+theorem lt_minus_l: \forall m,l,n:nat.
+ l < m \to m \le n \to n - m < n - l.
+apply nat_elim2
+ [intros.apply False_ind.apply (not_le_Sn_O ? H)
+ |intros.rewrite < minus_n_O.
+ auto
+ |intros.
+ generalize in match H2.
+ apply (nat_case n1)
+ [intros.apply False_ind.apply (not_le_Sn_O ? H3)
+ |intros.simplify.
+ apply H
+ [
+ apply lt_S_S_to_lt.
+ assumption
+ |apply le_S_S_to_le.assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_minus_r: \forall n,m,l:nat.
+ n \le l \to l < m \to l -n < m -n.
+intro.elim n
+ [applyS H1
+ |rewrite > eq_minus_S_pred.
+ rewrite > eq_minus_S_pred.
+ apply lt_pred
+ [unfold lt.apply le_plus_to_minus_r.applyS H1
+ |apply H[auto|assumption]
+ ]
+ ]
+qed.
+
+lemma lt_to_lt_O_minus : \forall m,n.
+ n < m \to O < m - n.
+intros.
+unfold. apply le_plus_to_minus_r. unfold in H. rewrite > sym_plus.
+rewrite < plus_n_Sm.
+rewrite < plus_n_O.
+assumption.
+qed.
+