fixed propagation under Fix/Lambda/Case of coercions, better names are

[helm.git] / matita / library / nat / div_and_mod.ma

diff --git a/matita/library/nat/div_and_mod.ma b/matita/library/nat/div_and_mod.ma
index 65ba0e9e5127a77ed8c69bb13f50300a058e3213..d7750e39ad7c03e4424d1d221e50710843d7e87a 100644 (file)
--- a/matita/library/nat/div_and_mod.ma
+++ b/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
@@ -215,11 +216,12 @@ 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.
@@ -330,3 +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.
+