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

1 files changed, 8 insertions, 11 deletions

diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v
index 30802f824..efe5b6847 100644
--- a/theories/ZArith/Zwf.v
+++ b/theories/ZArith/Zwf.v
@@ -29,7 +29,7 @@ Section wf_proof.
   (** The proof of well-foundness is classic: we do the proof by induction
       on a measure in nat, which is here [|x-c|] *)
 
-  Let f (z:Z) := Zabs_nat (z - c).
+  Let f (z:Z) := Z.abs_nat (z - c).
 
   Lemma Zwf_well_founded : well_founded (Zwf c).
     red in |- *; intros.
@@ -45,12 +45,12 @@ Section wf_proof.
     apply Acc_intro; intros.
     apply H.
     unfold Zwf in H1.
-    case (Zle_or_lt c y); intro; auto with zarith.
+    case (Z.le_gt_cases c y); intro; auto with zarith.
     left.
     red in H0.
     apply lt_le_trans with (f a); auto with arith.
-    unfold f in |- *.
-    apply Zabs.Zabs_nat_lt; omega.
+    unfold f.
+    apply Zabs2Nat.inj_lt; omega.
     apply (H (S (f a))); auto.
   Qed.
 
@@ -75,18 +75,15 @@ Section wf_proof_up.
   (** The proof of well-foundness is classic: we do the proof by induction
       on a measure in nat, which is here [|c-x|] *)
 
-  Let f (z:Z) := Zabs_nat (c - z).
+  Let f (z:Z) := Z.abs_nat (c - z).
 
   Lemma Zwf_up_well_founded : well_founded (Zwf_up c).
   Proof.
     apply well_founded_lt_compat with (f := f).
-    unfold Zwf_up, f in |- *.
+    unfold Zwf_up, f.
     intros.
-    apply Zabs.Zabs_nat_lt.
-    unfold Zminus in |- *. split.
-    apply Zle_left; intuition.
-    apply Zplus_lt_compat_l; unfold Zlt in |- *; rewrite <- Zcompare_opp;
-      intuition.
+    apply Zabs2Nat.inj_lt; try (apply Z.le_0_sub; intuition).
+    now apply Z.sub_lt_mono_l.
   Qed.
 
 End wf_proof_up.