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; ACL2 Univariate Polynomials over a Field books -- Polynomial Sums
;; Sums of Univariate Polynomials over a Field
; Copyright (C) 2006 John R. Cowles and Ruben A. Gamboa, University of
; Wyoming
; This book is free software; you can redistribute it and/or modify
; it under the terms of the GNU General Public License as published by
; the Free Software Foundation; either version 2 of the License, or
; (at your option) any later version.
; This book is distributed in the hope that it will be useful,
; but WITHOUT ANY WARRANTY; without even the implied warranty of
; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
; GNU General Public License for more details.
; You should have received a copy of the GNU General Public License
; along with this book; if not, write to the Free Software
; Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
;; Modified by J. Cowles
;; Last modified July 2006 (for ACL2 Version 3.0).
;; Based on
;;; ----------------------------------------------------------------
;;; Suma de polinomios
;;;
;;; Autores:
;;;
;;; Inmaculada Medina Bulo
;;; Francisco Palomo Lozano
;;;
;;; Descripción:
;;;
;;; Desarrollo de la suma de polinomios definida simplemente como la
;;; concatenación de las listas de monomios que los integran. Las
;;; propiedades de la concatenación de listas permiten establecer la
;;; base para realizar demostraciones de propiedades sobre los
;;; polinomios más complicadas que incorporan la igualdad
;;; semántica. Se demuestra que los polinomios con la operación de
;;; suma forman un monoide conmutativo.
;;; ----------------------------------------------------------------
#|
To certify this book, first, create a world with the following packages:
(in-package "ACL2")
(defconst *import-symbols*
(set-difference-eq
(union-eq *acl2-exports*
*common-lisp-symbols-from-main-lisp-package*)
'(null + * - < = / commutativity-of-* associativity-of-*
commutativity-of-+ associativity-of-+ distributivity)))
(defpkg "FLD"
*import-symbols*)
(defpkg "FUTER"
*import-symbols*)
(defpkg "FUMON"
(union-eq *import-symbols*
'(FLD::fdp FUTER::terminop)))
(defpkg "FUPOL"
(union-eq *import-symbols*
'(FUTER::naturalp FUTER::terminop FUMON::monomio FUMON::coeficiente
FUMON::termino FUMON::monomiop)))
(certify-book "fusuma"
5
nil ;;compile-flg
)
|#
(in-package "FUPOL")
;;(include-book "forma-normal")
(include-book "fuforma-normal"
:load-compiled-file nil)
;;; ------------------
;;; Suma de polinomios
;;; ------------------
(defun + (p q)
(cond ((and (not (polinomiop p)) (not (polinomiop q)))
(nulo))
((not (polinomiop p))
q)
((not (polinomiop q))
p)
(t
(append p q))))
;;; Clausura
(defthm polinomiop-append
(implies (and (polinomiop (double-rewrite p))
(polinomiop (double-rewrite q)))
(polinomiop (append p q)))
:rule-classes (:type-prescription :rewrite))
(defthm polinomiop-+
(polinomiop (+ p q))
:rule-classes (:type-prescription :rewrite))
;;; ----------------------
;;; Propiedades de la suma
;;; ----------------------
;;; Neutro
(defthm |(0 p) = p|
(equal (append (nulo) p) p))
(defthm |0 + p = p|
(= (+ (nulo) p) p))
(defthm |(p 0) = p|
(implies (polinomiop (double-rewrite p))
(equal (append p (nulo)) p)))
(defthm |p + 0 = p|
(= (+ p (nulo)) p))
;;; Asociatividad
(defthm |(m +M p) + q =P m +M (p + q)|
(equal (+ (+M m p) q) (+M m (+ p q)))
:hints (("Goal"
:use polinomiop-append)))
(defthm |p + q =e mp(p) +M (resto(p) + q)-1|
(implies (and (polinomiop p)
(not (nulop p)))
(equal (+ p q) (+M (primero p) (+ (resto p) q))))
:rule-classes nil)
(defthm |p + q =e mp(p) +M (resto(p) + q)|
(implies (and (polinomiop (double-rewrite p))
(not (nulop p)))
(equal (+ p q) (+M (primero p) (+ (resto p) q))))
:hints (("Goal"
:use |p + q =e mp(p) +M (resto(p) + q)-1|)))
(local (in-theory (disable = + +M)))
(defthm |(p + q) + r = p + (q + r)|
(= (+ (+ p q) r) (+ p (+ q r)))
:hints (("Goal" :induct (fn p))
("Subgoal *1/1" :in-theory (enable = + +M))))
;;; Conmutatividad
(defthm |q + p = mp(p) +M (q + resto(p))|
(implies (and (polinomiop (double-rewrite p))
(not (nulop p)))
(= (+ q p) (+M (primero p) (+ q (resto p)))))
:hints (("Goal" :induct (fn q))
("Subgoal *1/1" :in-theory (enable = + +M))))
(defthm |p + q = q + p|
(= (+ p q) (+ q p))
:hints (("Goal" :induct (fn p))
("Subgoal *1/1" :in-theory (enable = + +M))))
|
