proposition: Lemma concave_function_atN f x y t : concave_function_at f x y t -> forall k, (0 <= k)%coqR -> concave_function_at (fun x => f x * k)%coqR x y t.
proof: Proof. move=> H k k0; rewrite /concave_function_at /convex_function_at. rewrite conv_leoppD leoppP avgRE. rewrite /leconv /= -avgR_mulDl. exact: leR_wpmul2r. Qed.
imports: HB(structures) mathcomp(all_ssreflect ssralg fingroup perm matrix) mathcomp(mathcomp_extra boolp classical_sets Rstruct) mathcomp(ssrnum ereal) mathcomp(lra Rstruct reals) Reals ssrR Reals_ext Ranalysis_ext ssr_ext ssralg_ext logb realType_ext fdist