[Haskell-cafe] From function over expression (+, *) derive function over expression (+)

[Radek Micek]

Hello.

I have two types for expression:

data Expr = Add Expr Expr | Mul Expr Expr | Const Int

data AExpr = AAdd AExpr AExpr | AConst Int

The first one supports addition and multiplication and the second
only addition.

I can write a function to simplify the first expression:

simplify :: Expr -> Expr
simplify = {- replaces:
"a*1" and "1*a" by "a",
"a+0" and "0+a" by a -}

And I would like to use the function simplify for the second type
AExpr. What can I do is to convert AExpr to Expr, simplify it and
convert it back. But I don't like this solution because
conversions take some time.

I would prefer following: I say to the compiler that AAdd is like Add
and AConst is like Const and the compiler derives function
asimplify for AExpr.

Is it possible to do this? In fact I want to have two distinct types
where one is "extension" of the second (Expr is extension of AExpr)
and I want to define a function for the extended type (Expr) and
use it for the original type (AExpr). I assume that the function won't
introduce Mul to the expression which had no Mul.

Thanks in advance

Radek Micek