Missing file added

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lib/coq/number/Coprime.v

+(* This file is generated by Why3's Coq-realize driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import BuiltIn.
+Require Import ZOdiv.
+Require BuiltIn.
+Require int.Int.
+Require int.Abs.
+Require int.EuclideanDivision.
+Require int.ComputerDivision.
+Require number.Parity.
+Require number.Divisibility.
+Require number.Gcd.
+Require number.Prime.
+
+(* Why3 assumption *)
+Definition coprime (a:Z) (b:Z): Prop := ((number.Gcd.gcd a b) = 1%Z).
+
+Lemma coprime_is_Zrel_prime :
+  forall a b, coprime a b <-> Znumtheory.rel_prime a b.
+intros.
+unfold coprime.
+unfold Znumtheory.rel_prime.
+split; intro h.
+rewrite <- h; apply Znumtheory.Zgcd_is_gcd.
+apply Znumtheory.Zis_gcd_gcd; auto with zarith.
+Qed.
+
+(* Why3 goal *)
+Lemma prime_coprime : forall (p:Z), (number.Prime.prime p) <->
+  ((2%Z <= p)%Z /\ forall (n:Z), ((1%Z <= n)%Z /\ (n < p)%Z) -> (coprime n
+  p)).
+intros p.
+(*
+Znumtheory.prime_intro:
+  forall p : int,
+  (1 < p)%Z ->
+  (forall n : int, (1 <= n < p)%Z -> Znumtheory.rel_prime n p) ->
+  Znumtheory.prime p
+*)
+rewrite Prime.prime_is_Zprime.
+split.
+intro h; inversion h; clear h.
+split; auto with zarith.
+intros n h.
+rewrite coprime_is_Zrel_prime.
+apply H0; auto.
+intros (h1,h2).
+constructor; auto with zarith.
+intros n h.
+rewrite <- coprime_is_Zrel_prime.
+apply h2; auto.
+Qed.
+
+(* Why3 goal *)
+Lemma Gauss : forall (a:Z) (b:Z) (c:Z), ((number.Divisibility.divides a
+  (b * c)%Z) /\ (coprime a b)) -> (number.Divisibility.divides a c).
+intros a b c (h1,h2).
+apply Znumtheory.Gauss with b; auto.
+rewrite <- coprime_is_Zrel_prime; auto.
+Qed.
+
+(* Why3 goal *)
+Lemma Euclid : forall (p:Z) (a:Z) (b:Z), ((number.Prime.prime p) /\
+  (number.Divisibility.divides p (a * b)%Z)) -> ((number.Divisibility.divides
+  p a) \/ (number.Divisibility.divides p b)).
+intros p a b (h1,h2).
+apply Znumtheory.prime_mult; auto.
+now rewrite <- Prime.prime_is_Zprime.
+Qed.