Lemma 6: LOSR monotones are WPICC monotones

Let $f(s,p)\in[0,\infty]$ be any function on scenarios and real behavior arrays. Assume it is finite on every nonsignaling correlation, zero on every local correlation, and satisfies $f(t,w(p))\le f(s,p)$ for every nonsignaling $p$ whenever $w$, on all real arrays, has the form

with probability-vector kernels and separate probability mass functions $\mu,\nu$ on arbitrary types. Then $f$ also satisfies the same inequality whenever $w$ agrees on all nonsignaling correlations with a probability mixture of deterministic transformations that either separately choose source inputs from target inputs and locally process the outputs, or first fix Alice’s input and, depending on her outcome, either choose Bob’s input from his target input while Alice’s output ignores Bob’s outcome, or fix Bob’s input and permit both outputs to depend on Bob’s outcome, or do the symmetric Bob-first construction. Each final output depends on its own target input and the stated available outcomes, and deterministic probabilities are sums of source entries with output indicators. The conclusion also includes the same finiteness and zero-value requirements. Scenarios have positive integer sizes; nonsignaling correlations are nonnegative normalized arrays with each marginal independent of the other party’s input; local correlations additionally have $p_{xy}^{ab}=\sum_\lambda\mu(\lambda)A_\lambda(a\mid x)B_\lambda(b\mid y)$ with probability-vector responses. The second class requires agreement only on nonsignaling correlations.