Are there any propositional proof systems which are not Cook-Reckhow proof systems? An abstract proof system is a polynomial time function $f$ whose
  range is equal to the set of tautologies. If $\tau$ is a tautology, then an $f$-proof of $\tau$ is any value $\pi$ such that $f(\pi) = \tau$.


Is there an example of a propositional  proof system which doesn't satisfy this definition?
Natural examples of propositional proof systems that do not fall under this definition are algebraic proof systems where the lines in the proof are arbitrary polynomials (not necessarily fully expanded). To verify the correctness of such proofs, among other things one has to test the identity of polynomials, which is not known to be possible in deterministic polynomial time. 

Cf. the paper "Algebraic propositional proof systems" by Toni Pitassi, in  DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 31, Descriptive Complexity and Finite Models, Immerman and Kolaitis (Eds.), pp. 215-244, 1996.