src/ZF/ex/Bin.ML
author lcp
Tue Aug 16 18:58:42 1994 +0200 (1994-08-16)
changeset 532 851df239ac8b
parent 515 abcc438e7c27
child 760 f0200e91b272
permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
     1 (*  Title: 	ZF/ex/Bin.ML
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 For Bin.thy.  Arithmetic on binary integers.
     7 *)
     8 
     9 open Bin;
    10 
    11 (*Perform induction on l, then prove the major premise using prems. *)
    12 fun bin_ind_tac a prems i = 
    13     EVERY [res_inst_tac [("x",a)] bin.induct i,
    14 	   rename_last_tac a ["1"] (i+3),
    15 	   ares_tac prems i];
    16 
    17 
    18 (** bin_rec -- by Vset recursion **)
    19 
    20 goal Bin.thy "bin_rec(Plus,a,b,h) = a";
    21 by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
    22 by (rewrite_goals_tac bin.con_defs);
    23 by (simp_tac rank_ss 1);
    24 val bin_rec_Plus = result();
    25 
    26 goal Bin.thy "bin_rec(Minus,a,b,h) = b";
    27 by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
    28 by (rewrite_goals_tac bin.con_defs);
    29 by (simp_tac rank_ss 1);
    30 val bin_rec_Minus = result();
    31 
    32 goal Bin.thy "bin_rec(w$$x,a,b,h) = h(w, x, bin_rec(w,a,b,h))";
    33 by (rtac (bin_rec_def RS def_Vrec RS trans) 1);
    34 by (rewrite_goals_tac bin.con_defs);
    35 by (simp_tac rank_ss 1);
    36 val bin_rec_Bcons = result();
    37 
    38 (*Type checking*)
    39 val prems = goal Bin.thy
    40     "[| w: bin;    \
    41 \       a: C(Plus);   b: C(Minus);       \
    42 \       !!w x r. [| w: bin;  x: bool;  r: C(w) |] ==> h(w,x,r): C(w$$x)  \
    43 \    |] ==> bin_rec(w,a,b,h) : C(w)";
    44 by (bin_ind_tac "w" prems 1);
    45 by (ALLGOALS 
    46     (asm_simp_tac (ZF_ss addsimps (prems@[bin_rec_Plus,bin_rec_Minus,
    47 					 bin_rec_Bcons]))));
    48 val bin_rec_type = result();
    49 
    50 (** Versions for use with definitions **)
    51 
    52 val [rew] = goal Bin.thy
    53     "[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Plus) = a";
    54 by (rewtac rew);
    55 by (rtac bin_rec_Plus 1);
    56 val def_bin_rec_Plus = result();
    57 
    58 val [rew] = goal Bin.thy
    59     "[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(Minus) = b";
    60 by (rewtac rew);
    61 by (rtac bin_rec_Minus 1);
    62 val def_bin_rec_Minus = result();
    63 
    64 val [rew] = goal Bin.thy
    65     "[| !!w. j(w)==bin_rec(w,a,b,h) |] ==> j(w$$x) = h(w,x,j(w))";
    66 by (rewtac rew);
    67 by (rtac bin_rec_Bcons 1);
    68 val def_bin_rec_Bcons = result();
    69 
    70 fun bin_recs def = map standard
    71 	([def] RL [def_bin_rec_Plus, def_bin_rec_Minus, def_bin_rec_Bcons]);
    72 
    73 (** Type checking **)
    74 
    75 val bin_typechecks0 = bin_rec_type :: bin.intrs;
    76 
    77 goalw Bin.thy [integ_of_bin_def]
    78     "!!w. w: bin ==> integ_of_bin(w) : integ";
    79 by (typechk_tac (bin_typechecks0@integ_typechecks@
    80 		 nat_typechecks@[bool_into_nat]));
