How to extract negative information from programs is an importantissue in logic programming. Here we address the problem forfunctional logic programs, from a proof-theoretic perspective. Thestarting point of our work is CRWL (Constructorbased ReWriting Logic), a well established theoretical framework forfunctional logic programming, whose fundamental notion is that ofnon-strict non-deterministic function. We present a proof calculus,CRWLF, which is able to deduce negativeinformation from CRWL-programs. In particular,CRWLF is able to prove ‘finite’ failure ofreduction within CRWL.