diff options
Diffstat (limited to 'theories/ZArith/Znumtheory.v')
| -rw-r--r-- | theories/ZArith/Znumtheory.v | 19 |
1 files changed, 9 insertions, 10 deletions
diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v index 715cdc7d..a1963446 100644 --- a/theories/ZArith/Znumtheory.v +++ b/theories/ZArith/Znumtheory.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Znumtheory.v,v 1.5.2.1 2004/07/16 19:31:22 herbelin Exp $ i*) +(*i $Id: Znumtheory.v 6984 2005-05-02 10:50:15Z herbelin $ i*) Require Import ZArith_base. Require Import ZArithRing. @@ -278,12 +278,12 @@ Lemma euclid_rec : (forall d:Z, Zis_gcd u3 v3 d -> Zis_gcd a b d) -> Euclid. Proof. intros v3 Hv3; generalize Hv3; pattern v3 in |- *. -apply Z_lt_rec. +apply Zlt_0_rec. clear v3 Hv3; intros. elim (Z_zerop x); intro. apply Euclid_intro with (u := u1) (v := u2) (d := u3). assumption. -apply H2. +apply H3. rewrite a0; auto with zarith. set (q := u3 / x) in *. assert (Hq : 0 <= u3 - q * x < x). @@ -297,9 +297,9 @@ apply (H (u3 - q * x) Hq (proj1 Hq) v1 v2 x (u1 - q * v1) (u2 - q * v2)). tauto. replace ((u1 - q * v1) * a + (u2 - q * v2) * b) with (u1 * a + u2 * b - q * (v1 * a + v2 * b)). -rewrite H0; rewrite H1; trivial. +rewrite H1; rewrite H2; trivial. ring. -intros; apply H2. +intros; apply H3. apply Zis_gcd_for_euclid with q; assumption. assumption. Qed. @@ -377,11 +377,11 @@ Definition Zgcd_pos : Proof. intros a Ha. apply - (Z_lt_rec + (Zlt_0_rec (fun a:Z => forall b:Z, {g : Z | 0 <= a -> Zis_gcd a b g /\ g >= 0})); try assumption. intro x; case x. -intros _ b; exists (Zabs b). +intros _ _ b; exists (Zabs b). elim (Z_le_lt_eq_dec _ _ (Zabs_pos b)). intros H0; split. apply Zabs_ind. @@ -393,7 +393,7 @@ intros _ b; exists (Zabs b). rewrite <- (Zabs_Zsgn b); rewrite <- H0; simpl in |- *. split; [ apply Zis_gcd_0 | idtac ]; auto with zarith. -intros p Hrec b. +intros p Hrec _ b. generalize (Z_div_mod b (Zpos p)). case (Zdiv_eucl b (Zpos p)); intros q r Hqr. elim Hqr; clear Hqr; intros; auto with zarith. @@ -405,8 +405,7 @@ split; auto. rewrite H. apply Zis_gcd_for_euclid2; auto. -intros p Hrec b. -exists 0; intros. +intros p _ H b. elim H; auto. Defined. |