    81 val integ_of_bin_type = result();
    82 
    83 goalw Bin.thy [bin_succ_def]
    84     "!!w. w: bin ==> bin_succ(w) : bin";
    85 by (typechk_tac (bin_typechecks0@bool_typechecks));
    86 val bin_succ_type = result();
    87 
    88 goalw Bin.thy [bin_pred_def]
    89     "!!w. w: bin ==> bin_pred(w) : bin";
    90 by (typechk_tac (bin_typechecks0@bool_typechecks));
    91 val bin_pred_type = result();
    92 
    93 goalw Bin.thy [bin_minus_def]
    94     "!!w. w: bin ==> bin_minus(w) : bin";
    95 by (typechk_tac ([bin_pred_type]@bin_typechecks0@bool_typechecks));
    96 val bin_minus_type = result();
    97 
    98 goalw Bin.thy [bin_add_def]
    99     "!!v w. [| v: bin; w: bin |] ==> bin_add(v,w) : bin";
   100 by (typechk_tac ([bin_succ_type,bin_pred_type]@bin_typechecks0@
   101 		 bool_typechecks@ZF_typechecks));
   102 val bin_add_type = result();
   103 
   104 goalw Bin.thy [bin_mult_def]
   105     "!!v w. [| v: bin; w: bin |] ==> bin_mult(v,w) : bin";
   106 by (typechk_tac ([bin_minus_type,bin_add_type]@bin_typechecks0@
   107 		 bool_typechecks));
   108 val bin_mult_type = result();
   109 
   110 val bin_typechecks = bin_typechecks0 @
   111     [integ_of_bin_type, bin_succ_type, bin_pred_type, 
   112      bin_minus_type, bin_add_type, bin_mult_type];
   113 
   114 val bin_ss = integ_ss 
   115     addsimps([bool_1I, bool_0I,
   116 	     bin_rec_Plus, bin_rec_Minus, bin_rec_Bcons] @ 
   117 	     bin_recs integ_of_bin_def @ bool_simps @ bin_typechecks);
   118 
   119 val typechecks = bin_typechecks @ integ_typechecks @ nat_typechecks @
   120                  [bool_subset_nat RS subsetD];
   121 
   122 (**** The carry/borrow functions, bin_succ and bin_pred ****)
   123 
   124 (** Lemmas **)
   125 
   126 goal Integ.thy 
   127     "!!z v. [| z $+ v = z' $+ v';  \
   128 \       z: integ; z': integ;  v: integ; v': integ;  w: integ |]   \
   129 \    ==> z $+ (v $+ w) = z' $+ (v' $+ w)";
   130 by (asm_simp_tac (integ_ss addsimps ([zadd_assoc RS sym])) 1);
   131 val zadd_assoc_cong = result();
   132 
   133 goal Integ.thy 
   134     "!!z v w. [| z: integ;  v: integ;  w: integ |]   \
   135 \    ==> z $+ (v $+ w) = v $+ (z $+ w)";
   136 by (REPEAT (ares_tac [zadd_commute RS zadd_assoc_cong] 1));
   137 val zadd_assoc_swap = result();
   138 
   139 val zadd_cong = 
   140     read_instantiate_sg (sign_of Integ.thy) [("t","op $+")] subst_context2;
   141 
   142 val zadd_kill = (refl RS zadd_cong);
   143 val zadd_assoc_swap_kill = zadd_kill RSN (4, zadd_assoc_swap RS trans);
   144 
   145 (*Pushes 'constants' of the form $#m to the right -- LOOPS if two!*)
   146 val zadd_assoc_znat = standard (znat_type RS zadd_assoc_swap);
   147 
   148 goal Integ.thy 
   149     "!!z w. [| z: integ;  w: integ |]   \
   150 \    ==> w $+ (z $+ (w $+ z)) = w $+ (w $+ (z $+ z))";
   151 by (REPEAT (ares_tac [zadd_kill, zadd_assoc_swap] 1));
   152 val zadd_swap_pairs = result();
   153 
