Program extraction for 2-random reals

Let be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in Kreuzer and Kohlenbach (J. Symb. Log. 77(3):853–895, 2012 . doi: 10.2178/jsl/1344862165 ), Kreuzer (Notre Dame J. Formal Log. 53(2):245–265, 2012 . doi: 10.1215/00294527-1715716 ) to this principle. Let be the finite type extension of . We obtain that one can extract primitive recursive realizers from proofs in , i.e., if then one can extract from the proof a primitive recursive term t ( f ) such that . As a consequence, we obtain that is -conservative over .