apply le_plus_l.apply le_O_n.
qed.
+theorem le_plus_n_r :\forall n,m:nat. m \le m + n.
+intros.rewrite > sym_plus.
+apply le_plus_n.
+qed.
+
theorem eq_plus_to_le: \forall n,m,p:nat.n=m+p \to m \le n.
intros.rewrite > H.
rewrite < sym_plus.
apply le_plus_n.
qed.
+theorem le_plus_to_le:
+\forall a,n,m. a + n \le a + m \to n \le m.
+intro.
+elim a
+ [assumption
+ |apply H.
+ apply le_S_S_to_le.assumption
+ ]
+qed.
+
(* times *)
theorem monotonic_le_times_r:
\forall n:nat.monotonic nat le (\lambda m. n * m).
elim (plus_n_O ?).apply le_n.
simplify.rewrite < sym_plus.apply le_plus_n.
qed.
+
+theorem le_times_to_le:
+\forall a,n,m. S O \le a \to a * n \le a * m \to n \le m.
+intro.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.
+ rewrite < times_n_O in H1.
+ generalize in match H1.
+ apply (lt_O_n_elim ? H).
+ intros.
+ simplify in H2.
+ apply (le_to_not_lt ? ? H2).
+ apply lt_O_S
+ |apply le_S_S.
+ apply H
+ [assumption
+ |rewrite < times_n_Sm in H2.
+ rewrite < times_n_Sm in H2.
+ apply (le_plus_to_le a).
+ assumption
+ ]
+ ]
+qed.
+
+(*0 and times *)
+theorem O_lt_const_to_le_times_const: \forall a,c:nat.
+O \lt c \to a \le a*c.
+intros.
+rewrite > (times_n_SO a) in \vdash (? % ?).
+apply le_times
+[ apply le_n
+| assumption
+]
+qed.