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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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(** * Properties of orders and multiplication for modules implementing [NZOrdAxiomsSig']  This file defines the [NZMulOrderProp] functor type, meant to be [Include]d in a module implementing [NZOrdAxiomsSig'] (see [Stdlib.Numbers.NatInt.NZAxioms]).  This gives important basic compatibility lemmas between [mul] and [lt], [le]. It also gives cancellation lemmas between [mul] and [eq].  Since it applies both to natural numbers and integers, some of these lemmas have nonnegativity conditions which could disappear when used with natural numbers.  Notice that [NZMulOrderProp] [Include]s [NZAddOrderProp]. *)
Module Type NZMulOrderProp (Import NZ : NZOrdAxiomsSig').
Include NZAddOrderProp NZ.

Theorem mul_lt_pred :
forall p q n m, S p == q -> (p * n < p * m <-> q * n + m < q * m + n).