   154 
   155 val carry_ss = bin_ss addsimps 
   156                (bin_recs bin_succ_def @ bin_recs bin_pred_def);
   157 
   158 goal Bin.thy
   159     "!!w. w: bin ==> integ_of_bin(bin_succ(w)) = $#1 $+ integ_of_bin(w)";
   160 by (etac bin.induct 1);
   161 by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
   162 by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
   163 by (etac boolE 1);
   164 by (ALLGOALS (asm_simp_tac (carry_ss addsimps [zadd_assoc])));
   165 by (REPEAT (ares_tac (zadd_swap_pairs::typechecks) 1));
   166 val integ_of_bin_succ = result();
   167 
   168 goal Bin.thy
   169     "!!w. w: bin ==> integ_of_bin(bin_pred(w)) = $~ ($#1) $+ integ_of_bin(w)";
   170 by (etac bin.induct 1);
   171 by (simp_tac (carry_ss addsimps [zadd_0_right]) 1);
   172 by (simp_tac (carry_ss addsimps [zadd_zminus_inverse]) 1);
   173 by (etac boolE 1);
   174 by (ALLGOALS 
   175     (asm_simp_tac 
   176      (carry_ss addsimps [zadd_assoc RS sym,
   177 			zadd_zminus_inverse, zadd_zminus_inverse2])));
   178 by (REPEAT (ares_tac ([zadd_commute, zadd_cong, refl]@typechecks) 1));
   179 val integ_of_bin_pred = result();
   180 
   181 (*These two results replace the definitions of bin_succ and bin_pred*)
   182 
   183 
   184 (*** bin_minus: (unary!) negation of binary integers ***)
   185 
   186 val bin_minus_ss =
   187     bin_ss addsimps (bin_recs bin_minus_def @
   188 		    [integ_of_bin_succ, integ_of_bin_pred]);
   189 
   190 goal Bin.thy
   191     "!!w. w: bin ==> integ_of_bin(bin_minus(w)) = $~ integ_of_bin(w)";
   192 by (etac bin.induct 1);
   193 by (simp_tac (bin_minus_ss addsimps [zminus_0]) 1);
   194 by (simp_tac (bin_minus_ss addsimps [zadd_0_right]) 1);
   195 by (etac boolE 1);
   196 by (ALLGOALS 
   197     (asm_simp_tac (bin_minus_ss addsimps [zminus_zadd_distrib, zadd_assoc])));
   198 val integ_of_bin_minus = result();
   199 
   200 
   201 (*** bin_add: binary addition ***)
   202 
   203 goalw Bin.thy [bin_add_def] "!!w. w: bin ==> bin_add(Plus,w) = w";
   204 by (asm_simp_tac bin_ss 1);
   205 val bin_add_Plus = result();
   206 
   207 goalw Bin.thy [bin_add_def] "!!w. w: bin ==> bin_add(Minus,w) = bin_pred(w)";
   208 by (asm_simp_tac bin_ss 1);
   209 val bin_add_Minus = result();
   210 
   211 goalw Bin.thy [bin_add_def] "bin_add(v$$x,Plus) = v$$x";
   212 by (simp_tac bin_ss 1);
   213 val bin_add_Bcons_Plus = result();
   214 
   215 goalw Bin.thy [bin_add_def] "bin_add(v$$x,Minus) = bin_pred(v$$x)";
   216 by (simp_tac bin_ss 1);
   217 val bin_add_Bcons_Minus = result();
   218 
   219 goalw Bin.thy [bin_add_def]
   220     "!!w y. [| w: bin;  y: bool |] ==> \
   221 \           bin_add(v$$x, w$$y) = \
   222 \           bin_add(v, cond(x and y, bin_succ(w), w)) $$ (x xor y)";
   223 by (asm_simp_tac bin_ss 1);
   224 val bin_add_Bcons_Bcons = result();
   225 
   226 val bin_add_rews = [bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus,
