[Axiom-developer] interesting free algebra text

[root]

http://www.johnwoodwark.com/inge/docs/icg.pdf

for those who want to carve out an interesting algebra area
(similar to cliff's units work). most interesting is the whole
question of intervals. ideally we should be able to define
functions with the proviso that it is only defined over a
given interval, as in:

     sin(x) suchthat x in [0.3..0.4]

there is a suchthat domain that i originally wrote to address
exactly this. it was part of the "proviso" effort. provisos are
the things you see trailing expressions like

    1/x provided x <> 0

and are a general purpose way of addressing functions. from my
research i discovered that almost all (about 80%) of the common
proviso forms written in textbooks can be expressed in terms of
intervals.

symbolic intervals open up an interesting area of research (guaranteed
to generate lots of PhDs). computationally they generate a graph.

consider the problem of computing a function symbolically and you
do a division by x somewhere along the path of computation. at that point
the computation splits into three parts:

     f(x) suchthat x in [x < 0]
     f(x) suchthat x in [x = 0]
     f(x) suchthat x in [x > 0]

further computation should carry these provisos. another division, say by
y, can generate 3 more branches:

     f(x) suchthat x in [x < 0] and y in [y < 0]
     f(x) suchthat x in [x = 0] and y in [y < 0]
     f(x) suchthat x in [x > 0] and y in [y < 0]

     f(x) suchthat x in [x < 0] and y in [y = 0]
     f(x) suchthat x in [x = 0] and y in [y = 0]
     f(x) suchthat x in [x > 0] and y in [y = 0]

     f(x) suchthat x in [x < 0] and y in [y > 0]
     f(x) suchthat x in [x = 0] and y in [y > 0]
     f(x) suchthat x in [x > 0] and y in [y > 0]

these can sometimes recombine when either you can prove that the
interval collapses, that the function is continuous at the point,
etc.

the introduction of provisos based on symbolic intervals can
generate a lifetime of fun.

t