+(*a variant of or_div_mod *)
+theorem or_div_mod1: \forall n,q. O < q \to
+(divides q (S n)) \land S n = (S (div n q)) * q \lor
+(\lnot (divides q (S n)) \land S n= (div n q) * q + S (n\mod q)).
+intros.elim (or_div_mod n q H);elim H1
+ [left.split
+ [apply (witness ? ? (S (n/q))).
+ rewrite > sym_times.assumption
+ |assumption
+ ]
+ |right.split
+ [intro.
+ apply (not_eq_O_S (n \mod q)).
+ (* come faccio a fare unfold nelleipotesi ? *)
+ cut ((S n) \mod q = O)
+ [rewrite < Hcut.
+ apply (div_mod_spec_to_eq2 (S n) q (div (S n) q) (mod (S n) q) (div n q) (S (mod n q)))
+ [apply div_mod_spec_div_mod.
+ assumption
+ |apply div_mod_spec_intro;assumption
+ ]
+ |apply divides_to_mod_O;assumption
+ ]
+ |assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [rewrite > plus_n_O.
+ rewrite < (divides_to_mod_O ? ? H H1).
+ apply sym_eq.
+ apply div_mod.
+ assumption
+ |elim H1.
+ generalize in match H2.
+ rewrite < H.
+ simplify.
+ intro.
+ rewrite > H3.
+ reflexivity
+ ]
+qed.
+
+theorem divides_div: \forall d,n. divides d n \to divides (n/d) n.
+intros.
+apply (witness ? ? d).
+apply sym_eq.
+apply divides_to_div.
+assumption.
+qed.
+
+theorem div_div: \forall n,d:nat. O < n \to divides d n \to
+n/(n/d) = d.
+intros.
+apply (inj_times_l1 (n/d))
+ [apply (lt_times_n_to_lt d)
+ [apply (divides_to_lt_O ? ? H H1).
+ |rewrite > divides_to_div;assumption
+ ]
+ |rewrite > divides_to_div
+ [rewrite > sym_times.
+ rewrite > divides_to_div
+ [reflexivity
+ |assumption
+ ]
+ |apply (witness ? ? d).
+ apply sym_eq.
+ apply divides_to_div.
+ assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
+O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
+intros.
+elim H1.
+rewrite > H2.
+rewrite > (sym_times c n1).
+cut(O \lt c)
+[ rewrite > (lt_O_to_div_times n1 c)
+ [ rewrite < assoc_times.
+ rewrite > (lt_O_to_div_times (a *n1) c)
+ [ reflexivity
+ | assumption
+ ]
+ | assumption
+ ]
+| apply (divides_to_lt_O c b);
+ assumption.
+]
+qed.
+
+theorem eq_div_plus: \forall n,m,d. O < d \to
+divides d n \to divides d m \to
+(n + m ) / d = n/d + m/d.
+intros.
+elim H1.
+elim H2.
+rewrite > H3.rewrite > H4.
+rewrite < distr_times_plus.
+rewrite > sym_times.
+rewrite > sym_times in ⊢ (? ? ? (? (? % ?) ?)).
+rewrite > sym_times in ⊢ (? ? ? (? ? (? % ?))).
+rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [reflexivity
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+qed.
+