src/HOL/List.ML
author nipkow
Tue Apr 08 10:48:42 1997 +0200 (1997-04-08)
changeset 2919 953a47dc0519
parent 2891 d8f254ad1ab9
child 3011 a3b73ba44a11
permissions -rw-r--r--
Dep. on Provers/nat_transitive
     1 (*  Title:      HOL/List
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1994 TU Muenchen
     5 
     6 List lemmas
     7 *)
     8 
     9 open List;
    10 
    11 AddIffs list.distinct;
    12 AddIffs list.inject;
    13 
    14 bind_thm("Cons_inject", (hd list.inject) RS iffD1 RS conjE);
    15 
    16 goal List.thy "!x. xs ~= x#xs";
    17 by (list.induct_tac "xs" 1);
    18 by (ALLGOALS Asm_simp_tac);
    19 qed_spec_mp "not_Cons_self";
    20 Addsimps [not_Cons_self];
    21 
    22 goal List.thy "(xs ~= []) = (? y ys. xs = y#ys)";
    23 by (list.induct_tac "xs" 1);
    24 by (Simp_tac 1);
    25 by (Asm_simp_tac 1);
    26 qed "neq_Nil_conv";
    27 
    28 
    29 (** list_case **)
    30 
    31 goal List.thy
    32  "P(list_case a f xs) = ((xs=[] --> P(a)) & \
    33 \                        (!y ys. xs=y#ys --> P(f y ys)))";
    34 by (list.induct_tac "xs" 1);
    35 by (ALLGOALS Asm_simp_tac);
    36 by (Blast_tac 1);
    37 qed "expand_list_case";
    38 
    39 val prems = goal List.thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
    40 by(list.induct_tac "xs" 1);
    41 by(REPEAT(resolve_tac prems 1));
    42 qed "list_cases";
    43 
    44 goal List.thy  "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
    45 by (list.induct_tac "xs" 1);
    46 by (Blast_tac 1);
    47 by (Blast_tac 1);
    48 bind_thm("list_eq_cases",
    49   impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp))))));
    50 
    51 
    52 (** @ - append **)
    53 
    54 goal List.thy "(xs@ys)@zs = xs@(ys@zs)";
    55 by (list.induct_tac "xs" 1);
    56 by (ALLGOALS Asm_simp_tac);
    57 qed "append_assoc";
    58 Addsimps [append_assoc];
    59 
    60 goal List.thy "xs @ [] = xs";
    61 by (list.induct_tac "xs" 1);
    62 by (ALLGOALS Asm_simp_tac);
    63 qed "append_Nil2";
    64 Addsimps [append_Nil2];
    65 
    66 goal List.thy "(xs@ys = []) = (xs=[] & ys=[])";
    67 by (list.induct_tac "xs" 1);
    68 by (ALLGOALS Asm_simp_tac);
    69 qed "append_is_Nil_conv";
    70 AddIffs [append_is_Nil_conv];
    71 
    72 goal List.thy "([] = xs@ys) = (xs=[] & ys=[])";
    73 by (list.induct_tac "xs" 1);
    74 by (ALLGOALS Asm_simp_tac);
    75 by(Blast_tac 1);
    76 qed "Nil_is_append_conv";
    77 AddIffs [Nil_is_append_conv];
    78 
    79 goal List.thy "(xs @ ys = xs @ zs) = (ys=zs)";
    80 by (list.induct_tac "xs" 1);
    81 by (ALLGOALS Asm_simp_tac);
    82 qed "same_append_eq";
    83 AddIffs [same_append_eq];
    84 
    85 goal List.thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
    86 by(list.induct_tac "xs" 1);
    87  br allI 1;
    88  by(list.induct_tac "ys" 1);
    89   by(ALLGOALS Asm_simp_tac);
    90 br allI 1;
    91 by(list.induct_tac "ys" 1);
    92  by(ALLGOALS Asm_simp_tac);
    93 qed_spec_mp "append1_eq_conv";
    94 AddIffs [append1_eq_conv];
    95 
    96 goal List.thy "xs ~= [] --> hd xs # tl xs = xs";
    97 by(list.induct_tac "xs" 1);
    98 by(ALLGOALS Asm_simp_tac);
    99 qed_spec_mp "hd_Cons_tl";
   100 Addsimps [hd_Cons_tl];
   101 
   102 goal List.thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
   103 by (list.induct_tac "xs" 1);
   104 by (ALLGOALS Asm_simp_tac);
   105 qed "hd_append";
