Quantification in free theorems (Was: [Haskell-cafe] Exercise in point free-style)

Janis Voigtlaender voigt at tcs.inf.tu-dresden.de
Mon Sep 4 04:21:43 EDT 2006

ajb at spamcop.net wrote:
> G'day all.
> Quoting Donald Bruce Stewart <dons at cse.unsw.edu.au>:
>>Get some free theorems:
>>    lambdabot> free f :: (b -> b) -> [b] -> [b]
>>    f . g = h . f => map f . f g = f h . map f
> I finally got around to fixing the name clash bug.  It now reports:
>     g . h = k . g => map g . f h = f k . map g
> Get your free theorems from:
>     http://andrew.bromage.org/darcs/freetheorems/

I find the omission of quantifications in the produced theorems
problematic. It certainly makes the output more readable in some cases,
as in the example above. But consider the following type:

   filter :: (a -> Bool) -> [a] -> [a]

For this, you produce the following theorem:

   g x = h (f x)
   $map f . filter g = filter h . $map f

Lacking any information about the scope of free variables, the only
reasonable assumption is that they are all implicitly forall-quantified
at the outermost level (as for types in Haskell). But then the above
theorem is wrong. Consider:

   g = const False
   x = 0
   h = even
   f = (+1)

Clearly, for these choices the precondition g x = h (f x) is fulfilled,
since (const False) 0 = False = even ((+1) 0). But the conclusion is not
fulfilled, because with Haskell's standard filter-function we have, for

   map f (filter g [1]) = [] /= [2] = filter h (map f [1])

The correct free theorem, as produced by the online tool at


(and after renaming variables to agree with your output) is as follows:

forall T1,T2 in TYPES. forall f :: T1 -> T2.
   forall g :: T1 -> Bool.
     forall h :: T2 -> Bool.
       (forall x :: T1.
          g x = h (f x))
       ==> forall xs :: [T1].
             map f (filter g xs) = filter h (map f xs)

The essential difference is, of course, that the x is (and must be)
locally quantified here, not globally. That is not reflected in the
other version above.

Ciao, Janis.

Dr. Janis Voigtlaender
mailto:voigt at tcs.inf.tu-dresden.de

More information about the Haskell-Cafe mailing list