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-rw-r--r--theories/NArith/Ndiv_def.v14
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. *)