ZArith + other : favor the use of modern names instead of compat notations

 - For instance, refl_equal --> eq_refl
 - Npos, Zpos, Zneg now admit more uniform qualified aliases
   N.pos, Z.pos, Z.neg.
 - A new module BinInt.Pos2Z with results about injections from
   positive to Z
 - A result about Z.pow pushed in the generic layer
 - Zmult_le_compat_{r,l} --> Z.mul_le_mono_nonneg_{r,l}
 - Using tactic Z.le_elim instead of Zle_lt_or_eq
 - Some cleanup in ring, field, micromega
   (use of "Equivalence", "Proper" ...)
 - Some adaptions in QArith (for instance changed Qpower.Qpower_decomp)
 - In ZMake and ZMake, functor parameters are now named NN and ZZ
   instead of N and Z for avoiding confusions

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15515 85f007b7-540e-0410-9357-904b9bb8a0f7

2 files changed, 22 insertions, 20 deletions

diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v
index e22cac7be..58fe090dd 100644
--- a/theories/PArith/BinPos.v
+++ b/theories/PArith/BinPos.v
@@ -1871,6 +1871,10 @@ Notation xI := xI (only parsing).
 Notation xO := xO (only parsing).
 Notation xH := xH (only parsing).
 
+Notation IsNul := Pos.IsNul (only parsing).
+Notation IsPos := Pos.IsPos (only parsing).
+Notation IsNeg := Pos.IsNeg (only parsing).
+
 Notation Psucc := Pos.succ (compat "8.3").
 Notation Pplus := Pos.add (compat "8.3").
 Notation Pplus_carry := Pos.add_carry (compat "8.3").
@@ -1882,9 +1886,6 @@ Notation nat_of_P := Pos.to_nat (compat "8.3").
 Notation P_of_succ_nat := Pos.of_succ_nat (compat "8.3").
 Notation Pdouble_minus_one := Pos.pred_double (compat "8.3").
 Notation positive_mask := Pos.mask (compat "8.3").
-Notation IsNul := Pos.IsNul (compat "8.3").
-Notation IsPos := Pos.IsPos (compat "8.3").
-Notation IsNeg := Pos.IsNeg (compat "8.3").
 Notation positive_mask_rect := Pos.mask_rect (compat "8.3").
 Notation positive_mask_ind := Pos.mask_ind (compat "8.3").
 Notation positive_mask_rec := Pos.mask_rec (compat "8.3").
@@ -2056,24 +2057,24 @@ Lemma Peqb_true_eq x y : Pos.eqb x y = true -> x=y.
 Proof. apply Pos.eqb_eq. Qed.
 Lemma Pcompare_eq_Gt p q : (p ?= q) = Gt <-> p > q.
 Proof. reflexivity. Qed.
-Lemma Pplus_one_succ_r p : Psucc p = p + 1.
+Lemma Pplus_one_succ_r p : Pos.succ p = p + 1.
 Proof (eq_sym (Pos.add_1_r p)).
-Lemma Pplus_one_succ_l p : Psucc p = 1 + p.
+Lemma Pplus_one_succ_l p : Pos.succ p = 1 + p.
 Proof (eq_sym (Pos.add_1_l p)).
-Lemma Pcompare_refl p : Pcompare p p Eq = Eq.
+Lemma Pcompare_refl p : Pos.compare_cont p p Eq = Eq.
 Proof (Pos.compare_cont_refl p Eq).
-Lemma Pcompare_Eq_eq : forall p q, Pcompare p q Eq = Eq -> p = q.
+Lemma Pcompare_Eq_eq : forall p q, Pos.compare_cont p q Eq = Eq -> p = q.
 Proof Pos.compare_eq.
-Lemma ZC4 p q : Pcompare p q Eq = CompOpp (Pcompare q p Eq).
+Lemma ZC4 p q : Pos.compare_cont p q Eq = CompOpp (Pos.compare_cont q p Eq).
 Proof (Pos.compare_antisym q p).
-Lemma Ppred_minus p : Ppred p = p - 1.
+Lemma Ppred_minus p : Pos.pred p = p - 1.
 Proof (eq_sym (Pos.sub_1_r p)).
 
