PositiveInteger and NonNegativeInteger are subdomains of Integer.
Domain is a domain whose values are all domains.
Categories are values of the domain SubDomain Domain.
For example:
\begin{axiom}
Integer
Ring
\end{axiom}
Unfortunately the original Axiom library did not include
definitions of these domains.
\begin{axiom}
[Integer,PrimeField(3)]
\end{axiom}
\begin{axiom}
[Ring,Field]
\end{axiom}
OpenAxiom has implemented Domain and replaced
SubDomain Domain with Category. Here we try to provide
something similar for FriCAS.
\begin{spad}
)abbrev domain DOMAIN Domain
Domain():SetCategory
== add
coerce(x:%):OutputForm ==
n:SExpression:=devaluate(x)$Lisp
#n=1 => car(n) pretend OutputForm
n pretend OutputForm
x = y ==
devaluate(x)$Lisp = devaluate(y)$Lisp
\end{spad}
\begin{spad}
)abbrev domain SUBDOM SubDomain
SubDomain(D:Domain):SetCategory
== add
coerce(x:%):OutputForm ==
n:SExpression:=devaluate(x)$Lisp
#n=1 => car(n) pretend OutputForm
n pretend OutputForm
x = y ==
devaluate(x)$Lisp = devaluate(y)$Lisp
\end{spad}
\begin{axiom}
[Integer,PrimeField(3)]
%.1
\end{axiom}
\begin{axiom}
[Ring,Field,IntegerNumberSystem]
\end{axiom}
Also we would like to be able to coerce both Domain and
SubDomain to InputForm
\begin{axiom}
Integer::InputForm
\end{axiom}
Coercions like these are a special case in the Axiom
interpreter. Here we modify the treatment for InputForm
in references to '(InputForm) below.
From: interp/i-coerce.boot
\begin{boot}
coerceInteractive(triple,t2) ==
-- bind flag for recording/reporting instantiations
-- (see recordInstantiation)
t1 := objMode triple
val := objVal triple
null(t2) or t2 = $EmptyMode => NIL
t2 = t1 => triple
t2 = '$NoValueMode => objNew(val,t2)
if t2 is [SubDomain,x,.] then t2:= x
-- JHD added category Aug 1996 for BasicMath
-- WSP modified for coercion to InputForm Sept 2008
t1 in ((Category) (Mode) (Domain) (SubDomain (Domain))) =>
t2 = $OutputForm => objNew(val,t2)
t2 = '(InputForm) => objNewWrap(val,t2)
NIL
t1 = '$NoValueMode =>
if $compilingMap then clearDependentMaps($mapName,nil)
throwKeyedMsg("S2IC0009",[t2,$mapName])
$insideCoerceInteractive: local := true
expr2 := EQUAL(t2,$OutputForm)
if expr2 then startTimingProcess 'print
else startTimingProcess 'coercion
-- next 2 lines handle cases like '"failed"
result :=
expr2 and (t1 = val) => objNew(val,$OutputForm)
expr2 and t1 is [Variable,var] => objNewWrap(var,$OutputForm)
t2 = (InputForm) => objNewWrap(val,t2)
coerceInt0(triple,t2)
if expr2 then stopTimingProcess 'print
else stopTimingProcess 'coercion
result
\end{boot}
Now we get
\begin{axiom}
a:=Integer::InputForm
interpret(a)$InputForm
unparse(a)$InputForm
b:=PrimeField(11)::InputForm
interpret(b)$InputForm
unparse(b)$InputForm
c:=DirectProduct(3,Fraction Integer)::InputForm
interpret(c)$InputForm
unparse(c)$InputForm
d:=Ring::InputForm
interpret(d)$InputForm
unparse(d)$InputForm
\end{axiom}
Here is a useful function for parsing input strings
\begin{axiom}
parse(s:String):InputForm == ncParseFromString(s)$Lisp pretend InputForm
parse("Integer")
\end{axiom}
There are still problems problems with values of domains in types
\begin{axiom}
)set output tex off
)set output algebra on
\end{axiom}
\begin{axiom}
CliffordAlgebra(4,Fraction Integer, quadraticForm matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]])
%::InputForm
parse("CliffordAlgebra(4,Fraction Integer, quadraticForm matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]])")
interpret(%)
%::InputForm
\end{axiom}
We can easily implement similar functionality at the Spad
(algebra library) level by extending the InputFormFunction1
package. But limitation of FriCAS that prevent us from passing
domain-values as parameters of functions mean we must do it
this way.
\begin{spad}
)abbrev package INFORM1 InputFormFunctions1
--)boot $noSubsumption := false
++ Tools for manipulating input forms
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 19 April 1991
++ Description: Tools for manipulating input forms.
