I saw that many introductions to Haskell contain proofs of properties of
implemented functions. This is probably due to the fact that pure
functions can be handled more easily than imperative programs with hidden
states. I wondered whether one can use proofs of Haskell functions for
testing. 

I found QuickCheck
 http://www.cs.chalmers.se/~rjmh/QuickCheck/
  which, as far as I understand, relies on random inputs and I found
 http://homepages.inf.ed.ac.uk/wadler/realworld/era.html
  which sounds like some GUI driven program.

Is there something that can be used for automatical testing, e.g. for
darcs' tests to check patch integrity?

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