression])$PrintableForm</spad>
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8594176356433254296-25px002.spad
using old system compiler.
PRODUCT abbreviates domain Product
------------------------------------------------------------------------
initializing NRLIB PRODUCT for Product
compiling into NRLIB PRODUCT
compiling exported coerce : % -> OutputForm
Time: 0 SEC.
compiling exported = : (%,%) -> Boolean Time: 0 SEC.
compiling exported makeprod : (A,B) -> % PRODUCT;makeprod;AB%;3 is replaced by CONS Time: 0 SEC.
compiling exported selectfirst : % -> A PRODUCT;selectfirst;%A;4 is replaced by QCAR Time: 0 SEC.
compiling exported selectsecond : % -> B PRODUCT;selectsecond;%B;5 is replaced by QCDR Time: 0 SEC.
**** Domain: A already in scope augmenting A: (Monoid) **** Domain: B already in scope augmenting B: (Monoid) compiling exported One : () -> % Time: 0 SEC.
compiling exported * : (%,%) -> % Time: 0 SEC.
*********** USER ERROR ****** available signatures for : NONE NEED : (%,(|NonNegativeInteger|)) -> ? ** comp fails at level 1 with expression: ** ((DEF ( |x| |p|) (NIL NIL NIL) (|construct| ( (|x| |acomp|) |p|) ( (|x| |bcomp|) |p|)))) **** level 1 ** x:= (DEF ( x p) (NIL NIL NIL) (construct ( (x acomp) p) ( (x bcomp) p))) m:= $EmptyMode f:= ((((B # #) (~= #) (= #) (|coerce| #) ...)))
>> Apparent user error: unspecified error
(1) -> X:=Product(IntegerMod 3,Set PF 3)
$$ Product \left( {{IntegerMod \left( {3} \right)}, \: {Set \left( {{PrimeField \left( {3} \right)}} \right)}} \right) \leqno(1) $$
Type: Type size()$X
$$ 24 \leqno(2) $$
Type: NonNegativeInteger [index(i)$X for i in 1..size()$X::PositiveInteger]
Compiling function G15 with type NonNegativeInteger -> Boolean
$$ \left[ {\left[ 1, \: {\left\{ \right\}} \right]}, \: {\left[ 1, \: {\left\{ 1 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 2 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 1, \: 2 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 0 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 1, \: 0 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 2, \: 0 \right\}} \right]}, \: {\left[ 1, \: {\left\{ 1, \: 2, \: 0 \right\}} \right]}, \: {\left[ 2, \: {\left\{ \right\}} \right]}, \: {\left[ 2, \: {\left\{ 1 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 2 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 1, \: 2 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 0 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 1, \: 0 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 2, \: 0 \right\}} \right]}, \: {\left[ 2, \: {\left\{ 1, \: 2, \: 0 \right\}} \right]}, \: {\left[ 0, \: {\left\{ \right\}} \right]}, \: {\left[ 0, \: {\left\{ 1 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 2 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 1, \: 2 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 0 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 1, \: 0 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 2, \: 0 \right\}} \right]}, \: {\left[ 0, \: {\left\{ 1, \: 2, \: 0 \right\}} \right]} \right] \leqno(3) $$
Type: List(Product(IntegerMod(3),Set(PrimeField(3)))) reduce(_and,[(lookup(index(i)$X)=i)::Boolean for i in 1..size()$X::PositiveInteger])
$$ true \leqno(4) $$
Type: Boolean lookup(makeprod(2,[2])$X)
The function makeprod is not implemented in Product(IntegerMod(3), Set(PrimeField(3))) .
(5) -> for i in 1.. repeat 0
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