   106 
   107 goal List.thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
   108 by(simp_tac (!simpset setloop(split_tac[expand_list_case])) 1);
   109 qed "tl_append";
   110 
   111 (** map **)
   112 
   113 goal List.thy
   114   "(!x. x : set_of_list xs --> f x = g x) --> map f xs = map g xs";
   115 by(list.induct_tac "xs" 1);
   116 by(ALLGOALS Asm_simp_tac);
   117 bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
   118 
   119 goal List.thy "map (%x.x) = (%xs.xs)";
   120 by (rtac ext 1);
   121 by (list.induct_tac "xs" 1);
   122 by (ALLGOALS Asm_simp_tac);
   123 qed "map_ident";
   124 Addsimps[map_ident];
   125 
   126 goal List.thy "map f (xs@ys) = map f xs @ map f ys";
   127 by (list.induct_tac "xs" 1);
   128 by (ALLGOALS Asm_simp_tac);
   129 qed "map_append";
   130 Addsimps[map_append];
   131 
   132 goalw List.thy [o_def] "map (f o g) xs = map f (map g xs)";
   133 by (list.induct_tac "xs" 1);
   134 by (ALLGOALS Asm_simp_tac);
   135 qed "map_compose";
   136 Addsimps[map_compose];
   137 
   138 goal List.thy "rev(map f xs) = map f (rev xs)";
   139 by (list.induct_tac "xs" 1);
   140 by (ALLGOALS Asm_simp_tac);
   141 qed "rev_map";
   142 
   143 (** rev **)
   144 
   145 goal List.thy "rev(xs@ys) = rev(ys) @ rev(xs)";
   146 by (list.induct_tac "xs" 1);
   147 by (ALLGOALS Asm_simp_tac);
   148 qed "rev_append";
   149 Addsimps[rev_append];
   150 
   151 goal List.thy "rev(rev l) = l";
   152 by (list.induct_tac "l" 1);
   153 by (ALLGOALS Asm_simp_tac);
   154 qed "rev_rev_ident";
   155 Addsimps[rev_rev_ident];
   156 
   157 
   158 (** mem **)
   159 
   160 goal List.thy "x mem (xs@ys) = (x mem xs | x mem ys)";
   161 by (list.induct_tac "xs" 1);
   162 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
   163 qed "mem_append";
   164 Addsimps[mem_append];
   165 
   166 goal List.thy "x mem [x:xs.P(x)] = (x mem xs & P(x))";
   167 by (list.induct_tac "xs" 1);
   168 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
   169 qed "mem_filter";
   170 Addsimps[mem_filter];
   171 
   172 (** set_of_list **)
   173 
   174 goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)";
   175 by (list.induct_tac "xs" 1);
   176 by (ALLGOALS Asm_simp_tac);
   177 by (Blast_tac 1);
   178 qed "set_of_list_append";
   179 Addsimps[set_of_list_append];
   180 
   181 goal thy "(x mem xs) = (x: set_of_list xs)";
   182 by (list.induct_tac "xs" 1);
   183 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
   184 by (Blast_tac 1);
   185 qed "set_of_list_mem_eq";
   186 
   187 goal List.thy "set_of_list l <= set_of_list (x#l)";
   188 by (Simp_tac 1);
   189 by (Blast_tac 1);
   190 qed "set_of_list_subset_Cons";
   191 
   192 goal List.thy "(set_of_list xs = {}) = (xs = [])";
   193 by(list.induct_tac "xs" 1);
   194 by(ALLGOALS Asm_simp_tac);
   195 qed "set_of_list_empty";
   196 Addsimps [set_of_list_empty];
   197 
   198 goal List.thy "set_of_list(rev xs) = set_of_list(xs)";
   199 by(list.induct_tac "xs" 1);
   200 by(ALLGOALS Asm_simp_tac);
   201 by(Blast_tac 1);
   202 qed "set_of_list_rev";
   203 Addsimps [set_of_list_rev];
   204 
   205 goal List.thy "set_of_list(map f xs) = f``(set_of_list xs)";
   206 by(list.induct_tac "xs" 1);
   207 by(ALLGOALS Asm_simp_tac);
   208 qed "set_of_list_map";
   209 Addsimps [set_of_list_map];
   210 
   211 
   212 (** list_all **)
   213 
