Local Notation "∑" := (msum plus 0).Definition swap n := if eq_nat_dec n i then j else
                       if eq_nat_dec n j then i else n.Fact swap_spec_i : swap i = j.Fact swap_spec_j : swap j = i.Fact swap_spec n : n <> i -> n <> j -> swap n = n.Fact swap_involutive n : swap (swap n) = n.Fact swap_inj n m : swap n = swap m -> n = m.

(* Specifications: *)
Theorem php_fun : exists i j, i < j <= n /\ f i = f j.