   227 		    bin_add_Bcons_Minus, bin_add_Bcons_Bcons,
   228 		    integ_of_bin_succ, integ_of_bin_pred];
   229 
   230 val bin_add_ss = bin_ss addsimps ([bool_subset_nat RS subsetD] @ bin_add_rews);
   231 
   232 goal Bin.thy
   233     "!!v. v: bin ==> \
   234 \         ALL w: bin. integ_of_bin(bin_add(v,w)) = \
   235 \                     integ_of_bin(v) $+ integ_of_bin(w)";
   236 by (etac bin.induct 1);
   237 by (simp_tac bin_add_ss 1);
   238 by (simp_tac bin_add_ss 1);
   239 by (rtac ballI 1);
   240 by (bin_ind_tac "wa" [] 1);
   241 by (asm_simp_tac (bin_add_ss addsimps [zadd_0_right]) 1);
   242 by (asm_simp_tac bin_add_ss 1);
   243 by (REPEAT (ares_tac (zadd_commute::typechecks) 1));
   244 by (etac boolE 1);
   245 by (asm_simp_tac (bin_add_ss addsimps [zadd_assoc, zadd_swap_pairs]) 2);
   246 by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill]@typechecks) 2));
   247 by (etac boolE 1);
   248 by (ALLGOALS (asm_simp_tac (bin_add_ss addsimps [zadd_assoc,zadd_swap_pairs])));
   249 by (REPEAT (ares_tac ([refl, zadd_kill, zadd_assoc_swap_kill RS sym]@
   250 		      typechecks) 1));
   251 val integ_of_bin_add_lemma = result();
   252 
   253 val integ_of_bin_add = integ_of_bin_add_lemma RS bspec;
   254 
   255 
   256 (*** bin_add: binary multiplication ***)
   257 
   258 val bin_mult_ss =
   259     bin_ss addsimps (bin_recs bin_mult_def @ 
   260 		       [integ_of_bin_minus, integ_of_bin_add]);
   261 
   262 
   263 val major::prems = goal Bin.thy
   264     "[| v: bin; w: bin |] ==>	\
   265 \    integ_of_bin(bin_mult(v,w)) = \
   266 \    integ_of_bin(v) $* integ_of_bin(w)";
   267 by (cut_facts_tac prems 1);
   268 by (bin_ind_tac "v" [major] 1);
   269 by (asm_simp_tac (bin_mult_ss addsimps [zmult_0]) 1);
   270 by (asm_simp_tac (bin_mult_ss addsimps [zmult_1,zmult_zminus]) 1);
   271 by (etac boolE 1);
   272 by (asm_simp_tac (bin_mult_ss addsimps [zadd_zmult_distrib]) 2);
   273 by (asm_simp_tac 
   274     (bin_mult_ss addsimps [zadd_zmult_distrib, zmult_1, zadd_assoc]) 1);
   275 by (REPEAT (ares_tac ([zadd_commute, zadd_assoc_swap_kill RS sym]@
   276 		      typechecks) 1));
   277 val integ_of_bin_mult = result();
   278 
   279 (**** Computations ****)
   280 
   281 (** extra rules for bin_succ, bin_pred **)
   282 
   283 val [bin_succ_Plus, bin_succ_Minus, _] = bin_recs bin_succ_def;
   284 val [bin_pred_Plus, bin_pred_Minus, _] = bin_recs bin_pred_def;
   285 
   286 goal Bin.thy "bin_succ(w$$1) = bin_succ(w) $$ 0";
   287 by (simp_tac carry_ss 1);
   288 val bin_succ_Bcons1 = result();
   289 
   290 goal Bin.thy "bin_succ(w$$0) = w$$1";
   291 by (simp_tac carry_ss 1);
   292 val bin_succ_Bcons0 = result();
   293 
   294 goal Bin.thy "bin_pred(w$$1) = w$$0";
   295 by (simp_tac carry_ss 1);
   296 val bin_pred_Bcons1 = result();
   297 
   298 goal Bin.thy "bin_pred(w$$0) = bin_pred(w) $$ 1";
   299 by (simp_tac carry_ss 1);
   300 val bin_pred_Bcons0 = result();
   301 
   302 (** extra rules for bin_minus **)
