include "arithmetics/primes.ma".
(* successor mod n *)
-definition S_mod: nat → nat → nat ≝
-λn,m:nat. (S m) \mod n.
+definition S_mod: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
+λn,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 n.
-definition congruent: nat → nat → nat → Prop ≝
-λn,m,p:nat. mod n p = mod m p.
+definition congruent: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
+λn,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/div_and_mod/mod.def(3)"\ 6mod\ 5/a\ 6 n p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/div_and_mod/mod.def(3)"\ 6mod\ 5/a\ 6 m p.
interpretation "congruent" 'congruent n m p = (congruent n m p).
non associative with precedence 45
for @{ 'congruent $n $m $p }.
-theorem congruent_n_n: ∀n,p:nat.n ≅_{p} n .
+theorem congruent_n_n: ∀n,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} n .
// qed.
theorem transitive_congruent:
- ∀p.transitive nat (λn,m.congruent n m p).
+ ∀p.\ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 (λn,m.\ 5a href="cic:/matita/arithmetics/congruence/congruent.def(4)"\ 6congruent\ 5/a\ 6 n m p).
/2/ qed.
-theorem le_to_mod: ∀n,m:nat. n < m → n = n \mod m.
-#n #m #ltnm @(div_mod_spec_to_eq2 n m O n (n/m) (n \mod m))
-% // @lt_mod_m_m @(ltn_to_ltO … ltnm)
+theorem le_to_mod: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m.
+#n #m #ltnm @(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod_spec_to_eq2.def(10)"\ 6div_mod_spec_to_eq2\ 5/a\ 6 n m \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 n (n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6m) (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 m))
+% // @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 @(\ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(6)"\ 6ltn_to_ltO\ 5/a\ 6 … ltnm)
qed.
-theorem mod_mod : ∀n,p:nat. O<p → n \mod p = (n \mod p) \mod p.
-#n #p #posp >(div_mod (n \mod p) p) in ⊢ (? ? % ?)
->(eq_div_O ? p) // @lt_mod_m_m //
+theorem mod_mod : ∀n,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6p → n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p) \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p.
+#n #p #posp >(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p) p) in ⊢ (? ? % ?)
+>(\ 5a href="cic:/matita/arithmetics/div_and_mod/eq_div_O.def(14)"\ 6eq_div_O\ 5/a\ 6 ? p) // @\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 //
qed.
-theorem mod_times_mod : ∀n,m,p:nat. O<p → O<m →
- n \mod p = (n \mod (m*p)) \mod p.
+theorem mod_times_mod : ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6p → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m →
+ n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 (m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p)) \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p.
#n #m #p #posp #posm
-@(div_mod_spec_to_eq2 n p (n/p) (n \mod p)
-(n/(m*p)*m + (n \mod (m*p)/p)))
- [@div_mod_spec_div_mod //
- |% [@lt_mod_m_m //] >distributive_times_plus_r
- >associative_plus <div_mod >associative_times <div_mod //
+@(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod_spec_to_eq2.def(10)"\ 6div_mod_spec_to_eq2\ 5/a\ 6 n p (n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p) (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)
+(n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6(m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 (m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p)\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)))
+ [@\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod_spec_div_mod.def(13)"\ 6div_mod_spec_div_mod\ 5/a\ 6 //
+ |% [@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 //] >\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus_r.def(9)"\ 6distributive_times_plus_r\ 5/a\ 6
+ >\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/nat/associative_times.def(10)"\ 6associative_times\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 //
]
qed.
-theorem congruent_n_mod_n : ∀n,p:nat. O < p →
- n ≅_{p} (n \mod p).
+theorem congruent_n_mod_n : ∀n,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p →
+ n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p).
/2/ qed.
-theorem congruent_n_mod_times : ∀n,m,p:nat. O < p → O < m →
- n ≅_{p} (n \mod (m*p)).
+theorem congruent_n_mod_times : ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m →
+ n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 (m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p)).
/2/ qed.
-theorem eq_times_plus_to_congruent: ∀n,m,p,r:nat. O< p →
- n = r*p+m → n ≅_{p} m .
+theorem eq_times_plus_to_congruent: ∀n,m,p,r:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p →
+ n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 r\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} m .
#n #m #p #r #posp #eqn
-@(div_mod_spec_to_eq2 n p (div n p) (mod n p) (r +(div m p)) (mod m p))
- [@div_mod_spec_div_mod //
- |% [@lt_mod_m_m //] >distributive_times_plus_r
- >associative_plus <div_mod //
+@(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod_spec_to_eq2.def(10)"\ 6div_mod_spec_to_eq2\ 5/a\ 6 n p (\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 n p) (\ 5a href="cic:/matita/arithmetics/div_and_mod/mod.def(3)"\ 6mod\ 5/a\ 6 n p) (r \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/div_and_mod/div.def(3)"\ 6div\ 5/a\ 6 m p)) (\ 5a href="cic:/matita/arithmetics/div_and_mod/mod.def(3)"\ 6mod\ 5/a\ 6 m p))
+ [@\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod_spec_div_mod.def(13)"\ 6div_mod_spec_div_mod\ 5/a\ 6 //
+ |% [@\ 5a href="cic:/matita/arithmetics/div_and_mod/lt_mod_m_m.def(12)"\ 6lt_mod_m_m\ 5/a\ 6 //] >\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus_r.def(9)"\ 6distributive_times_plus_r\ 5/a\ 6
+ >\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 //
]
qed.
-theorem divides_to_congruent: ∀n,m,p:nat. O < p → m ≤ n →
- p ∣(n - m) → n ≅_{p} m .
