Rework of GenericMinMax: new axiomatic, split logical/decidable parts, Leibniz part

 Moreover, instantiation like MinMax are now made without redefining
 generic properties (easier maintenance). We start using inner modules
 for qualifying (e.g. Z.max_comm).

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12638 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 49 insertions, 140 deletions

diff --git a/theories/Reals/Rminmax.v b/theories/Reals/Rminmax.v
index e78e89542..79343ee45 100644
--- a/theories/Reals/Rminmax.v
+++ b/theories/Reals/Rminmax.v
@@ -15,200 +15,109 @@ Local Open Scope R_scope.
 (** The functions [Rmax] and [Rmin] implement indeed
     a maximum and a minimum *)
 
-Lemma Rmax_spec : forall x y,
-  (x<y /\ Rmax x y = y)  \/  (y<=x /\ Rmax x y = x).
+Lemma Rmax_l : forall x y, y<=x -> Rmax x y = x.
 Proof.
  unfold Rmax. intros.
- destruct Rle_dec as [H|H]; [| apply Rnot_le_lt in H ];
+ destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
   unfold Rle in *; intuition.
 Qed.
 
-Lemma Rmin_spec : forall x y,
-  (x<y /\ Rmin x y = x)  \/  (y<=x /\ Rmin x y = y).
+Lemma Rmax_r : forall x y, x<=y -> Rmax x y = y.
+Proof.
+ unfold Rmax. intros.
+ destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
+  unfold Rle in *; intuition.
+Qed.
+
+Lemma Rmin_l : forall x y, x<=y -> Rmin x y = x.
+Proof.
+ unfold Rmin. intros.
+ destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
+  unfold Rle in *; intuition.
+Qed.
+
+Lemma Rmin_r : forall x y, y<=x -> Rmin x y = y.
 Proof.
  unfold Rmin. intros.
- destruct Rle_dec as [H|H]; [| apply Rnot_le_lt in H ];
+ destruct Rle_dec as [H'|H']; [| apply Rnot_le_lt in H' ];
   unfold Rle in *; intuition.
 Qed.
 
 Module RHasMinMax <: HasMinMax R_as_OT.
  Definition max := Rmax.
  Definition min := Rmin.
- Definition max_spec := Rmax_spec.
- Definition min_spec := Rmin_spec.
+ Definition max_l := Rmax_l.
+ Definition max_r := Rmax_r.
+ Definition min_l := Rmin_l.
+ Definition min_r := Rmin_r.
 End RHasMinMax.
 
-(** We obtain hence all the generic properties of max and min. *)
-
-Module Import RMinMaxProps := MinMaxProperties R_as_OT RHasMinMax.
-
-(** For some generic properties, we can have nicer statements here,
-    since underlying equality is Leibniz. *)
-
-Lemma Rmax_case_strong : forall n m (P:R -> Type),
-  (m<=n -> P n) -> (n<=m -> P m) -> P (Rmax n m).
-Proof. intros; apply max_case_strong; auto. congruence. Defined.
+Module R.
 
