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

2 files changed, 89 insertions, 288 deletions

diff --git a/theories/NArith/Nminmax.v b/theories/NArith/Nminmax.v
index c2fee9c7e..625fcce23 100644
--- a/theories/NArith/Nminmax.v
+++ b/theories/NArith/Nminmax.v
@@ -15,212 +15,112 @@ Local Open Scope N_scope.
 (** The functions [Nmax] and [Nmin] implement indeed
     a maximum and a minimum *)
 
-Lemma Nmax_spec : forall x y,
-  (x<y /\ Nmax x y = y)  \/  (y<=x /\ Nmax x y = x).
+Lemma Nmax_l : forall x y, y<=x -> Nmax x y = x.
 Proof.
- unfold Nlt, Nle, Nmax. intros.
- generalize (Ncompare_eq_correct x y).
- rewrite <- (Ncompare_antisym x y).
- destruct (x ?= y); simpl; auto; right; intuition; discriminate.
+ unfold Nle, Nmax. intros x y.
+ generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
+ destruct (x ?= y); intuition.
 Qed.
 
-Lemma Nmin_spec : forall x y,
-  (x<y /\ Nmin x y = x)  \/  (y<=x /\ Nmin x y = y).
+Lemma Nmax_r : forall x y, x<=y -> Nmax x y = y.
 Proof.
- unfold Nlt, Nle, Nmin. intros.
- generalize (Ncompare_eq_correct x y).
- rewrite <- (Ncompare_antisym x y).
- destruct (x ?= y); simpl; auto; right; intuition; discriminate.
+ unfold Nle, Nmax. intros x y. destruct (x ?= y); intuition.
 Qed.
 
-Module NHasMinMax <: HasMinMax N_as_OT.
- Definition max := Nmax.
- Definition min := Nmin.
- Definition max_spec := Nmax_spec.
- Definition min_spec := Nmin_spec.
-End NHasMinMax.
-
-(** We obtain hence all the generic properties of max and min. *)
-
-Module Import NMinMaxProps := MinMaxProperties N_as_OT NHasMinMax.
-
-(** For some generic properties, we can have nicer statements here,
-    since underlying equality is Leibniz. *)
-
-Lemma Nmax_case_strong : forall n m (P:N -> Type),
-  (m<=n -> P n) -> (n<=m -> P m) -> P (Nmax n m).
-Proof. intros; apply max_case_strong; auto. congruence. Defined.
-
-Lemma Nmax_case : forall n m (P:N -> Type),
-  P n -> P m -> P (Nmax n m).
-Proof. intros. apply Nmax_case_strong; auto. Defined.
-
-Lemma Nmax_monotone: forall f,
- (Proper (Nle ==> Nle) f) ->
- forall x y, Nmax (f x) (f y) = f (Nmax x y).
-Proof. intros; apply max_monotone; auto. congruence. Qed.
-
-Lemma Nmin_case_strong : forall n m (P:N -> Type),
-  (n<=m -> P n) -> (m<=n -> P m) -> P (Nmin n m).
-Proof. intros; apply min_case_strong; auto. congruence. Defined.
-
-Lemma Nmin_case : forall n m (P:N -> Type),
-  P n -> P m -> P (Nmin n m).
-Proof. intros. apply Nmin_case_strong; auto. Defined.
-
-Lemma Nmin_monotone: forall f,
- (Proper (Nle ==> Nle) f) ->
- forall x y, Nmin (f x) (f y) = f (Nmin x y).
-Proof. intros; apply min_monotone; auto. congruence. Qed.
-
-Lemma Nmax_min_antimonotone : forall f,
- Proper (Nle==>Nge) f ->
- forall x y, Nmax (f x) (f y) == f (Nmin x y).
+Lemma Nmin_l : forall x y, x<=y -> Nmin x y = x.
 Proof.
- intros f H. apply max_min_antimonotone. congruence.
- intros x x' Hx. red. specialize (H _ _ Hx). clear Hx.
- unfold Nle, Nge in *; contradict H. rewrite <- Ncompare_antisym, H; auto.
+ unfold Nle, Nmin. intros x y. destruct (x ?= y); intuition.
 Qed.
 
