Definition sgzE := (sgz0, sgz1, sgzN1).

(* Specifications: *)
Lemma sgz_sgr x : sgz (sgr x) = sgz x.
Lemma sgrz (n : int) : sgr n = sgz n.
Lemma sgz_id (x : R) : sgz (sgz x) = sgz x.
Lemma sgzN x : sgz (- x) = - sgz x.
Lemma sgzM x y : sgz (x * y) = sgz x * sgz y.
Lemma sgz_gt0 x : (sgz x > 0) = (x > 0).
Lemma sgz_lt0 x : (sgz x < 0) = (x < 0).
Lemma sgz_ge0 x : (sgz x >= 0) = (x >= 0).
Lemma sgz_le0 x : (sgz x <= 0) = (x <= 0).
Lemma sgz_smul x y : sgz (y *~ (sgz x)) = (sgz x) * (sgz y).
Lemma sgrMz m x : sgr (x *~ m) = sgr x *~ sgr m.