1;;; calc-rules.el --- rules for simplifying algebraic expressions in Calc 2 3;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, 4;; 2005, 2006, 2007 Free Software Foundation, Inc. 5 6;; Author: David Gillespie <daveg@synaptics.com> 7;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> 8 9;; This file is part of GNU Emacs. 10 11;; GNU Emacs is free software; you can redistribute it and/or modify 12;; it under the terms of the GNU General Public License as published by 13;; the Free Software Foundation; either version 2, or (at your option) 14;; any later version. 15 16;; GNU Emacs is distributed in the hope that it will be useful, 17;; but WITHOUT ANY WARRANTY; without even the implied warranty of 18;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 19;; GNU General Public License for more details. 20 21;; You should have received a copy of the GNU General Public License 22;; along with GNU Emacs; see the file COPYING. If not, write to the 23;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, 24;; Boston, MA 02110-1301, USA. 25 26;;; Commentary: 27 28;;; Code: 29 30;; This file is autoloaded from calc-ext.el. 31 32(require 'calc-ext) 33(require 'calc-macs) 34 35(defun calc-compile-rule-set (name rules) 36 (prog2 37 (message "Preparing rule set %s..." name) 38 (math-read-plain-expr rules t) 39 (message "Preparing rule set %s...done" name))) 40 41(defun calc-CommuteRules () 42 "CommuteRules" 43 (calc-compile-rule-set 44 "CommuteRules" "[ 45iterations(1), 46select(plain(a + b)) := select(plain(b + a)), 47select(plain(a - b)) := select(plain((-b) + a)), 48select(plain((1/a) * b)) := select(b / a), 49select(plain(a * b)) := select(b * a), 50select((1/a) / b) := select((1/b) / a), 51select(a / b) := select((1/b) * a), 52select((a^b) ^ c) := select((a^c) ^ b), 53select(log(a, b)) := select(1 / log(b, a)), 54select(plain(a && b)) := select(b && a), 55select(plain(a || b)) := select(b || a), 56select(plain(a = b)) := select(b = a), 57select(plain(a != b)) := select(b != a), 58select(a < b) := select(b > a), 59select(a > b) := select(b < a), 60select(a <= b) := select(b >= a), 61select(a >= b) := select(b <= a) ]")) 62 63(defun calc-JumpRules () 64 "JumpRules" 65 (calc-compile-rule-set 66 "JumpRules" "[ 67iterations(1), 68plain(select(x) = y) := 0 = select(-x) + y, 69plain(a + select(x) = y) := a = select(-x) + y, 70plain(a - select(x) = y) := a = select(x) + y, 71plain(select(x) + a = y) := a = select(-x) + y, 72plain(a * select(x) = y) := a = y / select(x), 73plain(a / select(x) = y) := a = select(x) * y, 74plain(select(x) / a = y) := 1/a = y / select(x), 75plain(a ^ select(2) = y) := a = select(sqrt(y)), 76plain(a ^ select(x) = y) := a = y ^ select(1/x), 77plain(select(x) ^ a = y) := a = log(y, select(x)), 78plain(log(a, select(x)) = y) := a = select(x) ^ y, 79plain(log(select(x), a) = y) := a = select(x) ^ (1/y), 80plain(y = select(x)) := y - select(x) = 0, 81plain(y = a + select(x)) := y - select(x) = a, 82plain(y = a - select(x)) := y + select(x) = a, 83plain(y = select(x) + a) := y - select(x) = a, 84plain(y = a * select(x)) := y / select(x) = a, 85plain(y = a / select(x)) := y * select(x) = a, 86plain(y = select(x) / a) := y / select(x) = 1/a, 87plain(y = a ^ select(2)) := select(sqrt(y)) = a, 88plain(y = a ^ select(x)) := y ^ select(1/x) = a, 89plain(y = select(x) ^ a) := log(y, select(x)) = a, 90plain(y = log(a, select(x))) := select(x) ^ y = a, 91plain(y = log(select(x), a)) := select(x) ^ (1/y) = a ]")) 92 93(defun calc-DistribRules () 94 "DistribRules" 95 (calc-compile-rule-set 96 "DistribRules" "[ 97iterations(1), 98x * select(a + b) := x*select(a) + x*b, 99x * select(sum(a,b,c,d)) := sum(x*select(a),b,c,d), 100x / select(a + b) := 1 / (select(a)/x + b/x), 101select(a + b) / x := select(a)/x + b/x, 102sum(select(a),b,c,d) / x := sum(select(a)/x,b,c,d), 103x ^ select(a + b) := x^select(a) * x^b, 104x ^ select(sum(a,b,c,d)) := prod(x^select(a),b,c,d), 105x ^ select(a * b) := (x^a)^select(b), 106x ^ select(a / b) := (x^a)^select(1/b), 107select(a + b) ^ n := select(x) 108 :: integer(n) :: n >= 2 109 :: let(x, expandpow(a+b,n)) 110 :: quote(matches(x,y+z)), 111select(a + b) ^ x := a*select(a+b)^(x-1) + b*select(a+b)^(x-1), 112select(a * b) ^ x := a^x * select(b)^x, 113select(prod(a,b,c,d)) ^ x := prod(select(a)^x,b,c,d), 114select(a / b) ^ x := select(a)^x / b^x, 115select(- a) ^ x := (-1)^x * select(a)^x, 116plain(-select(a + b)) := select(-a) - b, 117plain(-select(sum(a,b,c,d))) := sum(select(-a),b,c,d), 118plain(-select(a * b)) := select(-a) * b, 119plain(-select(a / b)) := select(-a) / b, 120sqrt(select(a * b)) := sqrt(select(a)) * sqrt(b), 121sqrt(select(prod(a,b,c,d))) := prod(sqrt(select(a)),b,c,d), 122sqrt(select(a / b)) := sqrt(select(a)) / sqrt(b), 123sqrt(select(- a)) := sqrt(-1) sqrt(select(a)), 124exp(select(a + b)) := exp(select(a)) / exp(-b) :: negative(b), 125exp(select(a + b)) := exp(select(a)) * exp(b), 126exp(select(sum(a,b,c,d))) := prod(exp(select(a)),b,c,d), 127exp(select(a * b)) := exp(select(a)) ^ b :: constant(b), 128exp(select(a * b)) := exp(select(a)) ^ b, 129exp(select(a / b)) := exp(select(a)) ^ (1/b), 130ln(select(a * b)) := ln(select(a)) + ln(b), 131ln(select(prod(a,b,c,d))) := sum(ln(select(a)),b,c,d), 132ln(select(a / b)) := ln(select(a)) - ln(b), 133ln(select(a ^ b)) := ln(select(a)) * b, 134log10(select(a * b)) := log10(select(a)) + log10(b), 135log10(select(prod(a,b,c,d))) := sum(log10(select(a)),b,c,d), 136log10(select(a / b)) := log10(select(a)) - log10(b), 137log10(select(a ^ b)) := log10(select(a)) * b, 138log(select(a * b), x) := log(select(a), x) + log(b,x), 139log(select(prod(a,b,c,d)),x) := sum(log(select(a),x),b,c,d), 140log(select(a / b), x) := log(select(a), x) - log(b,x), 141log(select(a ^ b), x) := log(select(a), x) * b, 142log(a, select(b)) := ln(a) / select(ln(b)), 143sin(select(a + b)) := sin(select(a)) cos(b) + cos(a) sin(b), 144sin(select(2 a)) := 2 sin(select(a)) cos(a), 145sin(select(n a)) := 2sin((n-1) select(a)) cos(a) - sin((n-2) a) 146 :: integer(n) :: n > 2, 147cos(select(a + b)) := cos(select(a)) cos(b) - sin(a) sin(b), 148cos(select(2 a)) := 2 cos(select(a))^2 - 1, 149cos(select(n a)) := 2cos((n-1) select(a)) cos(a) - cos((n-2) a) 150 :: integer(n) :: n > 2, 151tan(select(a + b)) := (tan(select(a)) + tan(b)) / 152 (1 - tan(a) tan(b)), 153tan(select(2 a)) := 2 tan(select(a)) / (1 - tan(a)^2), 154tan(select(n a)) := (tan((n-1) select(a)) + tan(a)) / 155 (1 - tan((n-1) a) tan(a)) 156 :: integer(n) :: n > 2, 157cot(select(a + b)) := (cot(select(a)) cot(b) - 1) / 158 (cot(a) + cot(b)), 159sinh(select(a + b)) := sinh(select(a)) cosh(b) + cosh(a) sinh(b), 160cosh(select(a + b)) := cosh(select(a)) cosh(b) + sinh(a) sinh(b), 161tanh(select(a + b)) := (tanh(select(a)) + tanh(b)) / 162 (1 + tanh(a) tanh(b)), 163coth(select(a + b)) := (coth(select(a)) coth(b) + 1) / 164 (coth(a) + coth(b)), 165x && select(a || b) := (x && select(a)) || (x && b), 166select(a || b) && x := (select(a) && x) || (b && x), 167! select(a && b) := (!a) || (!b), 168! select(a || b) := (!a) && (!b) ]")) 169 170(defun calc-MergeRules () 171 "MergeRules" 172 (calc-compile-rule-set 173 "MergeRules" "[ 174iterations(1), 175 (x*opt(a)) + select(x*b) := x * (a + select(b)), 176 (x*opt(a)) - select(x*b) := x * (a - select(b)), 177sum(select(x)*a,b,c,d) := x * sum(select(a),b,c,d), 178 (a/x) + select(b/x) := (a + select(b)) / x, 179 (a/x) - select(b/x) := (a - select(b)) / x, 180sum(a/select(x),b,c,d) := sum(select(a),b,c,d) / x, 181 (a/opt(b)) + select(c/d) := ((select(a)*d) + (b*c)) / (b*d), 182 (a/opt(b)) - select(c/d) := ((select(a)*d) - (b*c)) / (b*d), 183 (x^opt(a)) * select(x^b) := x ^ (a + select(b)), 184 (x^opt(a)) / select(x^b) := x ^ (a - select(b)), 185select(x^a) / (x^opt(b)) := x ^ (select(a) - b), 186prod(select(x)^a,b,c,d) := x ^ sum(select(a),b,c,d), 187select(x^a) / (x^opt(b)) := x ^ (select(a) - b), 188 (a^x) * select(b^x) := select((a * b) ^x), 189 (a^x) / select(b^x) := select((b / b) ^ x), 190select(a^x) / (b^x) := select((a / b) ^ x), 191prod(a^select(x),b,c,d) := select(prod(a,b,c,d) ^ x), 192 (a^x) * select(b^y) := select((a * b^(y-x)) ^x), 193 (a^x) / select(b^y) := select((b / b^(y-x)) ^ x), 194select(a^x) / (b^y) := select((a / b^(y-x)) ^ x), 195select(x^a) ^ b := x ^ select(a * b), 196 (x^a) ^ select(b) := x ^ select(a * b), 197select(sqrt(a)) ^ b := select(a ^ (b / 2)), 198sqrt(a) ^ select(b) := select(a ^ (b / 2)), 199sqrt(select(a) ^ b) := select(a ^ (b / 2)), 200sqrt(a ^ select(b)) := select(a ^ (b / 2)), 201sqrt(a) * select(sqrt(b)) := select(sqrt(a * b)), 202sqrt(a) / select(sqrt(b)) := select(sqrt(a / b)), 203select(sqrt(a)) / sqrt(b) := select(sqrt(a / b)), 204prod(select(sqrt(a)),b,c,d) := select(sqrt(prod(a,b,c,d))), 205exp(a) * select(exp(b)) := select(exp(a + b)), 206exp(a) / select(exp(b)) := select(exp(a - b)), 207select(exp(a)) / exp(b) := select(exp(a - b)), 208prod(select(exp(a)),b,c,d) := select(exp(sum(a,b,c,d))), 209select(exp(a)) ^ b := select(exp(a * b)), 210exp(a) ^ select(b) := select(exp(a * b)), 211ln(a) + select(ln(b)) := select(ln(a * b)), 212ln(a) - select(ln(b)) := select(ln(a / b)), 213select(ln(a)) - ln(b) := select(ln(a / b)), 214sum(select(ln(a)),b,c,d) := select(ln(prod(a,b,c,d))), 215b * select(ln(a)) := select(ln(a ^ b)), 216select(b) * ln(a) := select(ln(a ^ b)), 217select(ln(a)) / ln(b) := select(log(a, b)), 218ln(a) / select(ln(b)) := select(log(a, b)), 219select(ln(a)) / b := select(ln(a ^ (1/b))), 220ln(a) / select(b) := select(ln(a ^ (1/b))), 221log10(a) + select(log10(b)) := select(log10(a * b)), 222log10(a) - select(log10(b)) := select(log10(a / b)), 223select(log10(a)) - log10(b) := select(log10(a / b)), 224sum(select(log10(a)),b,c,d) := select(log10(prod(a,b,c,d))), 225b * select(log10(a)) := select(log10(a ^ b)), 226select(b) * log10(a) := select(log10(a ^ b)), 227select(log10(a)) / log10(b) := select(log(a, b)), 228log10(a) / select(log10(b)) := select(log(a, b)), 229select(log10(a)) / b := select(log10(a ^ (1/b))), 230log10(a) / select(b) := select(log10(a ^ (1/b))), 231log(a,x) + select(log(b,x)) := select(log(a * b,x)), 232log(a,x) - select(log(b,x)) := select(log(a / b,x)), 233select(log(a,x)) - log(b,x) := select(log(a / b,x)), 234sum(select(log(a,x)),b,c,d) := select(log(prod(a,b,c,d),x)), 235b * select(log(a,x)) := select(log(a ^ b,x)), 236select(b) * log(a,x) := select(log(a ^ b,x)), 237select(log(a,x)) / log(b,x) := select(log(a, b)), 238log(a,x) / select(log(b,x)) := select(log(a, b)), 239select(log(a,x)) / b := select(log(a ^ (1/b),x)), 240log(a,x) / select(b) := select(log(a ^ (1/b),x)), 241select(x && a) || (x && opt(b)) := x && (select(a) || b) ]")) 242 243(defun calc-NegateRules () 244 "NegateRules" 245 (calc-compile-rule-set 246 "NegateRules" "[ 247iterations(1), 248a + select(x) := a - select(-x), 249a - select(x) := a + select(-x), 250sum(select(x),b,c,d) := -sum(select(-x),b,c,d), 251a * select(x) := -a * select(-x), 252a / select(x) := -a / select(-x), 253select(x) / a := -select(-x) / a, 254prod(select(x),b,c,d) := (-1)^(d-c+1) * prod(select(-x),b,c,d), 255select(x) ^ n := select(-x) ^ a :: integer(n) :: n%2 = 0, 256select(x) ^ n := -(select(-x) ^ a) :: integer(n) :: n%2 = 1, 257select(x) ^ a := (-select(-x)) ^ a, 258a ^ select(x) := (1 / a)^select(-x), 259abs(select(x)) := abs(select(-x)), 260i sqrt(select(x)) := -sqrt(select(-x)), 261sqrt(select(x)) := i sqrt(select(-x)), 262re(select(x)) := -re(select(-x)), 263im(select(x)) := -im(select(-x)), 264conj(select(x)) := -conj(select(-x)), 265trunc(select(x)) := -trunc(select(-x)), 266round(select(x)) := -round(select(-x)), 267floor(select(x)) := -ceil(select(-x)), 268ceil(select(x)) := -floor(select(-x)), 269ftrunc(select(x)) := -ftrunc(select(-x)), 270fround(select(x)) := -fround(select(-x)), 271ffloor(select(x)) := -fceil(select(-x)), 272fceil(select(x)) := -ffloor(select(-x)), 273exp(select(x)) := 1 / exp(select(-x)), 274sin(select(x)) := -sin(select(-x)), 275cos(select(x)) := cos(select(-x)), 276tan(select(x)) := -tan(select(-x)), 277sec(select(x)) := sec(select(-x)), 278csc(select(x)) := -csc(select(-x)), 