/usr/share/acl2-7.1/books/meta/meta-plus-lessp.lisp is in acl2-books-source 7.1-1.
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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 | ; ACL2 books on arithmetic metafunctions
; Copyright (C) 1997 Computational Logic, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Written by: Matt Kaufmann and John Cowles
; Computational Logic, Inc.
; 1717 West Sixth Street, Suite 290
; Austin, TX 78703-4776 U.S.A.
(in-package "ACL2")
(include-book "term-defuns")
(local (include-book "term-lemmas" :load-compiled-file nil))
(defevaluator ev-plus-lessp ev-plus-lessp-list
((binary-+ x y)
(< x y)
(if x y z)))
(defun cancel_plus-lessp$1 (x)
(declare (xargs :guard (and (pseudo-termp x)
(consp x)
(not (equal (car x) 'quote)))
:guard-hints (("Goal" :expand
((pseudo-termp
(list* x1 (cons 'binary-+ x6) x4)))))))
(cond ((and (consp (cadr x))
(eq (car (cadr x)) 'binary-+))
(cond
((fringe-occur 'binary-+ (caddr x) (cadr x))
(list '<
(binary-op_tree-simple
'binary-+
0
(del (caddr x) (binary-op_fringe 'binary-+ (cadr x))))
*0*))
(t x)))
((and (consp (caddr x))
(eq (car (caddr x)) 'binary-+))
(cond
((fringe-occur 'binary-+ (cadr x) (caddr x))
(list '<
*0*
(binary-op_tree-simple
'binary-+
0
(del (cadr x) (binary-op_fringe 'binary-+ (caddr x))))))
(t x)))
(t x)))
(defun cancel_plus-lessp (x)
(declare (xargs :guard (pseudo-termp x)))
(if (and (consp x)
(eq (car x) '<))
(cond
((and (consp (cadr x))
(eq (car (cadr x)) 'binary-+)
(consp (caddr x))
(eq (car (caddr x)) 'binary-+))
(let* ((lt-side (binary-op_fringe 'binary-+ (cadr x)))
(rt-side (binary-op_fringe 'binary-+ (caddr x)))
(int (bagint lt-side rt-side)))
(if int
(list '<
(binary-op_tree-simple 'binary-+
0
(bagdiff lt-side int))
(binary-op_tree-simple 'binary-+
0
(bagdiff rt-side int)))
x)))
(t (cancel_plus-lessp$1 x)))
x))
(local (in-theory (disable binary-op_tree-simple)))
(local
(defthm ev-plus-lessp-binary-op_tree-simple-append
(implies (and (consp fringe1) (consp fringe2))
(equal (ev-plus-lessp (binary-op_tree-simple 'binary-+
0
(append fringe1 fringe2))
a)
(+ (ev-plus-lessp (binary-op_tree-simple 'binary-+
0
fringe1)
a)
(ev-plus-lessp (binary-op_tree-simple 'binary-+
0
fringe2)
a))))
:hints (("Goal" :induct (append fringe1 fringe2)))))
(local
(defthm ev-plus-lessp-binary-op_tree-simple-fringe
(equal (ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(binary-op_fringe 'binary-+ x))
a)
(ev-plus-lessp x a))))
(local (defthm equal-iff
(implies (and (booleanp x) (booleanp y))
(iff (equal x y)
(iff x y)))))
(local
(defthm plus-cancel-left
(equal (< (+ x y) (+ x z))
(< (fix y) (fix z)))))
(local
(defthm binary-op_tree-simple-plus-fringe-del-lemma
(implies (memb summand fringe)
(equal (+ (ev-plus-lessp summand a)
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a))
(fix (ev-plus-lessp (binary-op_tree-simple
'binary-+ 0 fringe)
a))))
:rule-classes nil
:hints (("Goal" :expand ((binary-op_tree-simple 'binary-+
0
(cdr fringe)))))))
(local
(defthm binary-op_tree-simple-plus-fringe-del-1
(implies (memb summand fringe)
(equal (< y
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a))
(< (+ y (ev-plus-lessp summand a))
(ev-plus-lessp (binary-op_tree-simple
'binary-+ 0 fringe)
a))))
:hints (("Goal" :use binary-op_tree-simple-plus-fringe-del-lemma))))
(local
(defthm binary-op_tree-simple-plus-fringe-del-2
(implies (memb summand fringe)
(equal (< (ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a)
y)
(< (ev-plus-lessp (binary-op_tree-simple
'binary-+ 0 fringe)
a)
(+ y (ev-plus-lessp summand a)))))
:hints (("Goal" :use binary-op_tree-simple-plus-fringe-del-lemma))))
(local
(defthm cancel_plus-lessp$1-property
(implies (and (consp x)
(equal (car x) '<))
(equal (ev-plus-lessp (cancel_plus-lessp$1 x) a)
(ev-plus-lessp x a)))))
(local
(in-theory (disable cancel_plus-lessp$1)))
(local
(defthm binary-op_tree-simple-plus-fringe-del
(implies (and (memb summand fringe)
(acl2-numberp y))
(equal (equal y
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a))
(and (acl2-numberp (ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a))
(equal (+ y (ev-plus-lessp summand a))
(fix (ev-plus-lessp (binary-op_tree-simple
'binary-+ 0 fringe)
a))))))
:hints (("Goal" :use binary-op_tree-simple-plus-fringe-del-lemma))))
(local
(defthm binary-op_tree-simple-plus-fringe-del-not-acl2-numberp
(implies (and (not (acl2-numberp
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(del summand fringe))
a)))
(memb summand fringe))
(equal (ev-plus-lessp
(binary-op_tree-simple
'binary-+
0
fringe)
a)
(fix (ev-plus-lessp summand a))))
:hints (("Goal" :expand ((del summand fringe)
(del summand (cdr fringe))
(binary-op_tree-simple 'binary-+
0 fringe))))))
(local
(encapsulate
()
(local
(defthm binary-op_tree-simple-opener-extra-1
(implies (and (consp fringe)
(not (consp (cdr fringe))))
(equal (binary-op_tree-simple 'binary-+ 0 fringe)
(car fringe)))))
(defthm cancel_plus-lessp-correct-lemma-1
(implies (subbagp fringe2 fringe1)
(equal
(+ (ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(bagdiff fringe1 fringe2))
a)
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
fringe2)
a))
(fix (ev-plus-lessp (binary-op_tree-simple
'binary-+
0
fringe1)
a))))
:rule-classes nil)))
(local
(defthm cancel_plus-lessp-correct-lemma
(equal
(<
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(bagdiff (binary-op_fringe 'binary-+ x1)
(bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
a)
(ev-plus-lessp (binary-op_tree-simple
'binary-+
0
(bagdiff (binary-op_fringe 'binary-+ x2)
(bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
a))
(< (ev-plus-lessp x1 a)
(ev-plus-lessp x2 a)))
:hints (("Goal" :use
((:instance cancel_plus-lessp-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-+ x1))
(fringe2 (bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
(:instance cancel_plus-lessp-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-+ x2))
(fringe2 (bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2)))))))))
(defthm cancel_plus-lessp-correct
(equal (ev-plus-lessp x a)
(ev-plus-lessp (cancel_plus-lessp x) a))
:rule-classes ((:meta :trigger-fns (<))))
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