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|
; ACL2 Univariate Polynomials over a Field books -- Unary Minus
;; Unary Minus 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
;;; ------------------------------------------------------------------
;;; Opuesto de polinomios
;;;
;;; Autores:
;;;
;;; Inmaculada Medina Bulo
;;; Francisco Palomo Lozano
;;;
;;; Descripci�n:
;;;
;;; Desarrollo del opuesto de un polinomio, que se define monomio a
;;; monomio. Su correcci�n se prueba demostrando que la funci�n que lo
;;; calcula produce el inverso aditivo. Para que �stas y otras
;;; propiedades sean incondicionales (carezcan de hip�tesis) se
;;; completa cuidadosamente la definici�n de la funci�n. Se demuestra
;;; que los polinomios con las operaciones de suma y opuesto forman un
;;; grupo 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 "fuopuesto"
5
nil ;;compile-flg
)
|#
(in-package "FUPOL")
;;(include-book "congruencias-suma")
(include-book "fucongruencias-suma"
:load-compiled-file nil)
;;; ---------------------
;;; Opuesto de polinomios
;;; ---------------------
;;; La siguiente desactivaci�n es necesaria para la verificaci�n de la
;;; protecci�n del opuesto
(in-theory (disable FUMON::monomio-coeficiente-termino))
(defun - (p)
(cond ((or (not (polinomiop p)) (nulop p))
(nulo))
(t
(+M (FUMON::- (primero p)) (- (resto p))))))
(in-theory (enable FUMON::monomio-coeficiente-termino))
;;; Clausura
(defthm polinomiop--
(polinomiop (- p))
:rule-classes (:type-prescription
:rewrite))
;;; Distributividad respecto de la suma de monomio y polinomio
(defthm |- p =e (- mp(p)) +M (- (resto(p)))|
(implies (and (polinomiop (double-rewrite p))
(not (nulop p)))
(equal (- p)
(+M (FUMON::- (primero p)) (- (resto p))))))
;;; Inverso de la suma
(defthm |m +M (- m +M 0) = 0|
(implies (monomiop (double-rewrite m))
(= (+M m (+M (FUMON::- m) (nulo)))
(nulo))))
(defthm |p + (- p) = 0|
(= (+ p (- p)) (nulo))
:hints (("Goal"
:in-theory (disable = + +M - FUMON::-)
:induct (fn p))
("Subgoal *1/1" :in-theory (enable = + +M))
("Subgoal *1/2"
:use ((:instance |p + q = q + p|
(p (+M (FUMON::- (primero p)) (- (resto p))))
(q (resto p)))))))
;;; Distributividad respecto de la suma en orden de monomio y
;;; polinomio
;; (defthm |- (m +Mo p) =e (- m) +Mo (- p)|
;; (implies (and (monomiop (double-rewrite m))
;; (polinomiop (double-rewrite p)))
;; (equal (- (+-monomio m p))
;; (+-monomio (FUMON::- m) (- p)))))
(defthm |- (m +Mo p) =P (- m) +Mo (- p)|
(implies (and (monomiop (double-rewrite m))
(polinomiop (double-rewrite p)))
(=P (- (+-monomio m p))
(+-monomio (FUMON::- m) (- p)))))
;;; -------------------------------------------------------
;;; Congruencia de la igualdad de polinomios con el opuesto
;;; -------------------------------------------------------
(defthm
|FUMON::nulop_-|
(implies (monomiop (double-rewrite m))
(equal (FUMON::nulop (FUMON::- m))
(FUMON::nulop m))))
(defthm
=P-implies-=P_-
(implies (=P p1 p2)
(=P (- p1)
(- p2)))
:rule-classes :congruence)
(defthm
Ordenadop_-
(implies (ordenadop p)
(ordenadop (- p))))
;; (local
;; (defthm |fn(- p) = - fn(p)|
;; (equal (fn (- p)) (- (fn p)))
;; :hints (("Goal" :in-theory (disable MON::- )))))
(local
(defthm |fn(- p) =P - fn(p)|
(=P (fn (- p)) (- (fn p)))
:hints (("Goal" :in-theory (disable FUMON::- )))))
;; (defcong = = (- p) 1)
(defthm
=-implies-=_-
(implies (= p1 p2)
(= (- p1)
(- p2)))
:rule-classes :congruence)
;;; ---------------
;;; Desactivaciones
;;; ---------------
(local (in-theory (disable polinomiop = + -)))
;;; -----------------------------------------------------------------------
;;; El inverso debe ser invisible para la primera operaci�n y para s� mismo
;;; -----------------------------------------------------------------------
(ACL2::set-invisible-fns-table ((+ -) (- -)))
;;; El opuesto de una suma de polinomios es la suma de los opuestos.
(defthm |p + r = q + r <=> p = q|
(implies (and (polinomiop (double-rewrite p))
(polinomiop (double-rewrite q)))
(iff (= (+ p r) (+ q r)) (= p q)))
:hints (("Goal"
:in-theory (disable |p + q = q + p| |p + 0 = p|)
:use ((:instance |p + q = q + p| (p (+ p r)) (q (- r)))
(:instance |p + q = q + p| (p (- r)) (q (+ q r)))
(:instance |p + 0 = p| (p q))
|p + 0 = p|))))
(local
(defthm |p + q = q <=> p = 0|
(implies (and (polinomiop (double-rewrite p))
(polinomiop (double-rewrite q)))
(iff (= (+ p q) q) (= p (nulo))))
:hints (("Goal"
:in-theory (disable |p + r = q + r <=> p = q|)
:use (:instance |p + r = q + r <=> p = q| (q (nulo)) (r q))))))
(local
(defthm |p + q = 0 <=> q = - p|
(implies (and (polinomiop (double-rewrite p))
(polinomiop (double-rewrite q)))
(iff (= (+ p q) (nulo)) (= q (- p))))
:hints (("Goal"
:in-theory (disable |p + r = q + r <=> p = q|)
:use (:instance |p + r = q + r <=> p = q| (p q) (q (- p)) (r p))))))
;;; Mezcla de asociatividad y conmutatividad
(defthm |p + (q + r) = q + (p + r)|
(= (+ p (+ q r)) (+ q (+ p r)))
:hints (("Goal"
:in-theory (disable |(p + q) + r = p + (q + r)|)
:use (|(p + q) + r = p + (q + r)|
(:instance |(p + q) + r = p + (q + r)| (p q) (q p))))))
;;; Idempotencia del inverso
(defthm |- (- p) = p|
(implies (polinomiop (double-rewrite p))
(= (- (- p)) p))
:hints (("Goal"
:in-theory (disable |p + q = 0 <=> q = - p|)
:use (:instance |p + q = 0 <=> q = - p| (p (- p)) (q p)))))
;;; Distributividad de la inversa sobre la primera operaci�n
(defthm |- (p + q) = (- p) + (- q)|
(= (- (+ p q)) (+ (- p) (- q)))
:hints (("Goal"
:in-theory (disable |p + q = 0 <=> q = - p|)
:use (:instance |p + q = 0 <=> q = - p| (p (+ p q)) (q (+ (- p) (- q)))))))
|
