X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=d7750e39ad7c03e4424d1d221e50710843d7e87a;hb=0ac6c9a8f9cb3206002178edcf2dfa761c7a94ef;hp=ccddcca3f14416507c5edabf52e985dbf7757a68;hpb=f275a4daa42f8ab455020ec5a7ce7bf69b460c37;p=helm.git

diff --git a/helm/software/matita/library/nat/div_and_mod.ma b/helm/software/matita/library/nat/div_and_mod.ma
index ccddcca3f..d7750e39a 100644
--- a/helm/software/matita/library/nat/div_and_mod.ma
+++ b/helm/software/matita/library/nat/div_and_mod.ma
@@ -17,6 +17,7 @@ set "baseuri" "cic:/matita/nat/div_and_mod".
 include "datatypes/constructors.ma".
 include "nat/minus.ma".
 
+
 let rec mod_aux p m n: nat \def
 match (leb m n) with
 [ true \Rightarrow m
@@ -193,14 +194,34 @@ unfold lt.apply le_S_S.apply le_O_n.
 rewrite < plus_n_O.rewrite < sym_times.reflexivity.
 qed.
 
+lemma div_plus_times: \forall m,q,r:nat. r < m \to  (q*m+r)/ m = q. 
+intros.
+apply (div_mod_spec_to_eq (q*m+r) m ? ((q*m+r) \mod m) ? r)
+  [apply div_mod_spec_div_mod.
+   apply (le_to_lt_to_lt ? r)
+    [apply le_O_n|assumption]
+  |apply div_mod_spec_intro[assumption|reflexivity]
+  ]
+qed.
+
+lemma mod_plus_times: \forall m,q,r:nat. r < m \to  (q*m+r) \mod m = r. 
+intros.
+apply (div_mod_spec_to_eq2 (q*m+r) m ((q*m+r)/ m) ((q*m+r) \mod m) q r)
+  [apply div_mod_spec_div_mod.
+   apply (le_to_lt_to_lt ? r)
+    [apply le_O_n|assumption]
+  |apply div_mod_spec_intro[assumption|reflexivity]
+  ]
+qed.
 (* some properties of div and mod *)
 theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
 intros.
-apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O).
-goal 15. (* ?11 is closed with the following tactics *)
-apply div_mod_spec_div_mod.
-unfold lt.apply le_S_S.apply le_O_n.
-apply div_mod_spec_times.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O);
+[2: apply div_mod_spec_div_mod.
+    unfold lt.apply le_S_S.apply le_O_n.
+|   skip
+|   apply div_mod_spec_times
+]
 qed.
 
 theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
@@ -311,3 +332,14 @@ let rec n_divides_aux p n m acc \def
 
 (* n_divides n m = <q,r> if m divides n q times, with remainder r *)
 definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
+
+
+(*a simple variant of div_times theorem *)
+theorem div_times_ltO: \forall a,b:nat. O \lt b \to
+a*b/b = a.
+intros.
+rewrite > sym_times.
+rewrite > (S_pred b H).
+apply div_times.
+qed.
+