src/HOL/List.thy
 author krauss Fri Nov 24 13:44:51 2006 +0100 (2006-11-24) changeset 21512 3786eb1b69d6 parent 21455 b6be1d1b66c5 child 21548 7c6216661e8a permissions -rw-r--r--
Lemma "fundef_default_value" uses predicate instead of set.
     1 (*  Title:      HOL/List.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow

     4 *)

     5

     6 header {* The datatype of finite lists *}

     7

     8 theory List

     9 imports PreList

    10 begin

    11

    12 datatype 'a list =

    13     Nil    ("[]")

    14   | Cons 'a  "'a list"    (infixr "#" 65)

    15

    16 subsection{*Basic list processing functions*}

    17

    18 consts

    19   "@" :: "'a list => 'a list => 'a list"    (infixr 65)

    20   filter:: "('a => bool) => 'a list => 'a list"

    21   concat:: "'a list list => 'a list"

    22   foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"

    23   foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"

    24   hd:: "'a list => 'a"

    25   tl:: "'a list => 'a list"

    26   last:: "'a list => 'a"

    27   butlast :: "'a list => 'a list"

    28   set :: "'a list => 'a set"

    29   map :: "('a=>'b) => ('a list => 'b list)"

    30   nth :: "'a list => nat => 'a"    (infixl "!" 100)

    31   list_update :: "'a list => nat => 'a => 'a list"

    32   take:: "nat => 'a list => 'a list"

    33   drop:: "nat => 'a list => 'a list"

    34   takeWhile :: "('a => bool) => 'a list => 'a list"

    35   dropWhile :: "('a => bool) => 'a list => 'a list"

    36   rev :: "'a list => 'a list"

    37   zip :: "'a list => 'b list => ('a * 'b) list"

    38   upt :: "nat => nat => nat list" ("(1[_..</_'])")

    39   remdups :: "'a list => 'a list"

    40   remove1 :: "'a => 'a list => 'a list"

    41   "distinct":: "'a list => bool"

    42   replicate :: "nat => 'a => 'a list"

    43   splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

    44

    45 abbreviation

    46   upto:: "nat => nat => nat list"  ("(1[_../_])") where

    47   "[i..j] == [i..<(Suc j)]"

    48

    49

    50 nonterminals lupdbinds lupdbind

    51

    52 syntax

    53   -- {* list Enumeration *}

    54   "@list" :: "args => 'a list"    ("[(_)]")

    55

    56   -- {* Special syntax for filter *}

    57   "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")

    58

    59   -- {* list update *}

    60   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")

    61   "" :: "lupdbind => lupdbinds"    ("_")

    62   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")

    63   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)

    64

    65 translations

    66   "[x, xs]" == "x#[xs]"

    67   "[x]" == "x#[]"

    68   "[x:xs . P]"== "filter (%x. P) xs"

    69

    70   "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"

    71   "xs[i:=x]" == "list_update xs i x"

    72

    73

    74 syntax (xsymbols)

    75   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")

    76 syntax (HTML output)

    77   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")

    78

    79

    80 text {*

    81   Function @{text size} is overloaded for all datatypes. Users may

    82   refer to the list version as @{text length}. *}

    83

    84 abbreviation

    85   length :: "'a list => nat" where

    86   "length == size"

    87

    88 primrec

    89   "hd(x#xs) = x"

    90

    91 primrec

    92   "tl([]) = []"

    93   "tl(x#xs) = xs"

    94

    95 primrec

    96   "last(x#xs) = (if xs=[] then x else last xs)"

    97

    98 primrec

    99   "butlast []= []"

   100   "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"

   101

   102 primrec

   103   "set [] = {}"

   104   "set (x#xs) = insert x (set xs)"

   105

   106 primrec

   107   "map f [] = []"

   108   "map f (x#xs) = f(x)#map f xs"

   109

   110 primrec

   111   append_Nil:"[]@ys = ys"

   112   append_Cons: "(x#xs)@ys = x#(xs@ys)"

   113

   114 primrec

   115   "rev([]) = []"

   116   "rev(x#xs) = rev(xs) @ [x]"

   117

   118 primrec

   119   "filter P [] = []"

   120   "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"

   121

   122 primrec

   123   foldl_Nil:"foldl f a [] = a"

   124   foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"

   125

   126 primrec

   127   "foldr f [] a = a"

   128   "foldr f (x#xs) a = f x (foldr f xs a)"

   129

   130 primrec

   131   "concat([]) = []"

   132   "concat(x#xs) = x @ concat(xs)"

   133

   134 primrec

   135   drop_Nil:"drop n [] = []"

   136   drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"

   137   -- {*Warning: simpset does not contain this definition, but separate

   138        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

   139

   140 primrec

   141   take_Nil:"take n [] = []"

   142   take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"

