(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/le_arith".
-
include "nat/times.ma".
include "nat/orders.ma".
(* plus *)
theorem monotonic_le_plus_r:
\forall n:nat.monotonic nat le (\lambda m.n + m).
-simplify.intros.elim n.
-simplify.assumption.
-simplify.apply le_S_S.assumption.
+simplify.intros.elim n
+ [simplify.assumption.
+ |simplify.apply le_S_S.assumption
+ ]
qed.
theorem le_plus_r: \forall p,n,m:nat. n \le m \to p + n \le p + m
theorem le_plus: \forall n1,n2,m1,m2:nat. n1 \le n2 \to m1 \le m2
\to n1 + m1 \le n2 + m2.
intros.
+(**
+auto.
+*)
+apply (transitive_le (plus n1 m1) (plus n1 m2) (plus n2 m2) ? ?);
+ [apply (monotonic_le_plus_r n1 m1 m2 ?).
+ apply (H1).
+ |apply (monotonic_le_plus_l m2 n1 n2 ?).
+ apply (H).
+ ]
+(* end auto($Revision$) proof: TIME=0.61 SIZE=100 DEPTH=100 *)
+(*
apply (trans_le ? (n2 + m1)).
apply le_plus_l.assumption.
apply le_plus_r.assumption.
+*)
qed.
theorem le_plus_n :\forall n,m:nat. m \le n + m.
]
qed.
+theorem le_S_times_SSO: \forall n,m.O < m \to
+n \le m \to S n \le (S(S O))*m.
+intros.
+simplify.
+rewrite > plus_n_O.
+simplify.rewrite > plus_n_Sm.
+apply le_plus
+ [assumption
+ |rewrite < plus_n_O.
+ 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.