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Replaced Definitions with Theorems and Lemmas and moved the auxiliary functions Forall_tail and Exists_impl to base. These two functions have been added to the latest version of the Coq List library.
Wrapped the definitions provided by this library within a module to prevent namespace collisions. Also marked the Forall_tail and Exists_impl proofs as local since I plan to eventually ask the library maintainers to add them to the Standard Library.Finally, I renamed pigeons.v to pigeonhole_principle.v for clarity.