By introducing a parallel extension rule that is aware of independence of the introduced extension variables, a calculus for
quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding
an uninterpreted predicate on bit-strings (analog to an oracle in relativised complexity classes) this statement can be made
precise in the sense that the height of the most shallow proof that a circuit can be evaluated is, up to an additive constant,
the height of that circuit.
The main tool for showing lower bounds on proof heights is a variant of an iteration principle studied by Takeuti. This reformulation
might be of independent interest, as it allows for polynomial size formulae in the relativised language that require proofs
of exponential height.