1 files changed, 0 insertions, 126 deletions
diff --git a/theories/NArith/Nminmax.v b/theories/NArith/Nminmax.v
deleted file mode 100644
index 58184a4f..00000000
--- a/theories/NArith/Nminmax.v
+++ /dev/null
@@ -1,126 +0,0 @@
-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011     *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-Require Import Orders BinNat Nnat NOrderedType GenericMinMax.
-
-(** * Maximum and Minimum of two [N] numbers *)
-
-Local Open Scope N_scope.
-
-(** The functions [Nmax] and [Nmin] implement indeed
-    a maximum and a minimum *)
-
-Lemma Nmax_l : forall x y, y<=x -> Nmax x y = x.
-Proof.
- unfold Nle, Nmax. intros x y.
- generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
- destruct (x ?= y); intuition.
-Qed.
-
-Lemma Nmax_r : forall x y, x<=y -> Nmax x y = y.
-Proof.
- unfold Nle, Nmax. intros x y. destruct (x ?= y); intuition.
-Qed.
-
-Lemma Nmin_l : forall x y, x<=y -> Nmin x y = x.
-Proof.
- unfold Nle, Nmin. intros x y. destruct (x ?= y); intuition.
-Qed.
-
-Lemma Nmin_r : forall x y, y<=x -> Nmin x y = y.
-Proof.
- unfold Nle, Nmin. intros x y.
- generalize (Ncompare_eq_correct x y). rewrite <- (Ncompare_antisym x y).
- destruct (x ?= y); intuition.
-Qed.
-
-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 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 max_0_r : forall n, Nmax n 0 = n.
-Proof. intros. rewrite N.max_comm. apply max_0_l. Qed.
-
-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 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 succ_max_distr :
- forall n m, Nsucc (Nmax n m) = Nmax (Nsucc n) (Nsucc m).
-Proof.
- intros. symmetry. apply max_monotone.
- intros x x'. unfold Nle.
- rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
- simpl; auto.
-Qed.
-
-Lemma succ_min_distr : forall n m, Nsucc (Nmin n m) = Nmin (Nsucc n) (Nsucc m).
-Proof.
- intros. symmetry. apply min_monotone.
- intros x x'. unfold Nle.
- rewrite 2 nat_of_Ncompare, 2 nat_of_Nsucc.
- simpl; auto.
-Qed.
-
-Lemma plus_max_distr_l : forall n m p, Nmax (p + n) (p + m) = p + Nmax n m.
-Proof.
- 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 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 plus_max_distr_l.
-Qed.
-
-Lemma plus_min_distr_l : forall n m p, Nmin (p + n) (p + m) = p + Nmin n m.
-Proof.
- 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 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 plus_min_distr_l.
-Qed.
-
-End N.
-\ No newline at end of file