proposition: Lemma natlthsnminussmsn ( n m : nat ) ( p : natlth m n ) : natlth ( sub ( S n ) ( S m ) ) ( S n ).
proof: Proof. revert m p. induction n. - intros m p. apply fromempty. apply (negnatlthn0 m p). - intros m p. destruct m. + assert ( sub ( S ( S n ) ) 1 = S n ) as f. { destruct n. * auto. * auto. } rewrite f. apply natlthnsn. + apply ( istransnatlth _ ( S n ) _ ). * apply IHn. assumption. * apply natlthnsn. Defined.
imports: UniMath.Tactics.EnsureStructuredProofs UniMath.Foundations.PartA UniMath.Foundations.Propositions UniMath.Foundations.NaturalNumbers UniMath.Algebra.RigsAndRings UniMath.Algebra.Domains_and_Fields UniMath.NumberSystems.Integers UniMath.Algebra.Monoids