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	Axiom Algebra
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SubDomain

Edit detail for SubDomain revision 7 of 7

	1 2 3 4 5 6 7
Editor: Bill Page Time: 2008/09/20 17:12:46 GMT-7
Note: problems with values of domains in types

removed:
-K := Fraction Integer
-g := matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]]
-D := CliffordAlgebra(4,K, quadraticForm g)
-D::InputForm
-unparse %

added:

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}

changed:
-package.
package. But limitation of FriCAS that prevent us from passing
domain-values as parameters of functions mean we must do it
this way.

removed:
-parse(s:String):InputForm == ncParseFromString(s)$Lisp pretend InputForm

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}


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l.137       #
             n=1 => car(n) pretend OutputForm
 Missing $ inserted.
<inserted text> 
                $
l.141 \end{spad}
                \newpage
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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}
                \newpage
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l.195 \end{boot}
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\T1/cmr/m/n/12 expr2 := EQUAL(t2,$\OML/cmm/m/it/12 OutputForm\OT1/cmr/m/n/12 )\
OML/cmm/m/it/12 ifexpr\OT1/cmr/m/n/12 2\OML/cmm/m/it/12 thenstartTimingProcess[
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OML/cmm/m/it/12 failed\OT1/cmr/m/n/12 "\OML/cmm/m/it/12 result \OT1/cmr/m/n/12 
:=
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\T1/cmr/m/n/12 f$\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
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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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