proposition: Lemma interval_left_unbounded_interior (X : set R) : is_interval X -> ~ has_lbound X -> has_ubound X -> X^° = [set r | r < sup X].
proof: Proof. move=> iX lX uX; rewrite eqEsubset; split; first exact: right_bounded_interior. rewrite -(open_subsetE _ (@open_lt _ _)) => r rsupX. move/has_lbPn : lX => /(_ r)[y Xy yr]. have hsX : has_sup X by split => //; exists y. have /sup_adherent/(_ hsX)[e Xe] : 0 < sup X - r by rewrite subr_gt0. by rewrite opprB addrCA subrr addr0 => re; apply: (iX y e); rewrite ?ltW. Qed.
imports: HB(structures) mathcomp(all_ssreflect ssralg ssrint ssrnum finmap matrix) mathcomp(rat interval zmodp vector fieldext falgebra) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality set_interval Rstruct) ereal reals signed topology prodnormedzmodule