(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/congruence".
-
include "nat/relevant_equations.ma".
include "nat/primes.ma".
definition congruent: nat \to nat \to nat \to Prop \def
\lambda n,m,p:nat. mod n p = mod m p.
+interpretation "congruent" 'congruent n m p =
+ (cic:/matita/nat/congruence/congruent.con n m p).
+
+notation < "hvbox(n break \cong\sub p m)"
+ (*non associative*) with precedence 45
+for @{ 'congruent $n $m $p }.
+
theorem congruent_n_n: \forall n,p:nat.congruent n n p.
intros.unfold congruent.reflexivity.
qed.
apply div_mod_spec_div_mod.assumption.
constructor 1.
apply lt_mod_m_m.assumption.
+(*cut (n = r * p + (m / p * p + m \mod p)).*)
+(*lapply (div_mod m p H).
+rewrite > sym_times.
+rewrite > distr_times_plus.
+(*rewrite > (sym_times p (m/p)).*)
+(*rewrite > sym_times.*)
+rewrite > assoc_plus.
+autobatch paramodulation.
+rewrite < div_mod.
+assumption.
+assumption.
+*)
rewrite > sym_times.
rewrite > distr_times_plus.
rewrite > sym_times.
theorem divides_to_congruent: \forall n,m,p:nat. O < p \to m \le n \to
divides p (n - m) \to congruent n m p.
intros.elim H2.
-apply (eq_times_plus_to_congruent n m p n2).
+apply (eq_times_plus_to_congruent n m p n1).
assumption.
rewrite < sym_plus.
apply minus_to_plus.assumption.