Much ado about nothing:

[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 ccddcca3f14416507c5edabf52e985dbf7757a68..537d3bcf0d11a4207f313f0103d069c1d9d87041 100644 (file)
--- a/matita/library/nat/div_and_mod.ma
+++ b/matita/library/nat/div_and_mod.ma
@@ -193,14 +193,34 @@ 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.