We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity $2^{\Omega \left(n\right)}$. ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function $2^{\Omega \left(n\right)}$ hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that if a minor variant...