Learning algorithms versus automatability of frege systems

We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system P, we prove that the following statements are equivalent, 1. Provable learning: P proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries. 2. Provable automatability: P proves efficiently that P is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower bounds. Here, P is sufficiently strong and well-behaved if I. -III. holds: I. P p-simulates Jeřabek's system WF (which strengthens the Extended Frege system EF by a surjective weak pigeonhole principle) ; II. P satisfies some basic properties of standard proof systems which p-simulate WF; III. P proves efficiently for some Boolean function h that h is hard on average for circuits of subexponential size. For example, if III. holds for P=WF, then Items 1 and 2 are equivalent for P=WF. If there is a function h NE coNE which is hard on average for circuits of size 2^n/4, for each sufficiently big n, then there is an explicit propositional proof system P satisfying properties I. -III. , i. e. the equivalence of Items 1 and 2 holds for P.