exercises ready

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

diff --git a/helm/software/matita/library/nat/div_and_mod.ma b/helm/software/matita/library/nat/div_and_mod.ma
index f7f2883d590a8118ae6a39a8c348a75dcb2ff565..538515a8cc806003287340ef9d1299b4625a17a0 100644 (file)
--- a/helm/software/matita/library/nat/div_and_mod.ma
+++ b/helm/software/matita/library/nat/div_and_mod.ma
@@ -12,12 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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
@@ -223,6 +220,7 @@ 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_intro[assumption|reflexivity]
   ]
 qed.
+
 (* some properties of div and mod *)
 theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
 intros.
@@ -312,16 +310,28 @@ rewrite > (div_mod ? (S O)) in \vdash (? ? ? %)
   ]
 qed.
 
-theorem le_div: \forall n,m. O < n \to m/n \le m.
+theorem or_div_mod: \forall n,q. O < q \to
+((S (n \mod q)=q) \land S n = (S (div n q)) * q \lor
+((S (n \mod q)<q) \land S n= (div n q) * q + S (n\mod q))).
 intros.
-rewrite > (div_mod m n) in \vdash (? ? %)
-  [apply (trans_le ? (m/n*n))
-    [rewrite > times_n_SO in \vdash (? % ?).
-     apply le_times
-      [apply le_n|assumption]
-    |apply le_plus_n_r
+elim (le_to_or_lt_eq ? ? (lt_mod_m_m n q H))
+  [right.split
+    [assumption
+    |rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
+    ]
+  |left.split
+    [assumption
+    |simplify.
+     rewrite > sym_plus.
+     rewrite < H1 in ⊢ (? ? ? (? ? %)).
+     rewrite < plus_n_Sm.
+     apply eq_f.
+     apply div_mod.
+     assumption
     ]
-  |assumption
   ]
 qed.
 
@@ -370,6 +380,7 @@ qed.
 variant inj_times_l1:\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q
 \def lt_O_to_injective_times_l.
 
+      
 (* n_divides computes the pair (div,mod) *)
 
 (* p is just an upper bound, acc is an accumulator *)