4 files changed, 110 insertions, 21 deletions

diff --git a/contrib/setoid_ring/Ring_zdiv.v b/contrib/setoid_ring/Ring_zdiv.v
index 9b102c28c..44874e547 100644
--- a/contrib/setoid_ring/Ring_zdiv.v
+++ b/contrib/setoid_ring/Ring_zdiv.v
@@ -40,10 +40,9 @@ destruct a; intros; simpl; auto.
   rewrite (Pcompare_Eq_eq _ _ H).
   assert (HH: forall p, Pminus p p = xH).
     intro p0; unfold Pminus; rewrite Pminus_mask_diag; auto.
-  simpl; auto.
-  generalize (Pplus_minus _ _ H).
-  case (p - b)%positive; intros; subst;
-  unfold Nplus; rewrite Pplus_comm; trivial.
+  apply Pcompare_Eq_eq in H. now rewrite Pminus_mask_diag.
+  pose proof (Pminus_mask_Gt p b H) as H1. destruct H1 as [d [H2 [H3 _]]].
+  rewrite H2. rewrite <- H3. unfold Nplus. now rewrite Pplus_comm.
 Qed.
 
 Theorem Z_of_N_plus: 
diff --git a/theories/NArith/BinNat.v b/theories/NArith/BinNat.v
index b26a22c9e..0a275c850 100644
--- a/theories/NArith/BinNat.v
+++ b/theories/NArith/BinNat.v
@@ -59,6 +59,16 @@ Definition Nsucc n :=
   | Npos p => Npos (Psucc p)
   end.
 
+(** Predecessor *)
+
+Definition Npred (n : N) := match n with
+| N0 => N0
+| Npos p => match p with
+  | xH => N0
+  | _ => Npos (Ppred p)
+  end
+end.
+
 (** Addition *)
 
 Definition Nplus n m :=
@@ -70,9 +80,20 @@ Definition Nplus n m :=
 
 Infix "+" := Nplus : N_scope.
 
-(** Substraction *)
+(** Subtraction *)
+
+Definition Nminus (n m : N) :=
+match n, m with
+| N0, _ => N0
+| n, N0 => n
+| Npos n', Npos m' =>
+  match Pminus_mask n' m' with
+  | IsPos p => Npos p
+  | _ => N0
+  end
+end.
 
-Definition Nminus (x y:N) :=
+(*Definition Nminus (x y:N) :=
   match x, y with
     | N0, _ => N0
     | x, N0 => x
@@ -81,7 +102,7 @@ Definition Nminus (x y:N) :=
         | Lt | Eq => N0
         | Gt => Npos (x' - y')
       end
-  end.
+      end.*)
 
 Infix "-" := Nminus : N_scope.
 
@@ -195,6 +216,15 @@ Proof.
 intros P a f; unfold Nrec; apply Nrect_step.
 Qed.
 
+(** Properties of successor and predecessor *)
+
+Theorem Npred_succ : forall n : N, Npred (Nsucc n) = n.
+Proof.
+destruct n as [| p]; simpl. reflexivity.
+case_eq (Psucc p); try (intros q H; rewrite <- H; now rewrite Ppred_succ).
+intro H; false_hyp H Psucc_not_one.
+Qed.
+
 (** Properties of addition *)
 
 Theorem Nplus_0_l : forall n:N, N0 + n = n.
@@ -254,9 +284,9 @@ intro n; pattern n in |- *; apply Nind; clear n; simpl in |- *.
   apply IHn; apply Nsucc_inj; assumption.
 Qed.
 
-(** Properties of substraction. *)
+(** Properties of subtraction. *)
 
-Lemma Nle_Nminus_N0 : forall n n':N, n <= n' -> n-n' = N0.
+(*Lemma Nle_Nminus_N0 : forall n n' : N, n <= n' -> n - n' = N0.
 Proof.
   destruct n; destruct n'; simpl; intros; auto.
   elim H; auto.
@@ -264,11 +294,39 @@ Proof.
   intros; destruct (Pcompare p p0 Eq); auto.
   elim H; auto.
 Qed.
+Proof.
+destruct n as [| p]; destruct n' as [| q]; simpl; intros; auto.
+now elim H.
+red in H; simpl in *.
+case_eq (Pcompare p q Eq); intro H1.
+assert (H2 : p = q); [now apply Pcompare_Eq_eq |]. now rewrite H2, Pminus_mask_diag.
+now rewrite Pminus_mask_Lt.
+false_hyp H1 H.
+Qed.*)
+
+Lemma Nminus_N0_Nle : forall n n' : N, n - n' = N0 <-> n <= n'.
+Proof.
+destruct n as [| p]; destruct n' as [| q]; unfold Nle; simpl;
+split; intro H; try discriminate; try reflexivity.
+now elim H.
+intro H1; apply Pminus_mask_Gt in H1. destruct H1 as [h [H1 _]].
+rewrite H1 in H; discriminate.
+case_eq (Pcompare p q Eq); intro H1; rewrite H1 in H; try now elim H.
+assert (H2 : p = q); [now apply Pcompare_Eq_eq |]. now rewrite H2, Pminus_mask_diag.
+now rewrite Pminus_mask_Lt.
+Qed.
+
+Theorem Nminus_0_r : forall n : N, n - N0 = n.
+Proof.
+now destruct n.
+Qed.
 
