Consequences of the provability of NP ⊆ P/poly

We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S2 proves NP ⊆ P/poly iff S 1 2 proves coNP ⊆ NP/O(log n). (iv) If S2 proves NP ⊆ P/poly then S 1 2 proves that the Polynomial Hierarchy collapses to P[log n]. (v) If S2 proves NP ⊆ P/poly then S2 proves that the Polynomial Hierarchy collapses to P.