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(* 02110-1301 USA                                                     *)
(** modprime. Some nice lemmas about div, mult, mod, prime, and gcd.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)

(* Specifications: *)
Lemma prime_div_or_gcd1 :
forall (p : nat) (a : Z),
Prime p -> ZDivides (Z_of_nat p) a \/ gcd (Z_of_nat p) a 1.
Lemma divmultgcd :
forall a b c : Z,
a <> 0%Z -> ZDivides a (b * c) -> gcd a b 1 -> ZDivides a c.
Lemma primedivmult :
forall p n m : nat,
Prime p -> Divides p (n * m) -> Divides p n \/ Divides p m.
Lemma mod_mult_inv_r :
forall (a : Z) (p : nat),
Prime p -> ~ Mod a 0 p -> exists ra : Z, Mod (a * ra) 1 p.
Lemma mod_mult_cancel_r :
forall (a b c : Z) (p : nat),
Prime p -> ~ Mod c 0 p -> Mod (a * c) (b * c) p -> Mod a b p.
Lemma mod_mult_0 :
forall (p : nat) (a b : Z),
Prime p -> Mod (a * b) 0 p -> Mod a 0 p \/ Mod b 0 p.
Lemma mod_not_exp_0 :
forall p : nat,
Prime p -> forall a : Z, ~ Mod a 0 p -> forall m : nat, ~ Mod (Exp a m) 0 p.