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, 10 insertions, 19 deletions

diff --git a/theories/Lists/List.v b/theories/Lists/List.v
index 7296d07ec..5e161ffab 100644
--- a/theories/Lists/List.v
+++ b/theories/Lists/List.v
@@ -619,30 +619,21 @@ Section Elts.
     end.
 
   (** Compatibility of count_occ with operations on list *)
-  Theorem count_occ_In : forall (l : list A) (x : A), In x l <-> count_occ l x > 0.
+  Theorem count_occ_In (l : list A) (x : A) : In x l <-> count_occ l x > 0.
   Proof.
-    induction l as [|y l].
-    simpl; intros; split; [destruct 1 | apply gt_irrefl].
-    simpl. intro x; destruct (eq_dec y x) as [Heq|Hneq].
-    rewrite Heq; intuition.
-    pose (IHl x). intuition.
+    induction l as [|y l]; simpl.
+    - split; [destruct 1 | apply gt_irrefl].
+    - destruct eq_dec as [->|Hneq]; rewrite IHl; intuition.
   Qed.
 
-  Theorem count_occ_inv_nil : forall (l : list A), (forall x:A, count_occ l x = 0) <-> l = [].
+  Theorem count_occ_inv_nil (l : list A) :
+    (forall x:A, count_occ l x = 0) <-> l = [].
   Proof.
     split.
-    (* Case -> *)
-    induction l as [|x l].
-    trivial.
-    intro H.
-    elim (O_S (count_occ l x)).
-    apply sym_eq.
-    generalize (H x).
-    simpl. destruct (eq_dec x x) as [|HF].
-    trivial.
-    elim HF; reflexivity.
-    (* Case <- *)
-    intro H; rewrite H; simpl; reflexivity.
+    - induction l as [|x l]; trivial.
+      intros H. specialize (H x). simpl in H.
+      destruct eq_dec as [_|NEQ]; [discriminate|now elim NEQ].
+    - now intros ->.
   Qed.
 
   Lemma count_occ_nil : forall (x : A), count_occ [] x = 0.