proposition: Lemma neg_tv_nondecreasing a b f : {in `[a, b] &, nondecreasing_fun (neg_tv a f)}.
proof: Proof. move=> x y xab yab xy; have ax : a <= x. by move: xab; rewrite in_itv //= => /andP []. rewrite /neg_tv lee_pmul2r // lee_subr_addl // addeCA -EFinB. rewrite [TV a y _](total_variationD _ ax xy) //. apply: leeD => //; apply: le_trans; last exact: total_variation_ge. by rewrite lee_fin ler_norm. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum finmap) mathcomp(matrix interval zmodp vector fieldext falgebra) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality) ereal reals signed topology prodnormedzmodule normedtype derive sequences real_interval HB(structures)