[Haskell-cafe] DSLs with {in,}equalities

[Andrew Hunter]

Several times now I've had to define an EDSL for working with
(vaguely) numeric expressions.  For stuff like 2*X+Y, this is easy,
looking pretty much like:

> data Expr = Const Integer | Plus Expr Expr | Times Expr Expr
>
> instance Num Expr where
> fromInterger = Const
> (+) = Plus
> (*) = Times

&c.  This lets me get a perfectly nice AST, which is what I want.
When I want to be able to express and work with inequalities and
equalities, this breaks.  Suppose I want to write 2*X + Y < 3.  I
either have to:

a) Hide Prelude.(<) and define a simple < that builds the AST term I want.
b) Come up with a new symbol for it that doesn't look totally awful.

Neither of these work decently well.  Hiding Eq and Ord operators,
which is what I effectively have to do for a), is pretty much a
nonstarter--we'll have to use them too much for that to be practical.

On the other hand, b) works...but is about as ugly as it gets.  We
have lots and lots of symbols that are already taken for important
purposes that are syntactically "near" <,<=,==, and the like: << and
>> and >>= for monads, >>> for arrows, etc.  There...are not good
choices that I know of for the symbols that don't defeat the purpose
of making a nice clean EDSL for expressions; I might as well use 3*X +
Y `lessthan` 3, which is just not cool.

Does anyone know of a good solution, here?  Are there good
substitutions for all the six operators that are important
(<,>,>=,<=,==,/=), that are close enough to be pretty-looking but not
used for other important modules?

Better yet, though a little harder, is there a nice type trick I'm not
thinking of?  This works for Num methods but not for Ord methods
because:

(+) :: (Num a) => a -> a -> a
(<) :: (Ord a) => a -> a -> Bool

i.e. the return type of comparisons is totally fixed.  I don't suppose
there's a good way to...well, I don't know what the *right* answer is,
but maybe define a new typeclass with a more flexible type for < that
lets both standard types return Bool and my expressions return Expr?
Any good solution would be appreciated.  

Thanks,
AHH