A substitution lemma for multiple context-free languages

In this paper, we present a necessary condition for an infinite language to be multiple context-free, which we call a Substitution Lemma. We apply it to show a sample selection of languages are not multiple context-free, including the word problem of the group F₂ × F₂. We also show that groups with multiple context-free word problem have decidable rational subset membership problem. Our result contrasts with previous work showing that the standard pumping lemma for context-free languages cannot be generalized to multiple context-free languages, and that weak variants of generalized Ogden’s lemma do not apply to multiple context-free languages.