(* *)
(**************************************************************************)
-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.
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.
+
+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.
+intros.
+rewrite > (times_n_SO a) in \vdash (? % ?).
+apply le_times
+[ apply le_n
+| assumption
+]
+qed.