Roger Hale wrote:
One set of cases that doesn't seem to have come up in discussion:
(1, 3, 2) >>-<< (83, 84, 81, 80, 85)
Should this give
(-82, -81, -79, -80, -85)
From an arithmetic point of view it should be exactly that. The
implementation might need to morph the code though, see below.
as by hallucinating $neutral - $x == $x? This latter $neutral in fact
doesn't exist among ordinary numbers, and I would call it algebraically
unnatural: for all (other) $n,
$n - ($a + $b) == ($n - $a) - $b
or, as you increase $a by $b, $n - $a decreases by $b (a sort of
contravariance), but
$neutral - ($a + $b) == $a + $b == ($neutral - $a) + $b
! This violates algebraic relations I would prefer to rely on, both in
my own reasoning and that of the compiler and other program-handling
programs.
Me too! The thing is that the field of the real numbers is build on the
operators + and * which are associative and commutative. The neutral
elements are 0 and 1 respectively. The non-associative operators - and /
are defined in terms of the inverse elements. Thus $a - $b == $a + (-$b)
and $a / $b == $a * (1/$b) if $b != 0. We could also bring in $b**-1 as
another way to find the multiplicative inverse. But that just pulls in
yet another operator and the question which axioms apply to it and how
it is hyperated.
For user-defined numerical types---that is ones that provide +, *, -, /,
**, etc---I would hope that these axioms hold. Actually I hope that
there is a set of roles that define the standard numerics and enforce
sanity as far as that is possible via the class composition mechanics.
So coming back to your example, operator Â-Â would call -Â on the RHS
and call Â+Â with the result. But I've no idea to which extent operator
  is a special runtime operator versus a term re-writing at compile
time.
--
TSa (Thomas SandlaÃ)
