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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 | ; ACL2 Arithmetic Rationals-with-axioms-PROVED book.
; Copyright (C) 1998 John R. Cowles, University of Wyoming
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by:
; John Cowles
; Department of Computer Science
; University of Wyoming
; Laramie, WY 82071-3682 U.S.A.
#| Summer 1997
Last modified 30 June 1998
Depends on the Arithmetic Inequalities book and the
Arithmetic Nonnegative Integer Mod and Gcd book.
Modified by Jared Davis in January 2014 to add xdoc section.
This book proves all the axioms in the
Arithmetic Rationals-with-axioms book
To certify in Arithmetic Book directory:
(certify-book "rationals-with-axioms-proved" 0 nil)
|#
#| This following is copied from the Arithmetic Rationals-with-axioms book:
; Written by:
; Matt Kaufmann, Bishop Brock, and John Cowles, with help from Art Flatau
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
(include-book "inequalities")
; The following three axioms seemed crucial for some of Matt
; Kaufmann's work, both in non-standard analysis and in a termination
; and guard verification exercise for axioms.lisp. After these three
; axioms is an informal argument that justifies them. Perhaps some
; day, someone will translate this informal proof to an Acl2 proof.
(defaxiom denominator-unary-minus
(implies (rationalp x)
(equal (denominator (- x))
(denominator x))))
(defaxiom numerator-goes-down-by-integer-division
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= 0 r))
(<= (numerator (* (/ x) r))
(numerator r)))
:rule-classes :linear)
(defaxiom denominator-plus
(implies (and (rationalp r)
(integerp i))
(equal (denominator (+ r i))
(denominator r)))
:rule-classes
(:rewrite
(:rewrite :corollary
(implies (and (rationalp r)
(integerp i))
(equal (denominator (+ i r))
(denominator r))))))
#|
Here are proofs of the three axioms above, using heavily some standard
facts about gcd.
Basic Fact:
If r = p/q where all are rational, p and q are integers, and q is
positive, then if g = gcd(|p|,q), we have (numerator r) = p/g,
(denominator r) = q/g.
Proof of Basic Fact. Let n1 = p/g, d1 = q/g, n = (numerator r), d =
(denominator r). Thus we have n1/d1 = n/d, and we know that n1 and d1
are relatively prime. Let d2 = gcd(d1,d). We will show d2 = d. Let
us write d = k*d2 (all positive integers) where k and d1 are
relatively prime. Now n1*d=n*d1, so since k divides d, k divides
n*d1, and hence since k and d1 are relatively prime, therefore k
divides n. But then since k divides both n and d, we have a
contradiction of lowest terms. Therefore k=1, and hence d = d2 =
gcd(d1,d). Thus we may write d1 = i*d for some positive integer i.
Now n1*d=n*d1, so n1*d=n*i*d, and hence n1 = n*i. So i divides both
n1 and d1, which implies i = 1, and hence we're done.
Corollary: gcd(|(numerator r)|,(denominator r)) = 1 for all rationals r.
Proof. Setting p = (numerator r) and q = (denominator r) above, we
obtain p = p/gcd(|p|,q) and hence p = 0 or else gcd(|p|,q) = 1. But
if p = 0 then (denominator r) = 1 and hence gcd(|p|,q) = 1 in that
case too.
Lemma about gcd (stated without proof).
gcd(|i+k*j|,j) = gcd(i,j) for integers i,j,k.
(defaxiom denominator-unary-minus
(implies (rationalp x)
(equal (denominator (- x))
(denominator x))))
Proof. Let x = p/q, where p and q are the numerator and denominator
of x. Then -x = (-p)/q, so (denominator -x) = q/gcd(|-p|,q) =
q/gcd(p,q) = q/1 = q.
(defaxiom denominator-plus
(implies (and (rationalp r)
(integerp i))
(equal (denominator (+ r i))
(denominator r))))
Proof. (+ r i) = (/ (+ n (* i d)) d), where n and d are the numerator
and denominator of r. Now we have (denominator (+ r i)) = d/gcd(|+ n
(* i d)|,d) by the Basic Fact. But using the lemma about gcd stated
above, we have gcd(|+ n (* i d)|,d) = gcd(|n|,d) = 1, and hence we're
done.
