diff options
Diffstat (limited to 'theories/NArith/Ndiv_def.v')
| -rw-r--r-- | theories/NArith/Ndiv_def.v | 14 |
1 files changed, 6 insertions, 8 deletions
diff --git a/theories/NArith/Ndiv_def.v b/theories/NArith/Ndiv_def.v index 2a3fd152a..0850a631e 100644 --- a/theories/NArith/Ndiv_def.v +++ b/theories/NArith/Ndiv_def.v @@ -14,7 +14,7 @@ Local Open Scope N_scope. Definition NPgeb (a:N)(b:positive) := match a with | 0 => false - | Npos na => match Pcompare na b Eq with Lt => false | _ => true end + | Npos na => match Pos.compare na b with Lt => false | _ => true end end. Local Notation "a >=? b" := (NPgeb a b) (at level 70). @@ -54,24 +54,22 @@ Lemma NPgeb_ge : forall a b, NPgeb a b = true -> a >= Npos b. Proof. destruct a; simpl; intros. discriminate. - unfold Nge, Ncompare. now destruct Pcompare. + unfold Nge, Ncompare. now destruct Pos.compare. Qed. Lemma NPgeb_lt : forall a b, NPgeb a b = false -> a < Npos b. Proof. destruct a; simpl; intros. red; auto. - unfold Nlt, Ncompare. now destruct Pcompare. + unfold Nlt, Ncompare. now destruct Pos.compare. Qed. Theorem NPgeb_correct: forall (a:N)(b:positive), if NPgeb a b then a = a - Npos b + Npos b else True. Proof. destruct a as [|a]; simpl; intros b; auto. - generalize (Pcompare_Eq_eq a b). - case_eq (Pcompare a b Eq); intros; auto. - rewrite H0; auto. + case Pos.compare_spec; intros; subst; auto. now rewrite Pminus_mask_diag. - destruct (Pminus_mask_Gt a b H) as [d [H2 [H3 _]]]. + destruct (Pminus_mask_Gt a b (Pos.lt_gt _ _ H)) as [d [H2 [H3 _]]]. rewrite H2. rewrite <- H3. simpl; f_equal; apply Pplus_comm. Qed. @@ -96,7 +94,7 @@ rewrite Nplus_comm. generalize (NPgeb_correct (2*a+1) p). rewrite GE. intros <-. rewrite <- (Nmult_1_l (Npos p)). rewrite <- Nmult_plus_distr_r. destruct a; auto. -red; simpl. apply Pcompare_eq_Lt; auto. +red; simpl. apply Pcompare_Gt_Lt; auto. Qed. (* Proofs of specifications for these euclidean divisions. *) |