-Lemma Rmax_case : forall n m (P:R -> Type),
-  P n -> P m -> P (Rmax n m).
-Proof. intros. apply Rmax_case_strong; auto. Defined.
-
-Lemma Rmax_monotone: forall f,
- (Proper (Rle ==> Rle) f) ->
- forall x y, Rmax (f x) (f y) = f (Rmax x y).
-Proof. intros; apply max_monotone; auto. congruence. Qed.
-
-Lemma Rmin_case_strong : forall n m (P:R -> Type),
-  (n<=m -> P n) -> (m<=n -> P m) -> P (Rmin n m).
-Proof. intros; apply min_case_strong; auto. congruence. Defined.
-
-Lemma Rmin_case : forall n m (P:R -> Type),
-  P n -> P m -> P (Rmin n m).
-Proof. intros. apply Rmin_case_strong; auto. Defined.
-
-Lemma Rmin_monotone: forall f,
- (Proper (Rle ==> Rle) f) ->
- forall x y, Rmin (f x) (f y) = f (Rmin x y).
-Proof. intros; apply min_monotone; auto. congruence. Qed.
-
-Lemma Rmax_min_antimonotone : forall f,
- Proper (Rle==>Rge) f ->
- forall x y, Rmax (f x) (f y) == f (Rmin x y).
-Proof.
- intros. apply max_min_antimonotone. congruence.
- intros z z' Hz; red; apply Rge_le; auto.
-Qed.
-
-Lemma Rmin_max_antimonotone : forall f,
- Proper (Rle==>Rge) f ->
- forall x y, Rmin (f x) (f y) == f (Rmax x y).
-Proof.
- intros. apply min_max_antimonotone. congruence.
- intros z z' Hz; red. apply Rge_le. auto.
-Qed.
-
-(** For the other generic properties, we make aliases,
-   since otherwise SearchAbout misses some of them
-   (bad interaction with an Include).
-   See GenericMinMax (or SearchAbout) for the statements. *)
-
-Definition Rmax_spec_le := max_spec_le.
-Definition Rmax_dec := max_dec.
-Definition Rmax_unicity := max_unicity.
-Definition Rmax_unicity_ext := max_unicity_ext.
-Definition Rmax_id := max_id.
-Notation Rmax_idempotent := Rmax_id (only parsing).
-Definition Rmax_assoc := max_assoc.
-Definition Rmax_comm := max_comm.
-Definition Rmax_l := max_l.
-Definition Rmax_r := max_r.
-Definition Zle_max_l := le_max_l.
-Definition Zle_max_r := le_max_r.
-Definition Rmax_le := max_le.
-Definition Rmax_le_iff := max_le_iff.
-Definition Rmax_lt_iff := max_lt_iff.
-Definition Rmax_lub_l := max_lub_l.
-Definition Rmax_lub_r := max_lub_r.
-Definition Rmax_lub := max_lub.
-Definition Rmax_lub_iff := max_lub_iff.
-Definition Rmax_lub_lt := max_lub_lt.
-Definition Rmax_lub_lt_iff := max_lub_lt_iff.
-Definition Rmax_le_compat_l := max_le_compat_l.
-Definition Rmax_le_compat_r := max_le_compat_r.
-Definition Rmax_le_compat := max_le_compat.
-
-Definition Rmin_spec_le := min_spec_le.
-Definition Rmin_dec := min_dec.
-Definition Rmin_unicity := min_unicity.
-Definition Rmin_unicity_ext := min_unicity_ext.
-Definition Rmin_id := min_id.
-Notation Rmin_idempotent := Rmin_id (only parsing).
-Definition Rmin_assoc := min_assoc.
-Definition Rmin_comm := min_comm.
-Definition Rmin_l := min_l.
-Definition Rmin_r := min_r.
-Definition Zle_min_l := le_min_l.
-Definition Zle_min_r := le_min_r.
-Definition Rmin_le := min_le.
-Definition Rmin_le_iff := min_le_iff.
-Definition Rmin_lt_iff := min_lt_iff.
-Definition Rmin_glb_l := min_glb_l.
-Definition Rmin_glb_r := min_glb_r.
-Definition Rmin_glb := min_glb.
-Definition Rmin_glb_iff := min_glb_iff.
-Definition Rmin_glb_lt := min_glb_lt.
-Definition Rmin_glb_lt_iff := min_glb_lt_iff.
-Definition Rmin_le_compat_l := min_le_compat_l.
-Definition Rmin_le_compat_r := min_le_compat_r.
-Definition Rmin_le_compat := min_le_compat.
-
-Definition Rmin_max_absorption := min_max_absorption.
-Definition Rmax_min_absorption := max_min_absorption.
-Definition Rmax_min_distr := max_min_distr.
-Definition Rmin_max_distr := min_max_distr.
-Definition Rmax_min_modular := max_min_modular.
-Definition Rmin_max_modular := min_max_modular.
-Definition Rmax_min_disassoc := max_min_disassoc.
+(** We obtain hence all the generic properties of max and min. *)
 
+Include UsualMinMaxProperties R_as_OT RHasMinMax.
 
 (** * Properties specific to the [R] domain *)
 
 (** Compatibilities (consequences of monotonicity) *)
 
-Lemma Rplus_max_distr_l : forall n m p, Rmax (p + n) (p + m) = p + Rmax n m.
+Lemma plus_max_distr_l : forall n m p, Rmax (p + n) (p + m) = p + Rmax n m.
 Proof.
- intros. apply Rmax_monotone.
+ intros. apply max_monotone.
  intros x y. apply Rplus_le_compat_l.
 Qed.
 