-Lemma Nmin_max_antimonotone : forall f,
- Proper (Nle==>Nge) f ->
- forall x y, Nmin (f x) (f y) == f (Nmax x y).
+Lemma Nmin_r : forall x y, y<=x -> Nmin x y = y.
 Proof.
- intros f H. apply min_max_antimonotone. congruence.
- intros z z' Hz; red. specialize (H _ _ Hz). clear Hz.
- unfold Nle, Nge in *. contradict H. rewrite <- Ncompare_antisym, H; auto.
+ unfold Nle, Nmin. intros x y.
+ generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
+ destruct (x ?= y); intuition.
 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 Nmax_spec_le := max_spec_le.
-Definition Nmax_dec := max_dec.
-Definition Nmax_unicity := max_unicity.
-Definition Nmax_unicity_ext := max_unicity_ext.
-Definition Nmax_id := max_id.
-Notation Nmax_idempotent := Nmax_id (only parsing).
-Definition Nmax_assoc := max_assoc.
-Definition Nmax_comm := max_comm.
-Definition Nmax_l := max_l.
-Definition Nmax_r := max_r.
-Definition Nle_max_l := le_max_l.
-Definition Nle_max_r := le_max_r.
-Definition Nmax_le := max_le.
-Definition Nmax_le_iff := max_le_iff.
-Definition Nmax_lt_iff := max_lt_iff.
-Definition Nmax_lub_l := max_lub_l.
-Definition Nmax_lub_r := max_lub_r.
-Definition Nmax_lub := max_lub.
-Definition Nmax_lub_iff := max_lub_iff.
-Definition Nmax_lub_lt := max_lub_lt.
-Definition Nmax_lub_lt_iff := max_lub_lt_iff.
-Definition Nmax_le_compat_l := max_le_compat_l.
-Definition Nmax_le_compat_r := max_le_compat_r.
-Definition Nmax_le_compat := max_le_compat.
-
-Definition Nmin_spec_le := min_spec_le.
-Definition Nmin_dec := min_dec.
-Definition Nmin_unicity := min_unicity.
-Definition Nmin_unicity_ext := min_unicity_ext.
-Definition Nmin_id := min_id.
-Notation Nmin_idempotent := Nmin_id (only parsing).
-Definition Nmin_assoc := min_assoc.
-Definition Nmin_comm := min_comm.
-Definition Nmin_l := min_l.
-Definition Nmin_r := min_r.
-Definition Nle_min_l := le_min_l.
-Definition Nle_min_r := le_min_r.
-Definition Nmin_le := min_le.
-Definition Nmin_le_iff := min_le_iff.
-Definition Nmin_lt_iff := min_lt_iff.
-Definition Nmin_glb_l := min_glb_l.
-Definition Nmin_glb_r := min_glb_r.
-Definition Nmin_glb := min_glb.
-Definition Nmin_glb_iff := min_glb_iff.
-Definition Nmin_glb_lt := min_glb_lt.
-Definition Nmin_glb_lt_iff := min_glb_lt_iff.
-Definition Nmin_le_compat_l := min_le_compat_l.
-Definition Nmin_le_compat_r := min_le_compat_r.
-Definition Nmin_le_compat := min_le_compat.
-
-Definition Nmin_max_absorption := min_max_absorption.
-Definition Nmax_min_absorption := max_min_absorption.
-Definition Nmax_min_distr := max_min_distr.
-Definition Nmin_max_distr := min_max_distr.
-Definition Nmax_min_modular := max_min_modular.
-Definition Nmin_max_modular := min_max_modular.
-Definition Nmax_min_disassoc := max_min_disassoc.
+Module NHasMinMax <: HasMinMax N_as_OT.
+ Definition max := Nmax.
+ Definition min := Nmin.
+ Definition max_l := Nmax_l.
+ Definition max_r := Nmax_r.
+ Definition min_l := Nmin_l.
+ Definition min_r := Nmin_r.
+End NHasMinMax.
 
+Module N.
 
+(** We obtain hence all the generic properties of max and min. *)
+
+Include UsualMinMaxProperties N_as_OT NHasMinMax.
 
