proposition: Lemma ln_idlt0_xlt1 : forall x, 0 < x < 1 -> ln_id x < 0.
proof: Proof. rewrite {2}(_ : 0 = ln_id 1); last by rewrite /ln_id ln_1 2!subRR. move=> x Hx. have lt01 : 0 < 1 by lra. apply (pderive_increasing lt01 derive_pt_ln_id_xle1_ge0). - by split; [apply Hx | apply ltRW, Hx]. - lra. - by apply Hx. Qed.
imports: mathcomp(all_ssreflect ssralg ssrnum) mathcomp(Rstruct) Reals Lra ssrR realType_ext Reals_ext Ranalysis_ext logb convex