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## F Proof of Theorem 3.3 

We use the notation given in the first two paragraphs of Appendix A of the main manuscript, including: q$_{2}$ = q − q$_{1}$ ; R$_{j}$ is the indicator of whether H$_{NR,j}$ was rejected for j = 1 , . . . , m , and R = ∑ m j =1 R$_{j}$ . In addition we define: I$_{0}$ = I$_{00}$ ∪ I$_{01}$ ; p$_{1}$ = ( p$_{11}$, . . . , p$_{1}$$_{m}$ ); R $_{1}$( p$_{1}$ ) is the set of hypotheses selected for follow-up based on p$_{1}$, R$_{1}$ ( p$_{1}$ ) = |R $_{1}$( p$_{1}$ ) | . 

Lemma F.1. Assume that the p -values across studies are independent, and the set of p -values within the follow-up study has property PRDS. Then for any valid selection rule, the following results hold: 

1. Given p$_{1}$, for j ∈ I$_{10}$ ∩ R $_{1}$( p$_{1}$ ) , 

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