 (** * Properties specific to the [positive] domain *)
 
 (** Simplifications *)
 
-Lemma Nmax_0_l : forall n, Nmax 0 n = n.
+Lemma max_0_l : forall n, Nmax 0 n = n.
 Proof.
  intros. unfold Nmax. rewrite <- Ncompare_antisym. generalize (Ncompare_0 n).
  destruct (n ?= 0); intuition.
 Qed.
 
-Lemma Nmax_0_r : forall n, Nmax n 0 = n.
-Proof. intros. rewrite max_comm. apply Nmax_0_l. Qed.
+Lemma max_0_r : forall n, Nmax n 0 = n.
+Proof. intros. rewrite N.max_comm. apply max_0_l. Qed.
 
-Lemma Nmin_0_l : forall n, Nmin 0 n = 0.
+Lemma min_0_l : forall n, Nmin 0 n = 0.
 Proof.
  intros. unfold Nmin. rewrite <- Ncompare_antisym. generalize (Ncompare_0 n).
  destruct (n ?= 0); intuition.
 Qed.
 
-Lemma Nmin_0_r : forall n, Nmin n 0 = 0.
-Proof. intros. rewrite min_comm. apply Nmin_0_l. Qed.
+Lemma min_0_r : forall n, Nmin n 0 = 0.
+Proof. intros. rewrite N.min_comm. apply min_0_l. Qed.
 
 (** Compatibilities (consequences of monotonicity) *)
 
-Lemma Nsucc_max_distr :
+Lemma succ_max_distr :
  forall n m, Nsucc (Nmax n m) = Nmax (Nsucc n) (Nsucc m).
 Proof.
- intros. symmetry. apply Nmax_monotone.
+ intros. symmetry. apply max_monotone.
  intros x x'. unfold Nle.
  rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
  simpl; auto.
 Qed.
 
-Lemma Nsucc_min_distr : forall n m, Nsucc (Nmin n m) = Nmin (Nsucc n) (Nsucc m).
+Lemma succ_min_distr : forall n m, Nsucc (Nmin n m) = Nmin (Nsucc n) (Nsucc m).
 Proof.
- intros. symmetry. apply Nmin_monotone.
+ intros. symmetry. apply min_monotone.
  intros x x'. unfold Nle.
  rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
  simpl; auto.
 Qed.
 
-Lemma Nplus_max_distr_l : forall n m p, Nmax (p + n) (p + m) = p + Nmax n m.
+Lemma plus_max_distr_l : forall n m p, Nmax (p + n) (p + m) = p + Nmax n m.
 Proof.
- intros. apply Nmax_monotone.
+ intros. apply max_monotone.
  intros x x'. unfold Nle.
  rewrite 2 nat_of_Ncompare, 2 nat_of_Nplus.
  rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
 Qed.
 
-Lemma Nplus_max_distr_r : forall n m p, Nmax (n + p) (m + p) = Nmax n m + p.
+Lemma plus_max_distr_r : forall n m p, Nmax (n + p) (m + p) = Nmax n m + p.
 Proof.
  intros. rewrite (Nplus_comm n p), (Nplus_comm m p), (Nplus_comm _ p).
- apply Nplus_max_distr_l.
+ apply plus_max_distr_l.
 Qed.
 
-Lemma Nplus_min_distr_l : forall n m p, Nmin (p + n) (p + m) = p + Nmin n m.
+Lemma plus_min_distr_l : forall n m p, Nmin (p + n) (p + m) = p + Nmin n m.
 Proof.
- intros. apply Nmin_monotone.
+ intros. apply min_monotone.
  intros x x'. unfold Nle.
  rewrite 2 nat_of_Ncompare, 2 nat_of_Nplus.
  rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
 Qed.
 