279cot(select(x)) := -cot(select(-x)), 280arcsin(select(x)) := -arcsin(select(-x)), 281arccos(select(x)) := 4 arctan(1) - arccos(select(-x)), 282arctan(select(x)) := -arctan(select(-x)), 283sinh(select(x)) := -sinh(select(-x)), 284cosh(select(x)) := cosh(select(-x)), 285tanh(select(x)) := -tanh(select(-x)), 286sech(select(x)) := sech(select(-x)), 287csch(select(x)) := -csch(select(-x)), 288coth(select(x)) := -coth(select(-x)), 289arcsinh(select(x)) := -arcsinh(select(-x)), 290arctanh(select(x)) := -arctanh(select(-x)), 291select(x) = a := select(-x) = -a, 292select(x) != a := select(-x) != -a, 293select(x) < a := select(-x) > -a, 294select(x) > a := select(-x) < -a, 295select(x) <= a := select(-x) >= -a, 296select(x) >= a := select(-x) <= -a, 297a < select(x) := -a > select(-x), 298a > select(x) := -a < select(-x), 299a <= select(x) := -a >= select(-x), 300a >= select(x) := -a <= select(-x), 301select(x) := -select(-x) ]")) 302 303(defun calc-InvertRules () 304 "InvertRules" 305 (calc-compile-rule-set 306 "InvertRules" "[ 307iterations(1), 308a * select(x) := a / select(1/x), 309a / select(x) := a * select(1/x), 310select(x) / a := 1 / (select(1/x) a), 311prod(select(x),b,c,d) := 1 / prod(select(1/x),b,c,d), 312abs(select(x)) := 1 / abs(select(1/x)), 313sqrt(select(x)) := 1 / sqrt(select(1/x)), 314ln(select(x)) := -ln(select(1/x)), 315log10(select(x)) := -log10(select(1/x)), 316log(select(x), a) := -log(select(1/x), a), 317log(a, select(x)) := -log(a, select(1/x)), 318arctan(select(x)) := simplify(2 arctan(1))-arctan(select(1/x)), 319select(x) = a := select(1/x) = 1/a, 320select(x) != a := select(1/x) != 1/a, 321select(x) < a := select(1/x) > 1/a, 322select(x) > a := select(1/x) < 1/a, 323select(x) <= a := select(1/x) >= 1/a, 324select(x) >= a := select(1/x) <= 1/a, 325a < select(x) := 1/a > select(1/x), 326a > select(x) := 1/a < select(1/x), 327a <= select(x) := 1/a >= select(1/x), 328a >= select(x) := 1/a <= select(1/x), 329select(x) := 1 / select(1/x) ]")) 330 331 332(defun calc-FactorRules () 333 "FactorRules" 334 (calc-compile-rule-set 335 "FactorRules" "[ 336thecoefs(x, [z, a+b, c]) := thefactors(x, [d x + d a/c, (c/d) x + (b/d)]) 337 :: z = a b/c :: let(d := pgcd(pcont(c), pcont(b))), 338thecoefs(x, [z, a, c]) := thefactors(x, [(r x + a/(2 r))^2]) 339 :: z = (a/2)^2/c :: let(r := esimplify(sqrt(c))) 340 :: !matches(r, sqrt(rr)), 341thecoefs(x, [z, 0, c]) := thefactors(x, [rc x + rz, rc x - rz]) 342 :: negative(z) 343 :: let(rz := esimplify(sqrt(-z))) :: !matches(rz, sqrt(rzz)) 344 :: let(rc := esimplify(sqrt(c))) :: !matches(rc, sqrt(rcc)), 345thecoefs(x, [z, 0, c]) := thefactors(x, [rz + rc x, rz - rc x]) 346 :: negative(c) 347 :: let(rz := esimplify(sqrt(z))) :: !matches(rz, sqrt(rzz)) 348 :: let(rc := esimplify(sqrt(-c))) :: !matches(rc, sqrt(rcc)) 349 ]")) 350;;(setq var-FactorRules 'calc-FactorRules) 351 352 353(defun calc-IntegAfterRules () 354 "IntegAfterRules" 355 (calc-compile-rule-set 356 "IntegAfterRules" "[ 357 opt(a) ln(x) + opt(b) ln(y) := 2 a