(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
(* *)
(**************************************************************************)
-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 }.
+interpretation "congruent" 'congruent n m p = (congruent n m p).
theorem congruent_n_n: \forall n,p:nat.congruent n n p.
intros.unfold congruent.reflexivity.
(*rewrite > (sym_times p (m/p)).*)
(*rewrite > sym_times.*)
rewrite > assoc_plus.
-auto paramodulation.
+autobatch paramodulation.
rewrite < div_mod.
assumption.
assumption.
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.