+theorem divides_to_congruent: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n →
+ p \ 5a title="divides" href="cic:/fakeuri.def(1)"\ 6∣\ 5/a\ 6(n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 m) → n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} m .
#n #m #p #posp #lemn * #l #eqpl
-@(eq_times_plus_to_congruent … l posp) /2/
+@(\ 5a href="cic:/matita/arithmetics/congruence/eq_times_plus_to_congruent.def(14)"\ 6eq_times_plus_to_congruent\ 5/a\ 6 … l posp) /2/
qed.
-theorem congruent_to_divides: ∀n,m,p:nat.O < p →
- n ≅_{p} m → p ∣ (n - m).
-#n #m #p #posp #congnm @(quotient ? ? ((n / p)-(m / p)))
->commutative_times >(div_mod n p) in ⊢ (??%?)
->(div_mod m p) in ⊢ (??%?) <(commutative_plus (m \mod p))
-<congnm <(minus_plus ? (n \mod p)) <minus_plus_m_m //
+theorem congruent_to_divides: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p →
+ n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} m → p \ 5a title="divides" href="cic:/fakeuri.def(1)"\ 6∣\ 5/a\ 6 (n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 m).
+#n #m #p #posp #congnm @(\ 5a href="cic:/matita/arithmetics/primes/divides.con(0,1,2)"\ 6quotient\ 5/a\ 6 ? ? ((n \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p)))
+>\ 5a href="cic:/matita/arithmetics/nat/commutative_times.def(8)"\ 6commutative_times\ 5/a\ 6 >(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 n p) in ⊢ (??%?)
+>(\ 5a href="cic:/matita/arithmetics/div_and_mod/div_mod.def(9)"\ 6div_mod\ 5/a\ 6 m p) in ⊢ (??%?) <(\ 5a href="cic:/matita/arithmetics/nat/commutative_plus.def(5)"\ 6commutative_plus\ 5/a\ 6 (m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p))
+<congnm <(\ 5a href="cic:/matita/arithmetics/nat/minus_plus.def(12)"\ 6minus_plus\ 5/a\ 6 ? (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)) <\ 5a href="cic:/matita/arithmetics/nat/minus_plus_m_m.def(6)"\ 6minus_plus_m_m\ 5/a\ 6 //
qed.
-theorem mod_times: ∀n,m,p:nat. O < p →
- n*m ≅_{p} (n \mod p)*(m \mod p).
-#n #m #p #posp @(eq_times_plus_to_congruent ?? p
- ((n / p)*p*(m / p) + (n / p)*(m \mod p) + (n \mod p)*(m / p))) //
-@(trans_eq ?? (((n/p)*p+(n \mod p))*((m/p)*p+(m \mod p))))
- [@eq_f2 //
- |@(trans_eq ? ? (((n/p)*p)*((m/p)*p) + (n/p)*p*(m \mod p) +
- (n \mod p)*((m / p)*p) + (n \mod p)*(m \mod p))) //
- >distributive_times_plus >distributive_times_plus_r
- >distributive_times_plus_r <associative_plus @eq_f2 //
+theorem mod_times: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p →
+ n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p).
+#n #m #p #posp @(\ 5a href="cic:/matita/arithmetics/congruence/eq_times_plus_to_congruent.def(14)"\ 6eq_times_plus_to_congruent\ 5/a\ 6 ?? p
+ ((n \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p) \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (n \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p) \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p))) //
+@(\ 5a href="cic:/matita/basics/logic/trans_eq.def(4)"\ 6trans_eq\ 5/a\ 6 ?? (((n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p))\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6((m\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p))))
+ [@\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6 //
+ |@(\ 5a href="cic:/matita/basics/logic/trans_eq.def(4)"\ 6trans_eq\ 5/a\ 6 ? ? (((n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6((m\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p) \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (n\ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p) \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6
+ (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6((m \ 5a title="natural divide" href="cic:/fakeuri.def(1)"\ 6/\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p) \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (n \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(m \ 5a title="natural remainder" href="cic:/fakeuri.def(1)"\ 6\mod\ 5/a\ 6 p))) //
+ >\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus.def(7)"\ 6distributive_times_plus\ 5/a\ 6 >\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus_r.def(9)"\ 6distributive_times_plus_r\ 5/a\ 6
+ >\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus_r.def(9)"\ 6distributive_times_plus_r\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f2.def(3)"\ 6eq_f2\ 5/a\ 6 //
]
qed.
-theorem congruent_times: ∀n,m,n1,m1,p. O < p →
- n ≅_{p} n1 → m ≅_{p} m1 → n*m ≅_{p} n1*m1 .
+theorem congruent_times: ∀n,m,n1,m1,p. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p →
+ n \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} n1 → m \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} m1 → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="congruent" href="cic:/fakeuri.def(1)"\ 6≅\ 5/a\ 6_{p} n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m1 .
#n #m #n1 #m1 #p #posp #congn #congm
-@(transitive_congruent … (mod_times n m p posp))
->congn >congm @sym_eq @mod_times //
+@(\ 5a href="cic:/matita/arithmetics/congruence/transitive_congruent.def(5)"\ 6transitive_congruent\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/congruence/mod_times.def(15)"\ 6mod_times\ 5/a\ 6 n m p posp))
+>congn >congm @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/congruence/mod_times.def(15)"\ 6mod_times\ 5/a\ 6 //
qed.
(*
assumption.
apply congruent_n_mod_n.assumption.
assumption.
-qed. *)
+qed. *)
\ No newline at end of file
projects / helm.git / blobdiff