-Lemma Rplus_max_distr_r : forall n m p, Rmax (n + p) (m + p) = Rmax n m + p.
+Lemma plus_max_distr_r : forall n m p, Rmax (n + p) (m + p) = Rmax n m + p.
 Proof.
  intros. rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).
- apply Rplus_max_distr_l.
+ apply plus_max_distr_l.
 Qed.
 
-Lemma Rplus_min_distr_l : forall n m p, Rmin (p + n) (p + m) = p + Rmin n m.
+Lemma plus_min_distr_l : forall n m p, Rmin (p + n) (p + m) = p + Rmin n m.
 Proof.
- intros. apply Rmin_monotone.
+ intros. apply min_monotone.
  intros x y. apply Rplus_le_compat_l.
 Qed.
 
-Lemma Rplus_min_distr_r : forall n m p, Rmin (n + p) (m + p) = Rmin n m + p.
+Lemma plus_min_distr_r : forall n m p, Rmin (n + p) (m + p) = Rmin n m + p.
 Proof.
  intros. rewrite (Rplus_comm n p), (Rplus_comm m p), (Rplus_comm _ p).
- apply Rplus_min_distr_l.
+ apply plus_min_distr_l.
 Qed.
 
 (** Anti-monotonicity swaps the role of [min] and [max] *)
 
-Lemma Ropp_max_distr : forall n m : R, -(Rmax n m) = Rmin (- n) (- m).
+Lemma opp_max_distr : forall n m : R, -(Rmax n m) = Rmin (- n) (- m).
 Proof.
- intros. symmetry. apply Rmin_max_antimonotone.
- exact Ropp_le_ge_contravar.
+ intros. symmetry. apply min_max_antimonotone.
+ do 3 red. intros; apply Rge_le. apply Ropp_le_ge_contravar; auto.
 Qed.
 
-Lemma Ropp_min_distr : forall n m : R, - (Rmin n m) = Rmax (- n) (- m).
+Lemma opp_min_distr : forall n m : R, - (Rmin n m) = Rmax (- n) (- m).
 Proof.
- intros. symmetry. apply Rmax_min_antimonotone.
- exact Ropp_le_ge_contravar.
+ intros. symmetry. apply max_min_antimonotone.
+ do 3 red. intros; apply Rge_le. apply Ropp_le_ge_contravar; auto.
 Qed.
 
-Lemma Rminus_max_distr_l : forall n m p, Rmax (p - n) (p - m) = p - Rmin n m.
+Lemma minus_max_distr_l : forall n m p, Rmax (p - n) (p - m) = p - Rmin n m.
 Proof.
- unfold Rminus. intros. rewrite Ropp_min_distr. apply Rplus_max_distr_l.
+ unfold Rminus. intros. rewrite opp_min_distr. apply plus_max_distr_l.
 Qed.
 
-Lemma Rminus_max_distr_r : forall n m p, Rmax (n - p) (m - p) = Rmax n m - p.
+Lemma minus_max_distr_r : forall n m p, Rmax (n - p) (m - p) = Rmax n m - p.
 Proof.
- unfold Rminus. intros. apply Rplus_max_distr_r.
+ unfold Rminus. intros. apply plus_max_distr_r.
 Qed.
 
-Lemma Rminus_min_distr_l : forall n m p, Rmin (p - n) (p - m) = p - Rmax n m.
+Lemma minus_min_distr_l : forall n m p, Rmin (p - n) (p - m) = p - Rmax n m.
 Proof.
- unfold Rminus. intros. rewrite Ropp_max_distr. apply Rplus_min_distr_l.
+ unfold Rminus. intros. rewrite opp_max_distr. apply plus_min_distr_l.
 Qed.
 
-Lemma Rminus_min_distr_r : forall n m p, Rmin (n - p) (m - p) = Rmin n m - p.
+Lemma minus_min_distr_r : forall n m p, Rmin (n - p) (m - p) = Rmin n m - p.
 Proof.
- unfold Rminus. intros. apply Rplus_min_distr_r.
+ unfold Rminus. intros. apply plus_min_distr_r.
 Qed.
+
+End R.