A reduction of proof complexity to computational complexity for AC⁰[p] Frege systems

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime $p$ (the so called $AC^0[p]$ Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over the finite filed with $p$ elements.