/usr/share/acl2-7.1/books/meta/meta-plus-equal.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 | ; 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-equal ev-plus-equal-list
((binary-+ x y)
(fix x)
(equal x y)
(acl2-numberp x)
(if x y z)))
(defun cancel_plus-equal$1 (x)
(declare (xargs :guard (and (pseudo-termp x)
(consp x))))
; Not all of
; (and (consp (cadr x))
; (eq (car (cadr x)) 'binary-+)
; (consp (caddr x))
; (eq (car (caddr x)) 'binary-+))
; hold.
(mv-let
(elt term)
(cond ((and (consp (cadr x))
(eq (car (cadr x)) 'binary-+))
(mv (caddr x) (cadr x)))
((and (consp (caddr x))
(eq (car (caddr x)) 'binary-+))
(mv (cadr x) (caddr x)))
(t (mv nil nil)))
(cond
((and elt (fringe-occur 'binary-+ elt term))
(list 'if
(list 'acl2-numberp elt)
(list 'equal
*0*
(binary-op_tree 'binary-+
0
'fix
(del elt (binary-op_fringe 'binary-+ term))))
*nil*))
(t x))))
(defun cancel_plus-equal (x)
(declare (xargs :guard (pseudo-termp x)))
(if (and (consp x)
(eq (car x) 'equal))
(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 'equal
(binary-op_tree 'binary-+
0
'fix
(bagdiff lt-side int))
(binary-op_tree 'binary-+
0
'fix
(bagdiff rt-side int)))
x)))
(t (cancel_plus-equal$1 x)))
x))
(local
(defthm acl2-numberp-ev-plus-equal
(acl2-numberp (ev-plus-equal (binary-op_tree 'binary-+ 0 'fix fringe) a))
:rule-classes :type-prescription))
(local (in-theory (disable binary-op_tree)))
(local
(defthm ev-plus-equal-binary-op_tree-append
(equal (ev-plus-equal (binary-op_tree 'binary-+
0 'fix
(append fringe1 fringe2))
a)
(+ (ev-plus-equal (binary-op_tree 'binary-+
0 'fix
fringe1)
a)
(ev-plus-equal (binary-op_tree 'binary-+
0 'fix
fringe2)
a)))
:hints (("Goal" :induct (append fringe1 fringe2)))))
(local
(defthm ev-plus-equal-binary-op_tree-fringe
(equal (ev-plus-equal (binary-op_tree 'binary-+
0 'fix
(binary-op_fringe 'binary-+ x))
a)
(fix (ev-plus-equal x a)))))
(local
(defthm plus-cancel-left
(equal (equal (+ x y) (+ x z))
(equal (fix y) (fix z)))))
(local
(defthm binary-op_tree-plus-fringe-del-lemma
(implies (memb summand fringe)
(equal (+ (ev-plus-equal summand a)
(ev-plus-equal (binary-op_tree 'binary-+
0 'fix
(del summand fringe))
a))
(ev-plus-equal (binary-op_tree 'binary-+
0 'fix
fringe)
a)))
:rule-classes nil
:hints (("Goal" :expand ((binary-op_tree 'binary-+
0 'fix
(cdr fringe)))))))
(local
(defthm binary-op_tree-plus-fringe-del
(implies (and (memb summand fringe)
(acl2-numberp y))
(equal (equal y
(ev-plus-equal (binary-op_tree 'binary-+
0 'fix
(del summand fringe))
a))
(equal (+ y (ev-plus-equal summand a))
(ev-plus-equal (binary-op_tree 'binary-+
0 'fix
fringe)
a))))
:hints (("Goal" :use binary-op_tree-plus-fringe-del-lemma))))
(local
(defthm cancel_plus-equal$1-property
(implies (and (consp x)
(equal (car x) 'equal))
(equal (ev-plus-equal (cancel_plus-equal$1 x) a)
(ev-plus-equal x a)))))
(local
(in-theory (disable cancel_plus-equal$1)))
(local
(encapsulate
()
(local
(defthm binary-op_tree-opener-extra-1
(implies (and (consp fringe)
(not (consp (cdr fringe))))
(equal (binary-op_tree 'binary-+ 0 op fringe)
(list op (car fringe))))))
(defthm cancel_equal-plus-correct-lemma-1
(implies (subbagp fringe2 fringe1)
(equal
(+ (ev-plus-equal (binary-op_tree
'binary-+
0
'fix
(bagdiff fringe1 fringe2))
a)
(ev-plus-equal (binary-op_tree
'binary-+
0
'fix
fringe2)
a))
(ev-plus-equal (binary-op_tree
'binary-+
0
'fix
fringe1)
a)))
:rule-classes nil)))
(local
(defthm cancel_equal-plus-correct-lemma
(equal
(equal
(ev-plus-equal (binary-op_tree
'binary-+
0
'fix
(bagdiff (binary-op_fringe 'binary-+ x1)
(bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
a)
(ev-plus-equal (binary-op_tree
'binary-+
0
'fix
(bagdiff (binary-op_fringe 'binary-+ x2)
(bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
a))
(equal (fix (ev-plus-equal x1 a))
(fix (ev-plus-equal x2 a))))
:hints (("Goal" :use
((:instance cancel_equal-plus-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-+ x1))
(fringe2 (bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2))))
(:instance cancel_equal-plus-correct-lemma-1
(fringe1 (binary-op_fringe 'binary-+ x2))
(fringe2 (bagint (binary-op_fringe 'binary-+ x1)
(binary-op_fringe 'binary-+ x2)))))))))
(defthm cancel_plus-equal-correct
(equal (ev-plus-equal x a)
(ev-plus-equal (cancel_plus-equal x) a))
:rule-classes ((:meta :trigger-fns (equal))))
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