Using OpenAxiom
\begin{axiom}
)version
\end{axiom}
\begin{spad}
)abbrev domain MORPH Morphism
T ==> SetCategory
Morphism(source:T, target:T): with
domain:%->T
codomain:%->T
coerce:(source->target) -> %
coerce:%->(source->target)
coerce:%->OutputForm
elt:(%,source)->target
== (source->target) add
-- Rep == (source->target)
Rep == SExpression
domain(p:%):T == source
codomain(p:%):T == target
coerce(r:(source->target)):% == per(r pretend SExpression)
coerce(p:%):(source->target) == (rep(p) pretend (source->target))
coerce(p:%):OutputForm == p pretend OutputForm
elt(f,x) == (rep(f) pretend (source->target))(x)
\end{spad}
The naive representation of Rep == (source->target) gives the
following error:
Error: Caught fatal error [memory may be damaged]
...
Error signalled by EVALHOOK.
Backtrace: system:universal-error-handler > evalhook > lambda > lambda-closure > block > apply > APPLY
But using SExpression seems to work.
\begin{axiom}
)show Morphism(Float,Integer)
f:Morphism(Float,Integer)
f:=(x:Float):Integer +->wholePart(x)
f(1.1)
(f::(Float->Integer))(1.1)
domain f
codomain f
(f::(domain(f)->codomain(f)))(1.1)
g:Morphism(Integer,String):=(i:Integer):String +-> i::String
g(1985)
\end{axiom}
Next let's study how this sort of thing is accomplished in a similar domain.
\begin{spad}
)abbrev domain HOM Hom
++ Based on Automorphism by: Manuel Bronstein
++ Description:
++ Hom(R,S) is the domain of homomorphisms - Mapping(S,R).
Hom(R:SetCategory?,S:SetCategory?): Join(SetCategory?, Eltable(R, S)) with
morphism: (R -> S) -> %
++ morphism(f) returns the non-invertible morphism given by f.
== add
-- Why doesn't the obvious representation just (R->S) work?
Rep == SExpression?
morphism(f) == per(f pretend SExpression?)
elt(f, r) == (rep(f) pretend (R->S))(r)
--Rep == List(R -> S)
--morphism(f) == per([f #1]?)
--elt(f, r) == (rep(f)(1))(r)
--Rep := ((R, Integer) -> S)
--morphism(f) == (#2; f(#1))
--elt(f, r) == (g := f pretend ((R, Integer) -> S); g(r, 1))
f = g == EQ(f, g)$Lisp
coerce(f:%):OutputForm? == message("theMap ...")
\end{spad}
\begin{axiom}
p:=morphism((x:Integer):Float +-> x+1.0)$Hom(Integer,Float)
p(2)
p(-1)
\end{axiom}
Some or all expressions may not have rendered properly,
because Axiom returned the following error:
Error: export FRICAS=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13; 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/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8560717210793468152-25px.axm
/bin/sh: /usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13/bin/FRICASsys: not found
Some or all expressions may not have rendered properly,
because Latex returned the following error:
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LaTeX Warning: Characters dropped after `\end{axiom}' on input line 155.
You can't use `macro parameter character #' in horizontal mode.
l.171 --morphism(f) == per([f #
1])
You can't use `macro parameter character #' in horizontal mode.
l.175 ... --morphism(f) == (#
2; f(#1))
You can't use `macro parameter character #' in horizontal mode.
l.175 ...rphism(f) == (#2; f(#
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(see the transcript file for additional information)
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