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

3 files changed, 24 insertions, 34 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.
diff --git a/theories/Lists/ListTactics.v b/theories/Lists/ListTactics.v
index 3343aa6f3..447f420ea 100644
--- a/theories/Lists/ListTactics.v
+++ b/theories/Lists/ListTactics.v
@@ -60,7 +60,7 @@ Ltac Find_at a l :=
    match l with
    | nil     => fail 100 "anomaly: Find_at"
    | a :: _  => eval compute in n
-   | _ :: ?l => find (Psucc n) l
+   | _ :: ?l => find (Pos.succ n) l
    end
  in find 1%positive l.
 
diff --git a/theories/Lists/StreamMemo.v b/theories/Lists/StreamMemo.v
index 45490c623..b6f2d5ed2 100644
--- a/theories/Lists/StreamMemo.v
+++ b/theories/Lists/StreamMemo.v
@@ -32,10 +32,10 @@ Fixpoint memo_get (n:nat) (l:Stream A) : A :=
 Theorem memo_get_correct: forall n, memo_get n memo_list = f n.
 Proof.
 assert (F1: forall n m, memo_get n (memo_make m) = f (n + m)).
-  induction n as [| n Hrec]; try (intros m; refine (refl_equal _)).
+{ induction n as [| n Hrec]; try (intros m; reflexivity).
   intros m; simpl; rewrite Hrec.
-  rewrite plus_n_Sm; auto.
-intros n; apply trans_equal with (f (n + 0)); try exact (F1 n 0).
+  rewrite plus_n_Sm; auto. }
+intros n; transitivity (f (n + 0)); try exact (F1 n 0).
 rewrite <- plus_n_O; auto.
 Qed.
 
@@ -57,11 +57,10 @@ Definition imemo_list := let f0 := f 0 in
 
 Theorem imemo_get_correct: forall n, memo_get n imemo_list = f n.
 Proof.
-assert (F1: forall n m,
-    memo_get n (imemo_make (f m)) = f (S (n + m))).
-  induction n as [| n Hrec]; try (intros m; exact (sym_equal (Hg_correct m))).
-  simpl; intros m; rewrite <- Hg_correct; rewrite Hrec; rewrite <- plus_n_Sm; auto.
-destruct n as [| n]; try apply refl_equal.
+assert (F1: forall n m, memo_get n (imemo_make (f m)) = f (S (n + m))).
+{ induction n as [| n Hrec]; try (intros m; exact (eq_sym (Hg_correct m))).
+  simpl; intros m; rewrite <- Hg_correct, Hrec, <- plus_n_Sm; auto. }
+destruct n as [| n]; try reflexivity.
 unfold imemo_list; simpl; rewrite F1.
 rewrite <- plus_n_O; auto.
 Qed.
@@ -82,7 +81,7 @@ Inductive memo_val: Type :=
 
 Fixpoint is_eq (n m : nat) : {n = m} + {True} :=
   match n, m return {n = m} + {True} with
-  | 0, 0 =>left True (refl_equal 0)
+  | 0, 0 =>left True (eq_refl 0)
   | 0, S m1 => right (0 = S m1) I
   | S n1, 0 => right (S n1 = 0) I
   | S n1, S m1 =>
@@ -98,7 +97,7 @@ match v with
     match is_eq n m with
     | left H =>
        match H in (eq _ y) return (A y -> A n) with
-        | refl_equal => fun v1 : A n => v1
+        | eq_refl => fun v1 : A n => v1
        end
     | right _ => fun _ : A m => f n
     end x
@@ -115,7 +114,7 @@ Proof.
 intros n; unfold dmemo_get, dmemo_list.
 rewrite (memo_get_correct memo_val mf n); simpl.
 case (is_eq n n); simpl; auto; intros e.
-assert (e = refl_equal n).
+assert (e = eq_refl n).
  apply eq_proofs_unicity.
  induction x as [| x Hx]; destruct y as [| y].
  left; auto.
@@ -144,7 +143,7 @@ Proof.
 intros n; unfold dmemo_get, dimemo_list.
 rewrite (imemo_get_correct memo_val mf mg); simpl.
 case (is_eq n n); simpl; auto; intros e.
-assert (e = refl_equal n).
+assert (e = eq_refl n).
  apply eq_proofs_unicity.
  induction x as [| x Hx]; destruct y as [| y].
  left; auto.
@@ -169,11 +168,11 @@ Open Scope Z_scope.
 Fixpoint tfact (n: nat) :=
   match n with
    | O => 1
-   | S n1 => Z_of_nat n * tfact n1
+   | S n1 => Z.of_nat n * tfact n1
   end.
 
 Definition lfact_list :=
-  dimemo_list _ tfact (fun n z => (Z_of_nat (S  n) * z)).
+  dimemo_list _ tfact (fun n z => (Z.of_nat (S  n) * z)).
 
 Definition lfact n := dmemo_get _ tfact n lfact_list.