Lean proofs for "Sorting networks on 15 and 16 inputs need 55 and 60 comparators, and 20 inputs need 81"

This record contains the Lean 4 proofs of the paper "Sorting networks on 15 and 16 inputs need 55 and 60 comparators, and 20 inputs need 81" (doi:10.5281/zenodo.24156530), about S(n), the least number of comparators in a sorting network on n channels. Assuming that the bound tables of the S(14) record (doi:10.5281/zenodo.23632985) are valid, it proves S(15) >= 55 and S(16) = 60, where the lower bound S(16) >= 60 matches the upper bound given by Green's network. From S(16) >= 60 it also derives S(20) >= 81 and the last row of bounds in Table 6 for 17 <= n <= 27. STATEMENTS.md states every theorem and every hypothesis in full and is the authoritative description of what is proved. For the files and how to check them, see sections 4 and 5 of README.md.