(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/gcd".
-
include "nat/primes.ma".
include "nat/lt_arith.ma".
theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
p \divides (m \mod n) \to p \divides n \to p \divides m.
intros.elim H1.elim H2.
-apply (witness p m ((n1*(m / n))+n2)).
+apply (witness p m ((n2*(m / n))+n1)).
rewrite > distr_times_plus.
rewrite < H3.
rewrite < assoc_times.
generalize in match H1.
rewrite > H3.
intro.
-cut (O < n2)
- [elim (gcd_times_SO_to_gcd_SO n n n2 ? ? H4)
- [cut (gcd n (n*n2) = n)
+cut (O < n1)
+ [elim (gcd_times_SO_to_gcd_SO n n n1 ? ? H4)
+ [cut (gcd n (n*n1) = n)
[apply (lt_to_not_eq (S O) n)
[assumption|rewrite < H4.assumption]
|apply gcd_n_times_nm.assumption
|apply (trans_lt ? (S O))[apply le_n|assumption]
|assumption
]
- |elim (le_to_or_lt_eq O n2 (le_O_n n2));
+ |elim (le_to_or_lt_eq O n1 (le_O_n n1));
[assumption
|apply False_ind.
apply (le_to_not_lt n (S O))
apply divides_to_le
[rewrite > H4.apply lt_O_S
|apply divides_d_gcd
- [apply (witness ? ? n2).reflexivity
+ [apply (witness ? ? n1).reflexivity
|apply divides_n_n
]
]
|right.
cut (\exists a,b. a*n - b*p = (S O) \lor b*p - a*n = (S O))
[elim Hcut1.elim H3.elim H4
- [(* first case *)
- rewrite > (times_n_SO q).rewrite < H5.
+ [rewrite > (times_n_SO q).rewrite < H5.
rewrite > distr_times_minus.
+ elim H1.
+ autobatch by witness;
+ (*
rewrite > (sym_times q (a1*p)).
rewrite > (assoc_times a1).
- elim H1.
- (*
- rewrite > H6.
- applyS (witness n (n*(q*a-a1*n2)) (q*a-a1*n2))
- reflexivity. *);
- applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
- (*
+ applyS (witness n ? ? (refl_eq ? ?)).
rewrite < (sym_times n).rewrite < assoc_times.
rewrite > (sym_times q).rewrite > assoc_times.
rewrite < (assoc_times a1).rewrite < (sym_times n).
|(* second case *)
rewrite > (times_n_SO q).rewrite < H5.
rewrite > distr_times_minus.
+ elim H1. autobatch by witness;
+ (*
rewrite > (sym_times q (a1*p)).
rewrite > (assoc_times a1).
- elim H1.rewrite > H6.
+ rewrite > H6.
+ applyS (witness n ? ? (refl_eq ? ?)).
rewrite < sym_times.rewrite > assoc_times.
rewrite < (assoc_times q).
rewrite < (sym_times n).
rewrite < distr_times_minus.
- apply (witness ? ? (n2*a1-q*a)).reflexivity
+ apply (witness ? ? (n1*a1-q*a)).reflexivity
+ *)
](* end second case *)
|rewrite < (prime_to_gcd_SO n p)
[apply eq_minus_gcd|assumption|assumption
|cut (\exists a,b. a*n - b*m = (S O) \lor b*m - a*n = (S O))
[elim Hcut1.elim H4.elim H5
[(* first case *)
+ elim H2.
rewrite > (times_n_SO p).rewrite < H6.
rewrite > distr_times_minus.
- rewrite > (sym_times p (a1*m)).
- rewrite > (assoc_times a1).
+ autobatch by witness, divides_minus.
+ |(* second case *)
elim H2.
- applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
- |(* second case *)
rewrite > (times_n_SO p).rewrite < H6.
rewrite > distr_times_minus.
- rewrite > (sym_times p (a1*m)).
- rewrite > (assoc_times a1).
- elim H2.
- applyS (witness n ? ? (refl_eq ? ?)).
+ autobatch by witness, divides_minus.
](* end second case *)
|rewrite < H1.apply eq_minus_gcd.
]
elim (divides_times_to_divides ? ? ? H H2)
[apply False_ind.apply H1.assumption
|elim H5.
- apply (witness ? ? n1).
+ autobatch by transitive_divides, H5, reflexive_divides,divides_times.
+ (*
+ apply (witness ? ? n2).
rewrite > sym_times in ⊢ (? ? ? (? % ?)).
rewrite > assoc_times.
- rewrite < H6.assumption
+ autobatch.
+ (*rewrite < H6.assumption*)
+ *)
]
qed.
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