-Lemma Nminus_N0_Nle : forall n n':N, n-n' = N0 -> n <= n'.
+Theorem Nminus_succ_r : forall n m : N, n - (Nsucc m) = Npred (n - m).
 Proof.
-  destruct n; destruct n'; red; simpl; intros; try discriminate.
-  destruct (Pcompare p p0 Eq); discriminate.
+destruct n as [| p]; destruct m as [| q]; try reflexivity.
+now destruct p.
+simpl. rewrite Pminus_mask_succ_r, Pminus_mask_carry_spec.
+now destruct (Pminus_mask p q) as [| r |]; [| destruct r |].
 Qed.
 
 (** Properties of multiplication *)
@@ -348,13 +406,6 @@ destruct n; destruct m; simpl; auto.
 exact (Pcompare_antisym p p0 Eq).
 Qed.
 
-(** 0 is the least natural number *)
-
-Theorem Ncompare_0 : forall n : N, Ncompare n N0 <> Lt.
-Proof.
-destruct n; discriminate.
-Qed.
-
 Theorem Ncompare_n_Sm :
   forall n m : N, Ncompare n (Nsucc m) = Lt <-> Ncompare n m = Lt \/ n = m.
 Proof.
@@ -367,6 +418,13 @@ pose proof (proj2 (Pcompare_p_Sq p q));
 assert (p = q <-> Npos p = Npos q); [split; congruence | tauto].
 Qed.
 
+(** 0 is the least natural number *)
+
+Theorem Ncompare_0 : forall n : N, Ncompare n N0 <> Lt.
+Proof.
+destruct n; discriminate.
+Qed.
+
 (** Dividing by 2 *)
 
 Definition Ndiv2 (n:N) :=
diff --git a/theories/NArith/BinPos.v b/theories/NArith/BinPos.v
index 28c6fdb6d..fafc3285b 100644
--- a/theories/NArith/BinPos.v
+++ b/theories/NArith/BinPos.v
@@ -980,6 +980,35 @@ Proof.
   destruct p; compute; auto.
 Qed.
 
+Definition Ppred_mask (p : positive_mask) :=
+match p with
+| IsPos xH => IsNul
+| IsPos q => IsPos (Ppred q)
+| IsNul => IsNeg
+| IsNeg => IsNeg
+end.
+
+Lemma Pminus_mask_succ_r :
+  forall p q : positive, Pminus_mask p (Psucc q) = Pminus_mask_carry p q.
+Proof.
+induction p; destruct q; simpl in *; (now try rewrite IHp) || (now destruct p).
+Qed.
+
+Theorem Pminus_mask_carry_spec :
+  forall p q : positive, Pminus_mask_carry p q = Ppred_mask (Pminus_mask p q).
+Proof.
+induction p; destruct q; simpl; try reflexivity;
+try rewrite IHp; try now destruct (Pminus_mask p q) as [| r |]; [| destruct r |].
+now destruct p.
+Qed.
+
+Theorem Pminus_succ_r : forall p q : positive, p - (Psucc q) = Ppred (p - q).
+Proof.
+intros p q; unfold Pminus. rewrite Pminus_mask_succ_r.
+rewrite Pminus_mask_carry_spec.
+now destruct (Pminus_mask p q) as [| r |]; [| destruct r |].
+Qed.
+
 Lemma double_eq_zero_inversion :
   forall p:positive_mask, Pdouble_mask p = IsNul -> p = IsNul.
 Proof.
diff --git a/theories/NArith/Nnat.v b/theories/NArith/Nnat.v
index 76a3d616c..55c3a3b78 100644
--- a/theories/NArith/Nnat.v
+++ b/theories/NArith/Nnat.v
@@ -120,8 +120,11 @@ Proof.
   destruct a; destruct a'; simpl; auto with arith.
   case_eq (Pcompare p p0 Eq); simpl; intros.
   rewrite (Pcompare_Eq_eq _ _ H); auto with arith.
-  symmetry; apply not_le_minus_0.
-  generalize (nat_of_P_lt_Lt_compare_morphism _ _ H); auto with arith.
+  rewrite Pminus_mask_diag. simpl. apply minus_n_n.
+  rewrite Pminus_mask_Lt. pose proof (nat_of_P_lt_Lt_compare_morphism _ _ H). simpl.
+  symmetry; apply not_le_minus_0. auto with arith. assumption.
+  pose proof (Pminus_mask_Gt p p0 H) as H1. destruct H1 as [q [H1 _]]. rewrite H1; simpl.
+  replace q with (Pminus p p0) by (unfold Pminus; now rewrite H1).
   apply nat_of_P_minus_morphism; auto.
 Qed.