InputFormFunctions1(R:Type):with
packageCall: Symbol -> InputForm
++ packageCall(f) returns the input form corresponding to f$R.
interpret : InputForm -> R
++ interpret(f) passes f to the interpreter, and transforms
++ the result into an object of type R.
domainToInputForm: () -> InputForm
++ convert a domain to InputForm
== add
domainToInputForm():InputForm == devaluate(R)$Lisp
packageCall name ==
convert([convert($elt), domainToInputForm(),
convert name]$List(InputForm))@InputForm
interpret form ==
retract(interpret(convert([convert(@), form,
domainToInputForm()]$List(InputForm))@InputForm)$InputForm)$AnyFunctions1(R)
\end{spad}
\begin{axiom}
domainToInputForm()$INFORM1(PrimeField 7)
interpret(parse("x^2+1"))$INFORM1(Polynomial Float)
\end{axiom}
Some or all expressions may not have rendered properly,
because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4337775783186056780-25px.axm
/bin/sh: /usr/local/lib/fricas/target/x86_64-unknown-linux/bin/AXIOMsys: not found
Some or all expressions may not have rendered properly,
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l.137 #
n=1 => car(n) pretend OutputForm
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$
l.141 \end{spad}
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\T1/cmr/m/n/12 )abbrev do-main DO-MAIN Do-main Do-main():SetCategory == add coe
rce(x:n:SExpression:=devaluate(x)$\OML/cmm/m/it/12 Lispn \OT1/cmr/m/n/12 = 1 =\
OML/cmm/m/it/12 > car\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n\OT1/cmr/m/n/12 )\OML/c
mm/m/it/12 pretendOutputFormnpretendOutputFormx \OT1/cmr/m/n/12 =
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l.148 #
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l.152 \end{spad}
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See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
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l.163 \begin{boot}
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l.195 \end{boot}
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\T1/cmr/m/n/12 expr2 := EQUAL(t2,$\OML/cmm/m/it/12 OutputForm\OT1/cmr/m/n/12 )\
OML/cmm/m/it/12 ifexpr\OT1/cmr/m/n/12 2\OML/cmm/m/it/12 thenstartTimingProcess[
]printelsestartTimingProcess[]coercion \OMS/cmsy/m/n/12 ^^@ ^^@\OML/cmm/m/it/12
next\OT1/cmr/m/n/12 2\OML/cmm/m/it/12 lineshandlecaseslike[]\OT1/cmr/m/n/12 "\
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:=
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\T1/cmr/m/n/12 f$\OML/cmm/m/it/12 R:interpret \OT1/cmr/m/n/12 : \OML/cmm/m/it/1
2 InputForm\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > R \OT1/cmr/m/n/12 + +\OML/cm
m/m/it/12 interpret\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 f\OT1/cmr/m/n/12 )\OML/cmm
/m/it/12 passesftotheinterpreter; andtransforms \OT1/cmr/m/n/12 + +\OML/cmm/m/i
t/12 theresultintoanobjectoftypeR:domainToInputForm \OT1/cmr/m/n/12 :
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[]\T1/cmr/m/n/12 retract(interpret(convert([convert(@), form, domainT
oInputForm()]$\OML/cmm/m/it/12 List\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 InputForm\
OT1/cmr/m/n/12 ))@\OML/cmm/m/it/12 InputForm\OT1/cmr/m/n/12 )$\T1/cmr/m/n/12 In
putForm)$\OML/cmm/m/it/12 AnyFunctions\OT1/cmr/m/n/12 1(\OML/cmm/m/it/12 R\OT1/
cmr/m/n/12 )$
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