X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Flt_arith.ma;h=683ec262773cbe38541ba1499b6cf1eeb3775440;hb=6b61a9e6698a7c1936adf217b599e34e65a5e4c9;hp=623f30295922c7d6ff5fe5d23369b960f5e7c2dc;hpb=6db38e3d8e4083765f2fce40c7845c9827b9afd0;p=helm.git

diff --git a/helm/software/matita/library/nat/lt_arith.ma b/helm/software/matita/library/nat/lt_arith.ma
index 623f30295..683ec2627 100644
--- a/helm/software/matita/library/nat/lt_arith.ma
+++ b/helm/software/matita/library/nat/lt_arith.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/lt_arith".
-
 include "nat/div_and_mod.ma".
 
 (* plus *)
@@ -118,6 +116,17 @@ apply (nat_case1 a)
 ]
 qed.
 
+lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
+intros.
+elim (le_to_or_lt_eq O ? (le_O_n m))
+  [assumption
+  |apply False_ind.
+   rewrite < H1 in H.
+   rewrite < times_n_O in H.
+   apply (not_le_Sn_O ? H)
+  ]
+qed.
+
 (* times *)
 theorem monotonic_lt_times_r: 
 \forall n:nat.monotonic nat lt (\lambda m.(S n)*m).
@@ -239,26 +248,7 @@ apply (nat_case n)
   ]
 qed.
 
-theorem le_times_to_le_div: \forall a,b,c:nat.
-O \lt b \to (b*c) \le a \to c \le (a /b).
-intros.
-apply lt_S_to_le.
-apply (lt_times_n_to_lt b)
-  [assumption
-  |rewrite > sym_times.
-   apply (le_to_lt_to_lt ? a)
-    [assumption
-    |simplify.
-     rewrite > sym_plus.
-     rewrite > (div_mod a b) in ⊢ (? % ?)
-      [apply lt_plus_r.
-       apply lt_mod_m_m.
-       assumption
-      |assumption
-      ]
-    ]
-  ]
-qed.
+
 
 theorem nat_compare_times_l : \forall n,p,q:nat. 
 nat_compare p q = nat_compare ((S n) * p) ((S n) * q).