(defaxiom numerator-goes-down-by-integer-division
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= 0 r))
(<= (numerator (* (/ x) r))
(numerator r)))
:rule-classes :linear)
Let r = n/d, where n and d are its numerator and denominator.
Then (* (/ x) r) = n/(x*d), so by the Basic Fact,
(numerator (* (/ x) r)) = n/gcd(|n|,(x*d)) <= n.
|#
; Here are some axioms that Bishop wanted.
(encapsulate
()
(local
(defthm /-cancellation-on-right-temp
(implies (and (equal z (* x (/ y)))
(rationalp x)
(rationalp y)
(not (equal y 0)))
(equal (* y z) x))))
(defthm numerator-minus
(equal (numerator (- i))
(- (numerator i)))
:hints (("Goal" :use
((:instance rational-implies2
(x i))
(:instance rational-implies2
(x (- i)))
(:instance denominator-unary-minus
(x i)))
:in-theory (disable rational-implies2
denominator-unary-minus))))
)
(encapsulate
()
(local
(defthm numerator-/x-positive-case
(implies
(and (integerp x)
(< 0 x))
(equal (numerator (/ x)) 1))
:rule-classes nil
:hints
(("Goal" :use
((:instance rational-implies2 (x (/ x)))
(:instance lowest-terms (n (numerator (/ x)))
(r 1)
(q x)
(x (/ x))))
:in-theory (disable rational-implies2
*-r-denominator-r)))))
(defthm numerator-/x
(implies
(and (integerp x)
(not (equal x 0)))
(equal (numerator (/ x)) (signum x)))
:hints
(("Goal"
:use
(numerator-/x-positive-case
(:instance numerator-/x-positive-case
(x (- x)))))))
)
|#
(in-package "ACL2")
(include-book "inequalities")
(local (include-book "mod-gcd"))
(defsection more-rational-identities
:parents (arithmetic-1 numerator denominator)
:short "Rules about @(see numerator) and @(see denominator) in the
@('arithmetic/rationals') book."
(defthm Denominator-unary-minus
(implies (rationalp x)
(equal (denominator (- x))
(denominator x)))
:hints (("Goal" :use (:instance Unique-rationalp
(d (denominator x))
(n (- (numerator x)))))))
(defthm Numerator-goes-down-by-integer-division
(implies (and (integerp x)
(< 0 x)
(rationalp r))
(<= (abs (numerator (* (/ x) r)))
(abs (numerator r))))
:rule-classes ((:linear :corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(>= r 0))
(<= (numerator (* (/ x) r))
(numerator r))))
(:linear :corollary
(implies (and (integerp x)
(< 0 x)
(rationalp r)
(<= r 0))
(<= (numerator r)
(numerator (* (/ x) r))))))
:hints (("Goal" :use (:instance LEAST-numerator-denominator-<=
(n (numerator r))
(d (* x (denominator r)))))))
(defthm Denominator-plus
(implies (and (rationalp r)
(integerp i))
(equal (denominator (+ i r))
(denominator r)))
:hints (("Goal" :use ((:instance Unique-rationalp
(d (denominator r))
(n (+ (numerator r)(* i (denominator r)))))
(:instance Nonneg-int-gcd-abs
(x (numerator r))
(j i)
(n (denominator r)))))))
; Here are some axioms that Bishop wanted.
(defthm Numerator-minus
(equal (numerator (- i))
(- (numerator i)))
:hints (("Goal" :cases ((rationalp i)))
("Subgoal 1" :use (:instance Unique-rationalp
(d (denominator i))
(n (- (numerator i)))))))
(defthm Numerator-/x
(implies (and (integerp x)
(not (equal x 0)))
(equal (numerator (/ x))
(signum x)))
:hints (("Goal" :use (:instance Unique-rationalp
(n (signum x))
(d (abs x))))))
)
|