esimplify(arctanh(x-1)) 358 :: a + b = 0 :: nrat(x + y) = 2 || nrat(x - y) = 2, 359 a * (b + c) := a b + a c :: constant(a) 360 ]")) 361 362;;(setq var-IntegAfterRules 'calc-IntegAfterRules) 363 364 365(defun calc-FitRules () 366 "FitRules" 367 (calc-compile-rule-set 368 "FitRules" "[ 369 370schedule(1,2,3,4), 371iterations(inf), 372 373phase(1), 374e^x := exp(x), 375x^y := exp(y ln(x)) :: !istrue(constant(y)), 376x/y := x fitinv(y), 377fitinv(x y) := fitinv(x) fitinv(y), 378exp(a) exp(b) := exp(a + b), 379a exp(b) := exp(ln(a) + b) :: !hasfitvars(a), 380fitinv(exp(a)) := exp(-a), 381ln(a b) := ln(a) + ln(b), 382ln(fitinv(a)) := -ln(a), 383log10(a b) := log10(a) + log10(b), 384log10(fitinv(a)) := -log10(a), 385log(a,b) := ln(a)/ln(b), 386ln(exp(a)) := a, 387a*(b+c) := a*b + a*c, 388(a+b)^n := x :: integer(n) :: n >= 2 389 :: let(x, expandpow(a+b,n)) 390 :: quote(matches(x,y+z)), 391 392phase(1,2), 393fitmodel(y = x) := fitmodel(0, y - x), 394fitmodel(y, x+c) := fitmodel(y-c, x) :: !hasfitparams(c), 395fitmodel(y, x c) := fitmodel(y/c, x) :: !hasfitparams(c), 396fitmodel(y, x/(c opt(d))) := fitmodel(y c, x/d) :: !hasfitparams(c), 397fitmodel(y, apply(f,[x])) := fitmodel(yy, x) 398 :: hasfitparams(x) 399 :: let(FTemp() = yy, 400 solve(apply(f,[FTemp()]) = y, 401 FTemp())), 402fitmodel(y, apply(f,[x,c])) := fitmodel(yy, x) 403 :: !hasfitparams(c) 404 :: let(FTemp() = yy, 405 solve(apply(f,[FTemp(),c]) = y, 406 FTemp())), 407fitmodel(y, apply(f,[c,x])) := fitmodel(yy, x) 408 :: !hasfitparams(c) 409 :: let(FTemp() = yy, 410 solve(apply(f,[c,FTemp()]) = y, 411 FTemp())), 412 413phase(2,3), 414fitmodel(y, x) := fitsystem(y, [], [], fitpart(1,1,x)), 415fitpart(a,b,plain(x + y)) := fitpart(a,b,x) + fitpart(a,b,y), 416fitpart(a,b,plain(x - y)) := fitpart(a,b,x) + fitpart(-a,b,y), 417fitpart(a,b,plain(-x)) := fitpart(-a,b,x), 418fitpart(a,b,x opt(c)) := fitpart(a,x b,c) :: !hasfitvars(x), 419fitpart(a,x opt(b),c) := fitpart(x a,b,c) :: !hasfitparams(x), 420fitpart(a,x y + x opt(z),c) := fitpart(a,x*(y+z),c), 421fitpart(a,b,c) := fitpart2(a,b,c), 422 423phase(3), 424fitpart2(a1,b1,x) + fitpart2(a2,b2,x) := fitpart(1, a1 b1 + a2 b2, x), 425fitpart2(a1,x,c1) + fitpart2(a2,x,c2) := fitpart2(1, x, a1 c1 + a2 c2), 426 427phase(4), 428fitinv(x) := 1 / x, 429exp(x + ln(y)) := y exp(x), 430exp(x ln(y)) := y^x, 431ln(x) + ln(y) := ln(x y), 432ln(x) - ln(y) := ln(x/y), 433x*y + x*z := x*(y+z), 434fitsystem(y, xv, pv, fitpart2(a,fitparam(b),c) + opt(d)) 435 := fitsystem(y, rcons(xv, a c), 436 rcons(pv, fitdummy(b) = fitparam(b)), d) 437 :: b = vlen(pv)+1, 438fitsystem(y, xv, pv, fitpart2(a,b,c) + opt(d)) 439 := fitsystem(y, rcons(xv, a c), 440 rcons(pv, fitdummy(vlen(pv)+1) = b), d), 441fitsystem(y, xv, pv, 0) := fitsystem(y, xv, cons(fvh,fvt)) 442 :: !hasfitparams(xv) 443 :: let(cons(fvh,fvt), 444 solve(pv, table(fitparam(j), j, 1, 445 hasfitparams(pv)))), 446fitparam(n) = x := x ]")) 447 448(provide 'calc-rules) 449 450;;; arch-tag: 0ed54a52-38f3-4ed7-9ca7-b8ecf8f2febe 451;;; calc-rules.el ends here 452