  A fast consistency prover is a consistent poly-time axiomatized theory that
has short proofs of the finite consistency statements of any other poly-time
axiomatized theory. Kraj\'\i\v{c}ek and Pudl\'ak proved that the existence of
an optimal propositional proof system is equivalent to the existence of a fast
consistency prover. It is an easy observation that ${\sf NP}={\sf coNP}$
implies the existence of a fast consistency prover. The reverse implication is
an open question.
  In this paper we define the notion of an unlikely fast consistency prover and
prove that its existence is equivalent to ${\sf NP}={\sf coNP}$.
  Next it is proved that fast consistency provers do not exist if one considers
RE axiomatized theories rather than theories with an axiom set that is
recognizable in polynomial time.