-Lemma Nplus_min_distr_r : forall n m p, Nmin (n + p) (m + p) = Nmin n m + p.
+Lemma plus_min_distr_r : forall n m p, Nmin (n + p) (m + p) = Nmin n m + p.
 Proof.
  intros. rewrite (Nplus_comm n p), (Nplus_comm m p), (Nplus_comm _ p).
- apply Nplus_min_distr_l.
+ apply plus_min_distr_l.
 Qed.
+
+End N.
\ No newline at end of file
diff --git a/theories/NArith/Pminmax.v b/theories/NArith/Pminmax.v
index afae63f5a..3b3c58841 100644
--- a/theories/NArith/Pminmax.v
+++ b/theories/NArith/Pminmax.v
@@ -15,211 +15,112 @@ Local Open Scope positive_scope.
 (** The functions [Pmax] and [Pmin] implement indeed
     a maximum and a minimum *)
 
-Lemma Pmax_spec : forall x y,
-  (x<y /\ Pmax x y = y)  \/  (y<=x /\ Pmax x y = x).
+Lemma Pmax_l : forall x y, y<=x -> Pmax x y = x.
 Proof.
- unfold Plt, Ple, Pmax. intros.
- generalize (Pcompare_eq_iff x y). rewrite (ZC4 y x).
- destruct ((x ?= y) Eq); simpl; auto; right; intuition; discriminate.
+ unfold Ple, Pmax. intros x y.
+ rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y).
+ destruct ((x ?= y) Eq); intuition.
 Qed.
 
-Lemma Pmin_spec : forall x y,
-  (x<y /\ Pmin x y = x)  \/  (y<=x /\ Pmin x y = y).
+Lemma Pmax_r : forall x y, x<=y -> Pmax x y = y.
 Proof.
- unfold Plt, Ple, Pmin. intros.
- generalize (Pcompare_eq_iff x y). rewrite (ZC4 y x).
- destruct ((x ?= y) Eq); simpl; auto; right; intuition; discriminate.
+ unfold Ple, Pmax. intros x y. destruct ((x ?= y) Eq); intuition.
 Qed.
 
-Module PositiveHasMinMax <: HasMinMax Positive_as_OT.
- Definition max := Pmax.
- Definition min := Pmin.
- Definition max_spec := Pmax_spec.
- Definition min_spec := Pmin_spec.
-End PositiveHasMinMax.
-
-(** We obtain hence all the generic properties of max and min. *)
-
-Module Import NatMinMaxProps :=
- MinMaxProperties Positive_as_OT PositiveHasMinMax.
-
-
-(** For some generic properties, we can have nicer statements here,
-    since underlying equality is Leibniz. *)
-
-Lemma Pmax_case_strong : forall n m (P:positive -> Type),
-  (m<=n -> P n) -> (n<=m -> P m) -> P (Pmax n m).
-Proof. intros; apply max_case_strong; auto. congruence. Defined.
-
-Lemma Pmax_case : forall n m (P:positive -> Type),
-  P n -> P m -> P (Pmax n m).
-Proof. intros. apply Pmax_case_strong; auto. Defined.
-
-Lemma Pmax_monotone: forall f,
- (Proper (Ple ==> Ple) f) ->
- forall x y, Pmax (f x) (f y) = f (Pmax x y).
-Proof. intros; apply max_monotone; auto. congruence. Qed.
-
-Lemma Pmin_case_strong : forall n m (P:positive -> Type),
-  (n<=m -> P n) -> (m<=n -> P m) -> P (Pmin n m).
-Proof. intros; apply min_case_strong; auto. congruence. Defined.
-
-Lemma Pmin_case : forall n m (P:positive -> Type),
-  P n -> P m -> P (Pmin n m).
-Proof. intros. apply Pmin_case_strong; auto. Defined.
-
-Lemma Pmin_monotone: forall f,
- (Proper (Ple ==> Ple) f) ->
- forall x y, Pmin (f x) (f y) = f (Pmin x y).
-Proof. intros; apply min_monotone; auto. congruence. Qed.
-
-Lemma Pmax_min_antimonotone : forall f,
- Proper (Ple==>Pge) f ->
- forall x y, Pmax (f x) (f y) == f (Pmin x y).
+Lemma Pmin_l : forall x y, x<=y -> Pmin x y = x.
 Proof.
- intros f H. apply max_min_antimonotone. congruence.
- intros z z' Hz; red. specialize (H _ _ Hz). clear Hz.
- unfold Ple, Pge in *. contradict H. rewrite ZC4, H; auto.
+ unfold Ple, Pmin. intros x y. destruct ((x ?= y) Eq); intuition.
 Qed.
 
