function notation classes

4.12: adda/cstr/function notation classes:

. how does the type mgt express
parsing particulars?
eg, some infixes can be combined:
a*b*c = *(a,b,c),
whereas, a /b /c = *(a, /b, /c) .
so then (/) is not listable ...

types of declarations:
# unary:
eg, sin(a);

# infix listable:
*(a, ...) -- = a * b * c * ....
--. implies associativity;
ie, (a*b)*c = a*(b*c) = *(a,b,c).

# infix binary:
^(a,b) -- a^b^c = a^(b^c)
-- implies non-associativity;
ie, (a^b)^c =/= a^(b^c);
and, the given rule specifies which way to parse .
--. precedence is indicated by
the order in which operators are declared .
. can also be a prefix .
-- undeclared are assumed to be 2 args of the same type
(ie, the same supertype).

# infix or unary:
/(a=1,b) -- = 1 * /b;
-(a=0,b) -- = 0 + -b
-- /a = 1/a;

# nofix: no arg expected;
eg, pi, e, dimensions,
marks an EOterm
(is also trivially a prefix)
-- declared as having no args: x.t .

# postfix:
(x)!, -- factorial's declaration,
(having multiple parameters but of various types
counts as 1 arg of type record ):
(x.t, x.u)!.v .
[4.30: {postfix, prefix} are treated as separate functions,
like so:
()!, -- postfix
!, !() -- prefix .]

. each type class can overload an operator
as either {unary#prefix, unary#post}
or -- as in the case of Number.type --
{ infix (2 args)
, prefix (other numbers of args)
} . a symbol can be designated as
both pre- and post-fix
but cannot be both post- and in-fix .
. the one place math has a postfix is (x!).
in that case,
it's at the end of a term just like an infix
so the only way to be certain that a postfix
is not the infix found between terms
is by disallowing operators from being
both infix's and postfix's .
. postfix's can, however,
double as prefix's,
as can the infix's;
so, expecting a term and finding (!)
means the (!) is a prefix to that term;
whereas, expecting an infix or EOse
(end of subexpression) and finding (!)
means the (!) is a postfix .

. a subexpression can have multiple terms
separated by infix's;
terms are defined as being either:
nofix's: symbols accepting no args;
or a prefix followed by an arg .
. a subexpression is anything terminated by
an enclosure`end:
ie,
, . ; ) } .> end-of-file ] .