[O \Rightarrow n
| (S p) \Rightarrow mod_aux n n p].
-interpretation "natural remainder" 'module x y =
- (cic:/matita/nat/div_and_mod/mod.con x y).
+interpretation "natural remainder" 'module x y = (mod x y).
let rec div_aux p m n : nat \def
match (leb m n) with
[O \Rightarrow S n
| (S p) \Rightarrow div_aux n n p].
-interpretation "natural divide" 'divide x y =
- (cic:/matita/nat/div_and_mod/div.con x y).
+interpretation "natural divide" 'divide x y = (div x y).
theorem le_mod_aux_m_m:
\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
rewrite < sym_times.
rewrite > distr_times_minus.
rewrite > plus_minus.
+lapply(plus_to_minus ??? H3); demodulate all.
+(*
rewrite > sym_times.
rewrite < H5.
rewrite < sym_times.
apply plus_to_minus.
apply H3.
+*)
apply le_times_r.
apply lt_to_le.
apply H6.
theorem div_mod_spec_times : \forall n,m:nat.div_mod_spec ((S n)*m) (S n) m O.
intros.constructor 1.
-unfold lt.apply le_S_S.apply le_O_n.
-rewrite < plus_n_O.rewrite < sym_times.reflexivity.
+unfold lt.apply le_S_S.apply le_O_n. demodulate. reflexivity.
+(*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 n n (n / n) (n \mod n) (S O) O).
apply div_mod_spec_div_mod.assumption.
-constructor 1.assumption.
-rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+constructor 1.assumption. demodulate. reflexivity. (*
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.*)
qed.
theorem eq_div_O: \forall n,m. n < m \to n / m = O.
intros.
apply (div_mod_spec_to_eq2 n n (n / n) (n \mod n) (S O) O).
apply div_mod_spec_div_mod.assumption.
-constructor 1.assumption.
-rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+constructor 1.assumption. demodulate. reflexivity.(*
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.*)
qed.
theorem mod_S: \forall n,m:nat. O < m \to S (n \mod m) < m \to