-Lemma Pmin_max_antimonotone : forall f,
- Proper (Ple==>Pge) f ->
- forall x y, Pmin (f x) (f y) == f (Pmax x y).
+Lemma Pmin_r : forall x y, y<=x -> Pmin x y = y.
 Proof.
- intros f H. apply min_max_antimonotone. congruence.
- intros z z' Hz; red. specialize (H _ _ Hz). clear Hz.
- unfold Ple, Pge in *. contradict H. rewrite ZC4, H; auto.
+ unfold Ple, Pmin. intros x y.
+ rewrite (ZC4 y x). generalize (Pcompare_eq_iff x y).
+ destruct ((x ?= y) Eq); intuition.
 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 Pmax_spec_le := max_spec_le.
-Definition Pmax_dec := max_dec.
-Definition Pmax_unicity := max_unicity.
-Definition Pmax_unicity_ext := max_unicity_ext.
-Definition Pmax_id := max_id.
-Notation Pmax_idempotent := Pmax_id (only parsing).
-Definition Pmax_assoc := max_assoc.
-Definition Pmax_comm := max_comm.
-Definition Pmax_l := max_l.
-Definition Pmax_r := max_r.
-Definition Ple_max_l := le_max_l.
-Definition Ple_max_r := le_max_r.
-Definition Pmax_le := max_le.
-Definition Pmax_le_iff := max_le_iff.
-Definition Pmax_lt_iff := max_lt_iff.
-Definition Pmax_lub_l := max_lub_l.
-Definition Pmax_lub_r := max_lub_r.
-Definition Pmax_lub := max_lub.
-Definition Pmax_lub_iff := max_lub_iff.
-Definition Pmax_lub_lt := max_lub_lt.
-Definition Pmax_lub_lt_iff := max_lub_lt_iff.
-Definition Pmax_le_compat_l := max_le_compat_l.
-Definition Pmax_le_compat_r := max_le_compat_r.
-Definition Pmax_le_compat := max_le_compat.
-
-Definition Pmin_spec_le := min_spec_le.
-Definition Pmin_dec := min_dec.
-Definition Pmin_unicity := min_unicity.
-Definition Pmin_unicity_ext := min_unicity_ext.
-Definition Pmin_id := min_id.
-Notation Pmin_idempotent := Pmin_id (only parsing).
-Definition Pmin_assoc := min_assoc.
-Definition Pmin_comm := min_comm.
-Definition Pmin_l := min_l.
-Definition Pmin_r := min_r.
-Definition Ple_min_l := le_min_l.
-Definition Ple_min_r := le_min_r.
-Definition Pmin_le := min_le.
-Definition Pmin_le_iff := min_le_iff.
-Definition Pmin_lt_iff := min_lt_iff.
-Definition Pmin_glb_l := min_glb_l.
-Definition Pmin_glb_r := min_glb_r.
-Definition Pmin_glb := min_glb.
-Definition Pmin_glb_iff := min_glb_iff.
-Definition Pmin_glb_lt := min_glb_lt.
-Definition Pmin_glb_lt_iff := min_glb_lt_iff.
-Definition Pmin_le_compat_l := min_le_compat_l.
-Definition Pmin_le_compat_r := min_le_compat_r.
-Definition Pmin_le_compat := min_le_compat.
-
-Definition Pmin_max_absorption := min_max_absorption.
-Definition Pmax_min_absorption := max_min_absorption.
-Definition Pmax_min_distr := max_min_distr.
-Definition Pmin_max_distr := min_max_distr.
-Definition Pmax_min_modular := max_min_modular.
-Definition Pmin_max_modular := min_max_modular.
-Definition Pmax_min_disassoc := max_min_disassoc.
+Module PositiveHasMinMax <: HasMinMax Positive_as_OT.
+ Definition max := Pmax.
+ Definition min := Pmin.
+ Definition max_l := Pmax_l.
+ Definition max_r := Pmax_r.
+ Definition min_l := Pmin_l.
+ Definition min_r := Pmin_r.
+End PositiveHasMinMax.
+
 
+Module P.
+(** We obtain hence all the generic properties of max and min. *)
+
+Include UsualMinMaxProperties Positive_as_OT PositiveHasMinMax.
 
