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
(* 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.
(* 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.
+