   214 goal List.thy "list_all (%x.True) xs = True";
   215 by (list.induct_tac "xs" 1);
   216 by (ALLGOALS Asm_simp_tac);
   217 qed "list_all_True";
   218 Addsimps [list_all_True];
   219 
   220 goal List.thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
   221 by (list.induct_tac "xs" 1);
   222 by (ALLGOALS Asm_simp_tac);
   223 qed "list_all_append";
   224 Addsimps [list_all_append];
   225 
   226 goal List.thy "list_all P xs = (!x. x mem xs --> P(x))";
   227 by (list.induct_tac "xs" 1);
   228 by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
   229 by (Blast_tac 1);
   230 qed "list_all_mem_conv";
   231 
   232 
   233 (** filter **)
   234 
   235 goal List.thy "[x:xs@ys . P] = [x:xs . P] @ [y:ys . P]";
   236 by(list.induct_tac "xs" 1);
   237  by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
   238 qed "filter_append";
   239 Addsimps [filter_append];
   240 
   241 
   242 (** concat **)
   243 
   244 goal List.thy  "concat(xs@ys) = concat(xs)@concat(ys)";
   245 by (list.induct_tac "xs" 1);
   246 by (ALLGOALS Asm_simp_tac);
   247 qed"concat_append";
   248 Addsimps [concat_append];
   249 
   250 goal List.thy "rev(concat ls) = concat (map rev (rev ls))";
   251 by (list.induct_tac "ls" 1);
   252 by (ALLGOALS Asm_simp_tac);
   253 qed "rev_concat";
   254 
   255 (** length **)
   256 
   257 goal List.thy "length(xs@ys) = length(xs)+length(ys)";
   258 by (list.induct_tac "xs" 1);
   259 by (ALLGOALS Asm_simp_tac);
   260 qed"length_append";
   261 Addsimps [length_append];
   262 
   263 goal List.thy "length (map f l) = length l";
   264 by (list.induct_tac "l" 1);
   265 by (ALLGOALS Simp_tac);
   266 qed "length_map";
   267 Addsimps [length_map];
   268 
   269 goal List.thy "length(rev xs) = length(xs)";
   270 by (list.induct_tac "xs" 1);
   271 by (ALLGOALS Asm_simp_tac);
   272 qed "length_rev";
   273 Addsimps [length_rev];
   274 
   275 goal List.thy "(length xs = 0) = (xs = [])";
   276 by(list.induct_tac "xs" 1);
   277 by(ALLGOALS Asm_simp_tac);
   278 qed "length_0_conv";
   279 AddIffs [length_0_conv];
   280 
   281 goal List.thy "(0 < length xs) = (xs ~= [])";
   282 by(list.induct_tac "xs" 1);
   283 by(ALLGOALS Asm_simp_tac);
   284 qed "length_greater_0_conv";
   285 AddIffs [length_greater_0_conv];
   286 
   287 
   288 (** nth **)
   289 
   290 goal List.thy
   291   "!xs. nth n (xs@ys) = \
   292 \          (if n < length xs then nth n xs else nth (n - length xs) ys)";
   293 by(nat_ind_tac "n" 1);
   294  by(Asm_simp_tac 1);
   295  br allI 1;
   296  by(res_inst_tac [("xs","xs")]list_cases 1);
   297   by(ALLGOALS Asm_simp_tac);
   298 br allI 1;
   299 by(res_inst_tac [("xs","xs")]list_cases 1);
   300  by(ALLGOALS Asm_simp_tac);
   301 qed_spec_mp "nth_append";
   302 
   303 goal List.thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)";
   304 by (list.induct_tac "xs" 1);
   305 (* case [] *)
   306 by (Asm_full_simp_tac 1);
   307 (* case x#xl *)
   308 by (rtac allI 1);
   309 by (nat_ind_tac "n" 1);
   310 by (ALLGOALS Asm_full_simp_tac);
   311 qed_spec_mp "nth_map";
   312 Addsimps [nth_map];
   313 
   314 goal List.thy "!n. n < length xs --> list_all P xs --> P(nth n xs)";
   315 by (list.induct_tac "xs" 1);
   316 (* case [] *)
   317 by (Simp_tac 1);
   318 (* case x#xl *)
   319 by (rtac allI 1);
   320 by (nat_ind_tac "n" 1);
   321 by (ALLGOALS Asm_full_simp_tac);
   322 qed_spec_mp "list_all_nth";
