Definition mulr_cp1 := (mulr_ilte1, mulr_egte1).
(* ler and ^-1 *)

(* Specifications: *)
Lemma invr_gt0 x : (0 < x^-1) = (0 < x).
Lemma invr_ge0 x : (0 <= x^-1) = (0 <= x).
Definition exprn_gte0 := (exprn_ge0, exprn_gt0).
Lemma exprn_ile1 n x : 0 <= x -> x <= 1 -> x ^+ n <= 1.
Lemma exprn_ilt1 n x : 0 <= x -> x < 1 -> x ^+ n < 1 = (n != 0%N).
Definition exprn_ilte1 := (exprn_ile1, exprn_ilt1).
Lemma exprn_ege1 n x : 1 <= x -> 1 <= x ^+ n.
Lemma exprn_egt1 n x : 1 < x -> 1 < x ^+ n = (n != 0%N).
Definition exprn_egte1 := (exprn_ege1, exprn_egt1).