proposition: Lemma locator_left : locator (' s).
proof: Proof. intros q r ltqr. destruct (trichotomy _ q s) as [ltqs|[eqqs|ltsq]]. - apply inl. apply (strictly_order_preserving _); assumption. - rewrite eqqs in ltqr. apply inr, (strictly_order_preserving _); assumption. - apply inr, (strictly_order_preserving _), (transitivity ltsq ltqr); assumption. Qed.
imports: HoTT.Basics HoTT.DProp HoTT.BoundedSearch HoTT.Spaces.Finite.Fin HoTT.ExcludedMiddle HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.orders HoTT.Classes.interfaces.rationals HoTT.Classes.interfaces.cauchy HoTT.Classes.interfaces.archimedean HoTT.Classes.interfaces.round HoTT.Classes.interfaces.naturals HoTT.Classes.implementations.peano_naturals HoTT.Classes.orders.archimedean HoTT.Classes.orders.dec_fields HoTT.Classes.orders.lattices HoTT.Classes.theory.apartness HoTT.Classes.theory.rationals HoTT.Classes.orders.fields HoTT.Classes.theory.fields HoTT.Classes.theory.dec_fields