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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        *)
+(************************************************************************)
+(*                      Evgeny Makarov, INRIA, 2007                     *)
+(************************************************************************)
+
+(*i $Id: NAxioms.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+
+Require Export NZAxioms.
+
+Set Implicit Arguments.
+
+Module Type NAxiomsSig.
+Declare Module Export NZOrdAxiomsMod : NZOrdAxiomsSig.
+
+Delimit Scope NatScope with Nat.
+Notation N := NZ.
+Notation Neq := NZeq.
+Notation N0 := NZ0.
+Notation N1 := (NZsucc NZ0).
+Notation S := NZsucc.
+Notation P := NZpred.
+Notation add := NZadd.
+Notation mul := NZmul.
+Notation sub := NZsub.
+Notation lt := NZlt.
+Notation le := NZle.
+Notation min := NZmin.
+Notation max := NZmax.
+Notation "x == y"  := (Neq x y) (at level 70) : NatScope.
+Notation "x ~= y" := (~ Neq x y) (at level 70) : NatScope.
+Notation "0" := NZ0 : NatScope.
+Notation "1" := (NZsucc NZ0) : NatScope.
+Notation "x + y" := (NZadd x y) : NatScope.
+Notation "x - y" := (NZsub x y) : NatScope.
+Notation "x * y" := (NZmul x y) : NatScope.
+Notation "x < y" := (NZlt x y) : NatScope.
+Notation "x <= y" := (NZle x y) : NatScope.
+Notation "x > y" := (NZlt y x) (only parsing) : NatScope.
+Notation "x >= y" := (NZle y x) (only parsing) : NatScope.
+
+Open Local Scope NatScope.
+
+Parameter Inline recursion : forall A : Type, A -> (N -> A -> A) -> N -> A.
+Implicit Arguments recursion [A].
+
+Axiom pred_0 : P 0 == 0.
+
+Axiom recursion_wd : forall (A : Type) (Aeq : relation A),
+  forall a a' : A, Aeq a a' ->
+    forall f f' : N -> A -> A, fun2_eq Neq Aeq Aeq f f' ->
+      forall x x' : N, x == x' ->
+        Aeq (recursion a f x) (recursion a' f' x').
+
+Axiom recursion_0 :
+  forall (A : Type) (a : A) (f : N -> A -> A), recursion a f 0 = a.
+
+Axiom recursion_succ :
+  forall (A : Type) (Aeq : relation A) (a : A) (f : N -> A -> A),
+    Aeq a a -> fun2_wd Neq Aeq Aeq f ->
+      forall n : N, Aeq (recursion a f (S n)) (f n (recursion a f n)).
+
+(*Axiom dep_rec :
+  forall A : N -> Type, A 0 -> (forall n : N, A n -> A (S n)) -> forall n : N, A n.*)
+
+End NAxiomsSig.
+