 (** * Properties specific to the [positive] domain *)
 
 (** Simplifications *)
 
-Lemma Pmax_1_l : forall n, Pmax 1 n = n.
+Lemma max_1_l : forall n, Pmax 1 n = n.
 Proof.
  intros. unfold Pmax. rewrite ZC4. generalize (Pcompare_1 n).
  destruct (n ?= 1); intuition.
 Qed.
 
-Lemma Pmax_1_r : forall n, Pmax n 1 = n.
-Proof. intros. rewrite max_comm. apply Pmax_1_l. Qed.
+Lemma max_1_r : forall n, Pmax n 1 = n.
+Proof. intros. rewrite P.max_comm. apply max_1_l. Qed.
 
-Lemma Pmin_1_l : forall n, Pmin 1 n = 1.
+Lemma min_1_l : forall n, Pmin 1 n = 1.
 Proof.
  intros. unfold Pmin. rewrite ZC4. generalize (Pcompare_1 n).
  destruct (n ?= 1); intuition.
 Qed.
 
-Lemma Pmin_1_r : forall n, Pmin n 1 = 1.
-Proof. intros. rewrite min_comm. apply Pmin_1_l. Qed.
+Lemma min_1_r : forall n, Pmin n 1 = 1.
+Proof. intros. rewrite P.min_comm. apply min_1_l. Qed.
 
 (** Compatibilities (consequences of monotonicity) *)
 
-Lemma Psucc_max_distr :
+Lemma succ_max_distr :
  forall n m, Psucc (Pmax n m) = Pmax (Psucc n) (Psucc m).
 Proof.
- intros. symmetry. apply Pmax_monotone.
+ intros. symmetry. apply max_monotone.
  intros x x'. unfold Ple.
  rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism.
  simpl; auto.
 Qed.
 
-Lemma Psucc_min_distr : forall n m, Psucc (Pmin n m) = Pmin (Psucc n) (Psucc m).
+Lemma succ_min_distr : forall n m, Psucc (Pmin n m) = Pmin (Psucc n) (Psucc m).
 Proof.
- intros. symmetry. apply Pmin_monotone.
+ intros. symmetry. apply min_monotone.
  intros x x'. unfold Ple.
  rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_succ_morphism.
  simpl; auto.
 Qed.
 
-Lemma Pplus_max_distr_l : forall n m p, Pmax (p + n) (p + m) = p + Pmax n m.
+Lemma plus_max_distr_l : forall n m p, Pmax (p + n) (p + m) = p + Pmax n m.
 Proof.
- intros. apply Pmax_monotone.
+ intros. apply max_monotone.
  intros x x'. unfold Ple.
  rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism.
  rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
 Qed.
 
-Lemma Pplus_max_distr_r : forall n m p, Pmax (n + p) (m + p) = Pmax n m + p.
+Lemma plus_max_distr_r : forall n m p, Pmax (n + p) (m + p) = Pmax n m + p.
 Proof.
  intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p).
- apply Pplus_max_distr_l.
+ apply plus_max_distr_l.
 Qed.
 
-Lemma Pplus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m.
+Lemma plus_min_distr_l : forall n m p, Pmin (p + n) (p + m) = p + Pmin n m.
 Proof.
- intros. apply Pmin_monotone.
+ intros. apply min_monotone.
  intros x x'. unfold Ple.
  rewrite 2 nat_of_P_compare_morphism, 2 nat_of_P_plus_morphism.
  rewrite <- 2 Compare_dec.nat_compare_le. auto with arith.
 Qed.
 
-Lemma Pplus_min_distr_r : forall n m p, Pmin (n + p) (m + p) = Pmin n m + p.
+Lemma plus_min_distr_r : forall n m p, Pmin (n + p) (m + p) = Pmin n m + p.
 Proof.
  intros. rewrite (Pplus_comm n p), (Pplus_comm m p), (Pplus_comm _ p).
- apply Pplus_min_distr_l.
+ apply plus_min_distr_l.
 Qed.
+
+End P.
\ No newline at end of file