   323 
   324 goal List.thy "!n. n < length xs --> (nth n xs) mem xs";
   325 by (list.induct_tac "xs" 1);
   326 (* case [] *)
   327 by (Simp_tac 1);
   328 (* case x#xl *)
   329 by (rtac allI 1);
   330 by (nat_ind_tac "n" 1);
   331 (* case 0 *)
   332 by (Asm_full_simp_tac 1);
   333 (* case Suc x *)
   334 by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
   335 qed_spec_mp "nth_mem";
   336 Addsimps [nth_mem];
   337 
   338 
   339 (** take  & drop **)
   340 section "take & drop";
   341 
   342 goal thy "take 0 xs = []";
   343 by (list.induct_tac "xs" 1);
   344 by (ALLGOALS Asm_simp_tac);
   345 qed "take_0";
   346 
   347 goal thy "drop 0 xs = xs";
   348 by (list.induct_tac "xs" 1);
   349 by (ALLGOALS Asm_simp_tac);
   350 qed "drop_0";
   351 
   352 goal thy "take (Suc n) (x#xs) = x # take n xs";
   353 by (Simp_tac 1);
   354 qed "take_Suc_Cons";
   355 
   356 goal thy "drop (Suc n) (x#xs) = drop n xs";
   357 by (Simp_tac 1);
   358 qed "drop_Suc_Cons";
   359 
   360 Delsimps [take_Cons,drop_Cons];
   361 Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
   362 
   363 goal List.thy "!xs. length(take n xs) = min (length xs) n";
   364 by(nat_ind_tac "n" 1);
   365  by(ALLGOALS Asm_simp_tac);
   366 br allI 1;
   367 by(res_inst_tac [("xs","xs")]list_cases 1);
   368  by(ALLGOALS Asm_simp_tac);
   369 qed_spec_mp "length_take";
   370 Addsimps [length_take];
   371 
   372 goal List.thy "!xs. length(drop n xs) = (length xs - n)";
   373 by(nat_ind_tac "n" 1);
   374  by(ALLGOALS Asm_simp_tac);
   375 br allI 1;
   376 by(res_inst_tac [("xs","xs")]list_cases 1);
   377  by(ALLGOALS Asm_simp_tac);
   378 qed_spec_mp "length_drop";
   379 Addsimps [length_drop];
   380 
   381 goal List.thy "!xs. length xs <= n --> take n xs = xs";
   382 by(nat_ind_tac "n" 1);
   383  by(ALLGOALS Asm_simp_tac);
   384 br allI 1;
   385 by(res_inst_tac [("xs","xs")]list_cases 1);
   386  by(ALLGOALS Asm_simp_tac);
   387 qed_spec_mp "take_all";
   388 
   389 goal List.thy "!xs. length xs <= n --> drop n xs = []";
   390 by(nat_ind_tac "n" 1);
   391  by(ALLGOALS Asm_simp_tac);
   392 br allI 1;
   393 by(res_inst_tac [("xs","xs")]list_cases 1);
   394  by(ALLGOALS Asm_simp_tac);
   395 qed_spec_mp "drop_all";
   396 
   397 goal List.thy 
   398   "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
   399 by(nat_ind_tac "n" 1);
   400  by(ALLGOALS Asm_simp_tac);
   401 br allI 1;
   402 by(res_inst_tac [("xs","xs")]list_cases 1);
   403  by(ALLGOALS Asm_simp_tac);
   404 qed_spec_mp "take_append";
   405 Addsimps [take_append];
   406 
   407 goal List.thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
   408 by(nat_ind_tac "n" 1);
   409  by(ALLGOALS Asm_simp_tac);
   410 br allI 1;
   411 by(res_inst_tac [("xs","xs")]list_cases 1);
   412  by(ALLGOALS Asm_simp_tac);
   413 qed_spec_mp "drop_append";
   414 Addsimps [drop_append];
   415 
   416 goal List.thy "!xs n. take n (take m xs) = take (min n m) xs"; 
   417 by(nat_ind_tac "m" 1);
   418  by(ALLGOALS Asm_simp_tac);
   419 br allI 1;
   420 by(res_inst_tac [("xs","xs")]list_cases 1);
   421  by(ALLGOALS Asm_simp_tac);
   422 br allI 1;
   423 by(res_inst_tac [("n","n")]natE 1);
   424  by(ALLGOALS Asm_simp_tac);
   425 qed_spec_mp "take_take";
   426 
   427 goal List.thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
   428 by(nat_ind_tac "m" 1);
   429  by(ALLGOALS Asm_simp_tac);
