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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: EqNat.v,v 1.14.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+(** Equality on natural numbers *)
+
+Open Local Scope nat_scope.
+
+Implicit Types m n x y : nat.
+
+Fixpoint eq_nat n m {struct n} : Prop :=
+  match n, m with
+  | O, O => True
+  | O, S _ => False
+  | S _, O => False
+  | S n1, S m1 => eq_nat n1 m1
+  end.
+
+Theorem eq_nat_refl : forall n, eq_nat n n.
+induction n; simpl in |- *; auto.
+Qed.
+Hint Resolve eq_nat_refl: arith v62.
+
+Theorem eq_eq_nat : forall n m, n = m -> eq_nat n m.
+induction 1; trivial with arith.
+Qed.
+Hint Immediate eq_eq_nat: arith v62.
+
+Theorem eq_nat_eq : forall n m, eq_nat n m -> n = m.
+induction n; induction m; simpl in |- *; contradiction || auto with arith.
+Qed.
+Hint Immediate eq_nat_eq: arith v62.
+
+Theorem eq_nat_elim :
+ forall n (P:nat -> Prop), P n -> forall m, eq_nat n m -> P m.
+intros; replace m with n; auto with arith.
+Qed.
+
+Theorem eq_nat_decide : forall n m, {eq_nat n m} + {~ eq_nat n m}.
+induction n.
+destruct m as [| n].
+auto with arith.
+intros; right; red in |- *; trivial with arith.
+destruct m as [| n0].
+right; red in |- *; auto with arith.
+intros.
+simpl in |- *.
+apply IHn.
+Defined.
+
+Fixpoint beq_nat n m {struct n} : bool :=
+  match n, m with
+  | O, O => true
+  | O, S _ => false
+  | S _, O => false
+  | S n1, S m1 => beq_nat n1 m1
+  end.
+
+Lemma beq_nat_refl : forall n, true = beq_nat n n.
+Proof.
+  intro x; induction x; simpl in |- *; auto.
+Qed.
+
+Definition beq_nat_eq : forall x y, true = beq_nat x y -> x = y.
+Proof.
+  double induction x y; simpl in |- *.
+    reflexivity.
+    intros; discriminate H0.
+    intros; discriminate H0.
+    intros; case (H0 _ H1); reflexivity.
+Defined.