   303 
   304 val [bin_minus_Plus, bin_minus_Minus, _] = bin_recs bin_minus_def;
   305 
   306 goal Bin.thy "bin_minus(w$$1) = bin_pred(bin_minus(w) $$ 0)";
   307 by (simp_tac bin_minus_ss 1);
   308 val bin_minus_Bcons1 = result();
   309 
   310 goal Bin.thy "bin_minus(w$$0) = bin_minus(w) $$ 0";
   311 by (simp_tac bin_minus_ss 1);
   312 val bin_minus_Bcons0 = result();
   313 
   314 (** extra rules for bin_add **)
   315 
   316 goal Bin.thy 
   317     "!!w. w: bin ==> bin_add(v$$1, w$$1) = bin_add(v, bin_succ(w)) $$ 0";
   318 by (asm_simp_tac bin_add_ss 1);
   319 val bin_add_Bcons_Bcons11 = result();
   320 
   321 goal Bin.thy 
   322     "!!w. w: bin ==> bin_add(v$$1, w$$0) = bin_add(v,w) $$ 1";
   323 by (asm_simp_tac bin_add_ss 1);
   324 val bin_add_Bcons_Bcons10 = result();
   325 
   326 goal Bin.thy 
   327     "!!w y.[| w: bin;  y: bool |] ==> bin_add(v$$0, w$$y) = bin_add(v,w) $$ y";
   328 by (asm_simp_tac bin_add_ss 1);
   329 val bin_add_Bcons_Bcons0 = result();
   330 
   331 (** extra rules for bin_mult **)
   332 
   333 val [bin_mult_Plus, bin_mult_Minus, _] = bin_recs bin_mult_def;
   334 
   335 goal Bin.thy "bin_mult(v$$1, w) = bin_add(bin_mult(v,w)$$0, w)";
   336 by (simp_tac bin_mult_ss 1);
   337 val bin_mult_Bcons1 = result();
   338 
   339 goal Bin.thy "bin_mult(v$$0, w) = bin_mult(v,w)$$0";
   340 by (simp_tac bin_mult_ss 1);
   341 val bin_mult_Bcons0 = result();
   342 
   343 
   344 (*** The computation simpset ***)
   345 
   346 val bin_comp_ss = integ_ss 
   347     addsimps [bin_succ_Plus, bin_succ_Minus,
   348 	     bin_succ_Bcons1, bin_succ_Bcons0,
   349 	     bin_pred_Plus, bin_pred_Minus,
   350 	     bin_pred_Bcons1, bin_pred_Bcons0,
   351 	     bin_minus_Plus, bin_minus_Minus,
   352 	     bin_minus_Bcons1, bin_minus_Bcons0,
   353 	     bin_add_Plus, bin_add_Minus, bin_add_Bcons_Plus, 
   354 	     bin_add_Bcons_Minus, bin_add_Bcons_Bcons0, 
   355 	     bin_add_Bcons_Bcons10, bin_add_Bcons_Bcons11,
   356 	     bin_mult_Plus, bin_mult_Minus,
   357 	     bin_mult_Bcons1, bin_mult_Bcons0]
   358     setsolver (type_auto_tac ([bool_1I, bool_0I] @ bin_typechecks0));
   359 
   360 (*** Examples of performing binary arithmetic by simplification ***)
   361 
   362 proof_timing := true;
   363 (*All runtimes below are on a SPARCserver 10*)
   364 
   365 (* 13+19 = 32 *)
   366 goal Bin.thy
   367     "bin_add(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = Plus$$1$$0$$0$$0$$0$$0";
   368 by (simp_tac bin_comp_ss 1);	(*0.6 secs*)
   369 result();
   370 
   371 bin_add(binary_of_int 13, binary_of_int 19);
   372 
   373 (* 1234+5678 = 6912 *)
   374 goal Bin.thy
   375     "bin_add(Plus$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0, \
   376 \	     Plus$$1$$0$$1$$1$$0$$0$$0$$1$$0$$1$$1$$1$$0) = \
   377 \    Plus$$1$$1$$0$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0";
   378 by (simp_tac bin_comp_ss 1);	(*2.6 secs*)
   379 result();
