Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
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Let $${2-\textsf{RAN}}$$ be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in 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 $${{\textsf{WKL}_0^\omega}}$$ be the finite type extension of $${\textsf{WKL}_0}$$ . We obtain that one can extract primitive recursive realizers from proofs in $${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN}}$$ , i.e., if $${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN} \, {\vdash} \, \forall{f}\, {\exists}{x} A_{qf}(f,x)}$$ then one can extract from the proof a primitive recursive term t(f) such that $${A_{qf}(f,t(f))}$$ . As a consequence, we obtain that $${{\textsf{WKL}_0}+ \Pi^0_1 - {\textsf{CP}} + 2-\textsf{RAN}}$$ is $${\Pi^0_3}$$ -conservative over $${\textsf{RCA}_0}$$ .