   430 br allI 1;
   431 by(res_inst_tac [("xs","xs")]list_cases 1);
   432  by(ALLGOALS Asm_simp_tac);
   433 qed_spec_mp "drop_drop";
   434 
   435 goal List.thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
   436 by(nat_ind_tac "m" 1);
   437  by(ALLGOALS Asm_simp_tac);
   438 br allI 1;
   439 by(res_inst_tac [("xs","xs")]list_cases 1);
   440  by(ALLGOALS Asm_simp_tac);
   441 qed_spec_mp "take_drop";
   442 
   443 goal List.thy "!xs. take n (map f xs) = map f (take n xs)"; 
   444 by(nat_ind_tac "n" 1);
   445 by(ALLGOALS Asm_simp_tac);
   446 br allI 1;
   447 by(res_inst_tac [("xs","xs")]list_cases 1);
   448 by(ALLGOALS Asm_simp_tac);
   449 qed_spec_mp "take_map"; 
   450 
   451 goal List.thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
   452 by(nat_ind_tac "n" 1);
   453 by(ALLGOALS Asm_simp_tac);
   454 br allI 1;
   455 by(res_inst_tac [("xs","xs")]list_cases 1);
   456 by(ALLGOALS Asm_simp_tac);
   457 qed_spec_mp "drop_map";
   458 
   459 goal List.thy
   460   "!n i. i < n --> nth i (take n xs) = nth i xs";
   461 by(list.induct_tac "xs" 1);
   462  by(ALLGOALS Asm_simp_tac);
   463 by(strip_tac 1);
   464 by(res_inst_tac [("n","n")] natE 1);
   465  by(Blast_tac 1);
   466 by(res_inst_tac [("n","i")] natE 1);
   467 by(ALLGOALS (hyp_subst_tac THEN' Asm_full_simp_tac));
   468 qed_spec_mp "nth_take";
   469 Addsimps [nth_take];
   470 
   471 goal List.thy
   472   "!xs i. n + i < length xs --> nth i (drop n xs) = nth (n + i) xs";
   473 by(nat_ind_tac "n" 1);
   474  by(ALLGOALS Asm_simp_tac);
   475 br allI 1;
   476 by(res_inst_tac [("xs","xs")]list_cases 1);
   477  by(ALLGOALS Asm_simp_tac);
   478 qed_spec_mp "nth_drop";
   479 Addsimps [nth_drop];
   480 
   481 (** takeWhile & dropWhile **)
   482 
   483 goal List.thy
   484   "x:set_of_list xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
   485 by(list.induct_tac "xs" 1);
   486  by(Simp_tac 1);
   487 by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
   488 by(Blast_tac 1);
   489 bind_thm("takeWhile_append1", conjI RS (result() RS mp));
   490 Addsimps [takeWhile_append1];
   491 
   492 goal List.thy
   493   "(!x:set_of_list xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
   494 by(list.induct_tac "xs" 1);
   495  by(Simp_tac 1);
   496 by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
   497 bind_thm("takeWhile_append2", ballI RS (result() RS mp));
   498 Addsimps [takeWhile_append2];
   499 
   500 goal List.thy
   501   "x:set_of_list xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
   502 by(list.induct_tac "xs" 1);
   503  by(Simp_tac 1);
   504 by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
   505 by(Blast_tac 1);
   506 bind_thm("dropWhile_append1", conjI RS (result() RS mp));
   507 Addsimps [dropWhile_append1];
   508 
   509 goal List.thy
   510   "(!x:set_of_list xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
   511 by(list.induct_tac "xs" 1);
   512  by(Simp_tac 1);
   513 by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
   514 bind_thm("dropWhile_append2", ballI RS (result() RS mp));
   515 Addsimps [dropWhile_append2];
   516 
   517 goal List.thy "x:set_of_list(takeWhile P xs) --> x:set_of_list xs & P x";
   518 by(list.induct_tac "xs" 1);
   519  by(Simp_tac 1);
   520 by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1);
   521 qed_spec_mp"set_of_list_take_whileD";
   522