Fact RatK x P : @Rat (numq x, denq x) P = x.Definition fracq (x : int * int) := nosimpl (@Rat (_, _) (fracq_subproof x)).Fact ratz_frac n : ratz n = fracq (n, 1).Fact valqK x : fracq (valq x) = x.Fact scalq_key : unit.Definition scalq := locked_with scalq_key scalq_def.Definition zeroq := fracq (0, 1).Definition oneq := fracq (1, 1).Fact frac0q x : fracq (0, x) = zeroq.Fact fracq0  x : fracq (x, 0) = zeroq.Fact fracqP x : fracq_spec x x (fracq x).

(* Specifications: *)
Lemma sgr_denq x : sgr (denq x) = 1.