Program extraction for 2-random reals

Let 2-RAN be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in [10,9] to this principle. Let WKL0 be the finite type extension of WKL0. We obtain that one can extract primitive recursive realizers from proofs in WKL0 +Π 0 1-CP+ 2-RAN, i.e., if WKL0 + Π 0 1-CP+ 2-RAN ` ∀ f ∃xAqf( f ,x) then one can extract from the proof a primitive recursive term t( f ) such that Aqf( f , t( f )). As a consequence, we obtain that WKL0 +Π1-CP+2-RAN is Π 0 3-conservative over RCA0.