Follow-up concerning eqb / ltb / leb comparisons

 - All statement using reflect are made transparent.
   (Otherwise, since reflect isn't in Prop, extraction
    complains now about opaque Type definition).

 - remove two local Peqb_spec and Neqb_spec, now provided
   globally as Pos.eqb_spec and N.eqb_spec.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14232 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 9 insertions, 19 deletions

diff --git a/plugins/setoid_ring/Field_theory.v b/plugins/setoid_ring/Field_theory.v
index 6f4747dcf..40138526d 100644
--- a/plugins/setoid_ring/Field_theory.v
+++ b/plugins/setoid_ring/Field_theory.v
@@ -390,16 +390,6 @@ Qed.
 
   ***************************************************************************)
 
-Lemma Peqb_spec x y : Bool.reflect (x=y) (Pos.eqb x y).
-Proof.
- apply Bool.iff_reflect. symmetry. apply Pos.eqb_eq.
-Qed.
-
-Lemma Neqb_spec x y : Bool.reflect (x=y) (N.eqb x y).
-Proof.
- apply Bool.iff_reflect. symmetry. apply N.eqb_eq.
-Qed.
-
 (* equality test *)
 Fixpoint PExpr_eq (e1 e2 : PExpr C) {struct e1} : bool :=
  match e1, e2 with
@@ -434,7 +424,7 @@ intros l e1; elim e1.
 intros c1; intros e2; elim e2; simpl; (try (intros; discriminate)).
 intros c2; apply (morph_eq CRmorph).
 intros p1; intros e2; elim e2; simpl; (try (intros; discriminate)).
-intros p2; case Peqb_spec; intros; now subst.
+intros p2; case Pos.eqb_spec; intros; now subst.
 intros e3 rec1 e5 rec2 e2; case e2; simpl; (try (intros; discriminate)).
 intros e4 e6; generalize (rec1 e4); case (PExpr_eq e3 e4);
  (try (intros; discriminate)); generalize (rec2 e6); case (PExpr_eq e5 e6);
@@ -451,7 +441,7 @@ intros e3 rec e2; (case e2; simpl; (try (intros; discriminate))).
 intros e4; generalize (rec e4); case (PExpr_eq e3 e4);
  (try (intros; discriminate)); auto.
 intros e3 rec n3 e2;(case e2;simpl;(try (intros;discriminate))).
-intros e4 n4; case Neqb_spec; try discriminate; intros EQ H; subst.
+intros e4 n4; case N.eqb_spec; try discriminate; intros EQ H; subst.
 repeat rewrite  pow_th.(rpow_pow_N). rewrite (rec _ H);auto.
 Qed.
 
@@ -493,7 +483,7 @@ Proof.
  destruct n;simpl.
  rewrite pow_th.(rpow_pow_N);simpl;auto.
  fold (p =? 1)%positive.
- case Peqb_spec; intros H; (rewrite H || clear H).
+ case Pos.eqb_spec; intros H; (rewrite H || clear H).
  now rewrite  pow_th.(rpow_pow_N).
  destruct e;simpl;auto.
  repeat apply ceqb_rect;simpl;intros;rewrite pow_th.(rpow_pow_N);simpl.
@@ -526,7 +516,7 @@ induction e1;destruct e2; simpl in |- *;try reflexivity;
  try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; try reflexivity;
  try ring [(morph0 CRmorph) (morph1 CRmorph)].
  apply (morph_mul CRmorph).
-case Neqb_spec; intros H; try rewrite <- H; clear H.
+case N.eqb_spec; intros H; try rewrite <- H; clear H.
 rewrite NPEpow_correct. simpl.
 repeat rewrite pow_th.(rpow_pow_N).
 rewrite IHe1; destruct n;simpl;try ring.
diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v
index 437ce5726..d5d11ceaa 100644
--- a/theories/Bool/Bool.v
+++ b/theories/Bool/Bool.v
@@ -791,7 +791,7 @@ Qed.
 Lemma iff_reflect : forall P b, (P<->b=true) -> reflect P b.
 Proof.
  destr_bool; intuition.
-Qed.
+Defined.
 
 (** It would be nice to join [reflect_iff] and [iff_reflect]
     in a unique [iff] statement, but this isn't allowed since
@@ -802,7 +802,7 @@ Qed.
 Lemma reflect_dec : forall P b, reflect P b -> {P}+{~P}.
 Proof.
  destruct 1; auto.
-Qed.
+Defined.
 
 (** Reciprocally, from a decidability, we could state a
     [reflect] as soon as we have a [bool_of_sumbool]. *)
diff --git a/theories/Structures/Equalities.v b/theories/Structures/Equalities.v
index 933c4ea0e..eb5373859 100644
--- a/theories/Structures/Equalities.v
+++ b/theories/Structures/Equalities.v
@@ -196,7 +196,7 @@ Qed.
 Lemma eqb_spec x y : reflect (x==y) (x =? y).
 Proof.
 apply iff_reflect. symmetry. apply eqb_eq.
-Qed.
+Defined.
 
 (** Negated form of [eqb_eq] *)
 
diff --git a/theories/Structures/OrdersFacts.v b/theories/Structures/OrdersFacts.v
index a447e8fb5..2e9c0cf56 100644
--- a/theories/Structures/OrdersFacts.v
+++ b/theories/Structures/OrdersFacts.v
@@ -365,7 +365,7 @@ Include CompareBasedOrderFacts E C O.
 Lemma leb_spec0 x y : reflect (x<=y) (x<=?y).
 Proof.
  apply iff_reflect. symmetry. apply leb_le.
-Qed.
+Defined.
 
 Lemma leb_spec x y : BoolSpec (x<=y) (y<x) (x<=?y).
 Proof.
@@ -376,7 +376,7 @@ Qed.
 Lemma ltb_spec0 x y : reflect (x<y) (x<?y).
 Proof.
  apply iff_reflect. symmetry. apply ltb_lt.
-Qed.
+Defined.
 
 Lemma ltb_spec x y : BoolSpec (x<y) (y<=x) (x<?y).
 Proof.