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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        *)
-(************************************************************************)
-
-Require Export Axioms.
-Require Export EqAxioms.
-
-(** This file contains basic properties of addition with respect to equality *)
-
-(** Properties of Addition *)
-Lemma add_x_0 : (x:N)(x+zero)=x.
-EAuto 3 with num.
-Qed.
-Hints Resolve add_x_0 : num.
-
-Lemma add_x_Sy : (x,y:N)(x+(S y))=(S (x+y)).
-Intros x y; Apply eq_trans with (S y)+x; EAuto with num.
-Qed.
-
-Hints Resolve add_x_Sy : num.
-
-Lemma add_x_Sy_swap : (x,y:N)(x+(S y))=((S x)+y).
-EAuto with num.
-Qed.
-Hints Resolve add_x_Sy_swap : num.
-
-Lemma add_Sx_y_swap : (x,y:N)((S x)+y)=(x+(S y)).
-Auto with num.
-Qed.
-Hints Resolve add_Sx_y_swap : num.
-
-
-Lemma add_assoc_r : (x,y,z:N)(x+(y+z))=((x+y)+z).
-Auto with num.
-Qed.
-Hints Resolve add_assoc_r : num.
-
-Lemma add_x_y_z_perm : (x,y,z:N)((x+y)+z)=(y+(x+z)).
-EAuto with num.
-Qed.
-Hints Resolve add_x_y_z_perm : num.
-
-Lemma add_x_one_S : (x:N)(x+one)=(S x).
-Intros; Apply eq_trans with (x+(S zero)); EAuto with num.
-Qed.
-Hints Resolve add_x_one_S : num.
-
-Lemma add_one_x_S : (x:N)(one+x)=(S x).
-Intros; Apply eq_trans with (x+one); Auto with num.
-Qed.
-Hints Resolve add_one_x_S : num.
-
-