proposition: Lemma le_lt_dec : forall a b : nat, a <= b |_| b < a.
proof: Proof. induction a as [|a IHa]. - intros;left;apply zero_least. - intros [|b]. + right. apply lt_0_S. + destruct (IHa b). * left. apply le_S_S;trivial. * right. apply le_S_S. trivial. Defined.
imports: HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.naturals HoTT.Classes.interfaces.orders HoTT.Classes.theory.rings HoTT.Classes.orders.semirings HoTT.Classes.theory.apartness