We describe a general method how to construct from a propositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.
As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory V12 and hence, in particular, polynomially simulates the quantified propositional calculus G and the Πb1-consequences of S12 proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in S12. An iteration of the construction yields a proof system corresponding to T2 + Exp and, in principle, to much stronger theories.
"Implicit proofs." J. Symbolic Logic 69 (2) 387 - 397, June 2004. https://doi.org/10.2178/jsl/1082418532