1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
|
; ACL2 Univariate Polynomials over a Field books -- Sum Congruences
;; Congruences for 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
;;; ------------------------------------------------------------------
;;; Congruencia de la igualdad con la suma de polinomios
;;;
;;; Autores:
;;;
;;; Inmaculada Medina Bulo
;;; Francisco Palomo Lozano
;;;
;;; Descripción:
;;;
;;; Aquí se demuestran las congruencias de la igualdad de polinomios
;;; con la suma. Las demostraciones son complejas debido a que
;;; necesitan reglas expansivas. Estas reglas son peligrosas, ya que
;;; pueden producir fácilmente ciclos en el demostrador. Para
;;; restringir su aplicación caben dos opciones:
;;;
;;; 1. Desactivarlas y usarlas explícitamente donde sea necesario. Una
;;; variante es no generar la regla en absoluto (es decir, introducir
;;; el teorema con la clases de reglas vacía).
;;;
;;; 2. Restringir su aplicación sintácticamente para prevenir
;;; expansiones en cadena. Esto se puede lograr graciasa syntaxp.
;;;
;;; Elegimos la segunda opción porque se consigue un mayor grado de
;;; automatización y hace a las demostraciones menos sensibles a los
;;; cambios.
;;; ------------------------------------------------------------------
#|
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 "fucongruencias-suma"
5
nil ;;compile-flg
)
|#
(in-package "FUPOL")
;;(include-book "suma")
(include-book "fusuma"
:load-compiled-file nil)
;;; ----------------------------------------------------
;;; Congruencia de la igualdad de polinomios con la suma
;;; ----------------------------------------------------
;;; Segundo parámetro
;;; NOTA:
;;;
;;; Esta propiedad es expansiva; restringimos su aplicación sintácticamente
(defthm
polinomiop-implies-true-listp
(implies (polinomiop p)
(true-listp p))
:rule-classes :compound-recognizer)
(defthm
Right-identity-append
(implies (true-listp p)
(equal (append p nil) p)))
(defthm |p + q = p + fn(q)|
(implies (syntaxp (not (and (consp q) (eq (primero q) 'fn))))
(= (+ p q) (+ p (fn q)))))
(defthm
=P-implies-=P-append-1
(implies (=P p1 p2)
(=P (append p1 q)
(append p2 q)))
:rule-classes :congruence)
(defthm
=P-implies-=P-append-2
(implies (=P q1 q2)
(=P (append p q1)
(append p q2)))
:rule-classes :congruence)
(defthm
=P-implies-=P-fn
(implies (=P p1 p2)
(=P (fn p1)
(fn p2)))
:rule-classes :congruence)
;;(defcong = = (+ p q) 2)
(defthm
=-implies-=-+-2
(implies (= q1 q2)
(= (+ p q1)
(+ p q2)))
:rule-classes :congruence)
;;; Primer parámetro
;; (defcong = = (+ p q) 1
;; :hints (("Goal"
;; :in-theory (disable |p + q = q + p| + =)
;; :use (|p + q = q + p|
;; (:instance |p + q = q + p| (p p-equiv))))))
(defthm
=-implies-=-+-1
(implies (= p1 p2)
(= (+ p1 q)
(+ p2 q)))
:rule-classes :congruence
:hints (("Goal"
:in-theory (disable |p + q = q + p| + =)
:use ((:instance
|p + q = q + p|
(p p1))
(:instance
|p + q = q + p|
(p p2))))))
;;; NOTA:
;;;
;;; Esta propiedad es expansiva; restringimos su aplicación sintácticamente
(defthm |p + q = fn(p) + q|
(implies (syntaxp (not (and (consp p) (eq (primero p) 'fn))))
(= (+ p q) (+ (fn p) q)))
:hints (("Goal"
:in-theory (disable |p + q = p + fn(q)|)
:use ((:instance |p + q = p + fn(q)| (p q) (q p))))))
(defthm |fn(p) + fn(q) = p + q|
(= (+ (fn p) (fn q)) (+ p q)))
(in-theory (disable |p + q = p + fn(q)|
|p + q = fn(p) + q|
|fn(p) + fn(q) = p + q|))
|