 Lemma Pminus_mask_Gt p q :
   p > q ->
   exists h : positive,
-   Pminus_mask p q = IsPos h /\
-   q + h = p /\ (h = 1 \/ Pminus_mask_carry p q = IsPos (Ppred h)).
+   Pos.sub_mask p q = IsPos h /\
+   q + h = p /\ (h = 1 \/ Pos.sub_mask_carry p q = IsPos (Pos.pred h)).
 Proof.
  intros H. apply Pos.gt_lt in H.
  destruct (Pos.sub_mask_pos p q H) as (r & U).
diff --git a/theories/PArith/Pnat.v b/theories/PArith/Pnat.v
index 0fa6972b7..e9c77b05d 100644
--- a/theories/PArith/Pnat.v
+++ b/theories/PArith/Pnat.v
@@ -30,7 +30,7 @@ Qed.
 
 Theorem inj_add p q : to_nat (p + q) = to_nat p + to_nat q.
 Proof.
- revert q. induction p using Pind; intros q.
+ revert q. induction p using peano_ind; intros q.
  now rewrite add_1_l, inj_succ.
  now rewrite add_succ_l, !inj_succ, IHp.
 Qed.
@@ -410,23 +410,24 @@ Notation P_of_succ_nat_o_nat_of_P_eq_succ := Pos2SuccNat.id_succ (compat "8.3").
 Notation pred_o_P_of_succ_nat_o_nat_of_P_eq_id := Pos2SuccNat.pred_id (compat "8.3").
 
 Lemma nat_of_P_minus_morphism p q :
- Pcompare p q Eq = Gt -> Pos.to_nat (p - q) = Pos.to_nat p - Pos.to_nat q.
-Proof (fun H => Pos2Nat.inj_sub p q (ZC1 _ _ H)).
+ Pos.compare_cont p q Eq = Gt ->
+  Pos.to_nat (p - q) = Pos.to_nat p - Pos.to_nat q.
+Proof (fun H => Pos2Nat.inj_sub p q (Pos.gt_lt _ _ H)).
 
 Lemma nat_of_P_lt_Lt_compare_morphism p q :
- Pcompare p q Eq = Lt -> Pos.to_nat p < Pos.to_nat q.
+ Pos.compare_cont p q Eq = Lt -> Pos.to_nat p < Pos.to_nat q.
 Proof (proj1 (Pos2Nat.inj_lt p q)).
 
 Lemma nat_of_P_gt_Gt_compare_morphism p q :
- Pcompare p q Eq = Gt -> Pos.to_nat p > Pos.to_nat q.
+ Pos.compare_cont p q Eq = Gt -> Pos.to_nat p > Pos.to_nat q.
 Proof (proj1 (Pos2Nat.inj_gt p q)).
 
 Lemma nat_of_P_lt_Lt_compare_complement_morphism p q :
- Pos.to_nat p < Pos.to_nat q -> Pcompare p q Eq = Lt.
+ Pos.to_nat p < Pos.to_nat q -> Pos.compare_cont p q Eq = Lt.
 Proof (proj2 (Pos2Nat.inj_lt p q)).
 
 Definition nat_of_P_gt_Gt_compare_complement_morphism p q :
- Pos.to_nat p > Pos.to_nat q -> Pcompare p q Eq = Gt.
+ Pos.to_nat p > Pos.to_nat q -> Pos.compare_cont p q Eq = Gt.
 Proof (proj2 (Pos2Nat.inj_gt p q)).
 
 (** Old intermediate results about [Pmult_nat] *)
@@ -445,7 +446,7 @@ Proof.
 Qed.
 
 Lemma Pmult_nat_succ_morphism :
- forall p n, Pmult_nat (Psucc p) n = n + Pmult_nat p n.
+ forall p n, Pmult_nat (Pos.succ p) n = n + Pmult_nat p n.
 Proof.
  intros. now rewrite !Pmult_nat_mult, Pos2Nat.inj_succ.
 Qed.
@@ -457,7 +458,7 @@ Proof.
 Qed.
 
 Theorem Pmult_nat_plus_carry_morphism :
- forall p q n, Pmult_nat (Pplus_carry p q) n = n + Pmult_nat (p + q) n.
+ forall p q n, Pmult_nat (Pos.add_carry p q) n = n + Pmult_nat (p + q) n.
 Proof.
  intros. now rewrite Pos.add_carry_spec, Pmult_nat_succ_morphism.
 Qed.