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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: NMul.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
-
-Require Export NAdd.
-
-Module NMulPropFunct (Import NAxiomsMod : NAxiomsSig).
-Module Export NAddPropMod := NAddPropFunct NAxiomsMod.
-Open Local Scope NatScope.
-
-Theorem mul_wd :
-  forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> n1 * m1 == n2 * m2.
-Proof NZmul_wd.
-
-Theorem mul_0_l : forall n : N, 0 * n == 0.
-Proof NZmul_0_l.
-
-Theorem mul_succ_l : forall n m : N, (S n) * m == n * m + m.
-Proof NZmul_succ_l.
-
-(** Theorems that are valid for both natural numbers and integers *)
-
-Theorem mul_0_r : forall n, n * 0 == 0.
-Proof NZmul_0_r.
-
-Theorem mul_succ_r : forall n m, n * (S m) == n * m + n.
-Proof NZmul_succ_r.
-
-Theorem mul_comm : forall n m : N, n * m == m * n.
-Proof NZmul_comm.
-
-Theorem mul_add_distr_r : forall n m p : N, (n + m) * p == n * p + m * p.
-Proof NZmul_add_distr_r.
-
-Theorem mul_add_distr_l : forall n m p : N, n * (m + p) == n * m + n * p.
-Proof NZmul_add_distr_l.
-
-Theorem mul_assoc : forall n m p : N, n * (m * p) == (n * m) * p.
-Proof NZmul_assoc.
-
-Theorem mul_1_l : forall n : N, 1 * n == n.
-Proof NZmul_1_l.
-
-Theorem mul_1_r : forall n : N, n * 1 == n.
-Proof NZmul_1_r.
-
-(* Theorems that cannot be proved in NZMul *)
-
-(* In proving the correctness of the definition of multiplication on
-integers constructed from pairs of natural numbers, we'll need the
-following fact about natural numbers:
-
-a * n + u == a * m + v -> n + m' == n' + m -> a * n' + u = a * m' + v
-
-Here n + m' == n' + m expresses equality of integers (n, m) and (n', m'),
-since a pair (a, b) of natural numbers represents the integer a - b. On
-integers, the formula above could be proved by moving a * m to the left,
-factoring out a and replacing n - m by n' - m'. However, the formula is
-required in the process of constructing integers, so it has to be proved
-for natural numbers, where terms cannot be moved from one side of an
-equation to the other. The proof uses the cancellation laws add_cancel_l
-and add_cancel_r. *)
-
-Theorem add_mul_repl_pair : forall a n m n' m' u v : N,
-  a * n + u == a * m + v -> n + m' == n' + m -> a * n' + u == a * m' + v.
-Proof.
-intros a n m n' m' u v H1 H2.
-apply (@NZmul_wd a a) in H2; [| reflexivity].
-do 2 rewrite mul_add_distr_l in H2. symmetry in H2.
-pose proof (NZadd_wd _ _ H1 _ _ H2) as H3.
-rewrite (add_shuffle1 (a * m)), (add_comm (a * m) (a * n)) in H3.
-do 2 rewrite <- add_assoc in H3. apply -> add_cancel_l in H3.
-rewrite (add_assoc u), (add_comm (a * m)) in H3.
-apply -> add_cancel_r in H3.
-now rewrite (add_comm (a * n') u), (add_comm (a * m') v).
-Qed.
-
-End NMulPropFunct.
-