   380 
   381 bin_add(binary_of_int 1234, binary_of_int 5678);
   382 
   383 (* 1359-2468 = ~1109 *)
   384 goal Bin.thy
   385     "bin_add(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1,		\
   386 \	     Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = 	\
   387 \    Minus$$1$$0$$1$$1$$1$$0$$1$$0$$1$$0$$1$$1";
   388 by (simp_tac bin_comp_ss 1);	(*2.3 secs*)
   389 result();
   390 
   391 bin_add(binary_of_int 1359, binary_of_int ~2468);
   392 
   393 (* 93746-46375 = 47371 *)
   394 goal Bin.thy
   395     "bin_add(Plus$$1$$0$$1$$1$$0$$1$$1$$1$$0$$0$$0$$1$$1$$0$$0$$1$$0, \
   396 \	     Minus$$0$$1$$0$$0$$1$$0$$1$$0$$1$$1$$0$$1$$1$$0$$0$$1) = \
   397 \    Plus$$0$$1$$0$$1$$1$$1$$0$$0$$1$$0$$0$$0$$0$$1$$0$$1$$1";
   398 by (simp_tac bin_comp_ss 1);	(*3.9 secs*)
   399 result();
   400 
   401 bin_add(binary_of_int 93746, binary_of_int ~46375);
   402 
   403 (* negation of 65745 *)
   404 goal Bin.thy
   405     "bin_minus(Plus$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1$$1$$0$$1$$0$$0$$0$$1) = \
   406 \    Minus$$0$$1$$1$$1$$1$$1$$1$$1$$1$$0$$0$$1$$0$$1$$1$$1$$1";
   407 by (simp_tac bin_comp_ss 1);	(*0.6 secs*)
   408 result();
   409 
   410 bin_minus(binary_of_int 65745);
   411 
   412 (* negation of ~54321 *)
   413 goal Bin.thy
   414     "bin_minus(Minus$$0$$0$$1$$0$$1$$0$$1$$1$$1$$1$$0$$0$$1$$1$$1$$1) = \
   415 \    Plus$$0$$1$$1$$0$$1$$0$$1$$0$$0$$0$$0$$1$$1$$0$$0$$0$$1";
   416 by (simp_tac bin_comp_ss 1);	(*0.7 secs*)
   417 result();
   418 
   419 bin_minus(binary_of_int ~54321);
   420 
   421 (* 13*19 = 247 *)
   422 goal Bin.thy "bin_mult(Plus$$1$$1$$0$$1, Plus$$1$$0$$0$$1$$1) = \
   423 \               Plus$$1$$1$$1$$1$$0$$1$$1$$1";
   424 by (simp_tac bin_comp_ss 1);	(*1.5 secs*)
   425 result();
   426 
   427 bin_mult(binary_of_int 13, binary_of_int 19);
   428 
   429 (* ~84 * 51 = ~4284 *)
   430 goal Bin.thy
   431     "bin_mult(Minus$$0$$1$$0$$1$$1$$0$$0, Plus$$1$$1$$0$$0$$1$$1) = \
   432 \    Minus$$0$$1$$1$$1$$1$$0$$1$$0$$0$$0$$1$$0$$0";
   433 by (simp_tac bin_comp_ss 1);	(*2.6 secs*)
   434 result();
   435 
   436 bin_mult(binary_of_int ~84, binary_of_int 51);
   437 
   438 (* 255*255 = 65025;  the worst case for 8-bit operands *)
   439 goal Bin.thy
   440     "bin_mult(Plus$$1$$1$$1$$1$$1$$1$$1$$1, \
   441 \             Plus$$1$$1$$1$$1$$1$$1$$1$$1) = \
   442 \        Plus$$1$$1$$1$$1$$1$$1$$1$$0$$0$$0$$0$$0$$0$$0$$0$$1";
   443 by (simp_tac bin_comp_ss 1);	(*9.8 secs*)
   444 result();
   445 
   446 bin_mult(binary_of_int 255, binary_of_int 255);
   447 
   448 (* 1359 * ~2468 = ~3354012 *)
   449 goal Bin.thy
   450     "bin_mult(Plus$$1$$0$$1$$0$$1$$0$$0$$1$$1$$1$$1, 		\
   451 \	      Minus$$0$$1$$1$$0$$0$$1$$0$$1$$1$$1$$0$$0) = 	\
   452 \    Minus$$0$$0$$1$$1$$0$$0$$1$$1$$0$$1$$0$$0$$1$$0$$0$$1$$1$$0$$0$$1$$0$$0";
   453 by (simp_tac bin_comp_ss 1);	(*13.7 secs*)
   454 result();
   455 
   456 bin_mult(binary_of_int 1359, binary_of_int ~2468);