   143   -- {*Warning: simpset does not contain this definition, but separate

   144        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

   145

   146 primrec

   147   nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"

   148   -- {*Warning: simpset does not contain this definition, but separate

   149        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}

   150

   151 primrec

   152   "[][i:=v] = []"

   153   "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"

   154

   155 primrec

   156   "takeWhile P [] = []"

   157   "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"

   158

   159 primrec

   160   "dropWhile P [] = []"

   161   "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"

   162

   163 primrec

   164   "zip xs [] = []"

   165   zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"

   166   -- {*Warning: simpset does not contain this definition, but separate

   167        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}

   168

   169 primrec

   170   upt_0: "[i..<0] = []"

   171   upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"

   172

   173 primrec

   174   "distinct [] = True"

   175   "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"

   176

   177 primrec

   178   "remdups [] = []"

   179   "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"

   180

   181 primrec

   182   "remove1 x [] = []"

   183   "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"

   184

   185 primrec

   186   replicate_0: "replicate 0 x = []"

   187   replicate_Suc: "replicate (Suc n) x = x # replicate n x"

   188

   189 definition

   190   rotate1 :: "'a list \<Rightarrow> 'a list" where

   191   "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"

   192

   193 definition

   194   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where

   195   "rotate n = rotate1 ^ n"

   196

   197 definition

   198   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where

   199   "list_all2 P xs ys =

   200     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"

   201

   202 definition

   203   sublist :: "'a list => nat set => 'a list" where

   204   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"

   205

   206 primrec

   207   "splice [] ys = ys"

   208   "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"

   209     -- {*Warning: simpset does not contain the second eqn but a derived one. *}

   210

   211

   212 subsubsection {* @{const Nil} and @{const Cons} *}

   213

   214 lemma not_Cons_self [simp]:

   215   "xs \<noteq> x # xs"

   216 by (induct xs) auto

   217

   218 lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]

   219

   220 lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"

   221 by (induct xs) auto

   222

   223 lemma length_induct:

   224   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"

   225 by (rule measure_induct [of length]) iprover

   226

   227

   228 subsubsection {* @{const length} *}

   229

   230 text {*

   231   Needs to come before @{text "@"} because of theorem @{text

   232   append_eq_append_conv}.

   233 *}

   234

   235 lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"

   236 by (induct xs) auto

   237

   238 lemma length_map [simp]: "length (map f xs) = length xs"

   239 by (induct xs) auto

   240

   241 lemma length_rev [simp]: "length (rev xs) = length xs"

   242 by (induct xs) auto

   243

   244 lemma length_tl [simp]: "length (tl xs) = length xs - 1"

   245 by (cases xs) auto

   246

   247 lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"

   248 by (induct xs) auto

   249

   250 lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"

   251 by (induct xs) auto

   252

   253 lemma length_Suc_conv:

   254 "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"

   255 by (induct xs) auto

   256

   257 lemma Suc_length_conv:

   258 "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"

   259 apply (induct xs, simp, simp)

   260 apply blast

   261 done

   262

   263 lemma impossible_Cons [rule_format]:

   264   "length xs <= length ys --> xs = x # ys = False"

   265 apply (induct xs)

   266 apply auto

   267 done

   268

   269 lemma list_induct2[consumes 1]: "\<And>ys.

   270  \<lbrakk> length xs = length ys;

   271    P [] [];

   272    \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>

   273  \<Longrightarrow> P xs ys"

   274 apply(induct xs)

   275  apply simp

   276 apply(case_tac ys)

   277  apply simp

   278 apply(simp)

   279 done

   280

   281 subsubsection {* @{text "@"} -- append *}

   282

   283 lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"

   284 by (induct xs) auto

   285

   286 lemma append_Nil2 [simp]: "xs @ [] = xs"

   287 by (induct xs) auto

   288

   289 lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"

   290 by (induct xs) auto

   291

   292 lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"

   293 by (induct xs) auto

   294

   295 lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"

   296 by (induct xs) auto

   297

   298 lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"

   299 by (induct xs) auto

   300

   301 lemma append_eq_append_conv [simp]:

   302  "!!ys. length xs = length ys \<or> length us = length vs

   303  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"

   304 apply (induct xs)

   305  apply (case_tac ys, simp, force)

   306 apply (case_tac ys, force, simp)

   307 done

   308

   309 lemma append_eq_append_conv2: "!!ys zs ts.

   310  (xs @ ys = zs @ ts) =

   311  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"

   312 apply (induct xs)

   313  apply fastsimp

   314 apply(case_tac zs)

   315  apply simp

   316 apply fastsimp

   317 done

   318

   319 lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"

   320 by simp

   321

   322 lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"

   323 by simp

   324

   325 lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"

   326 by simp

   327

   328 lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"

   329 using append_same_eq [of _ _ "[]"] by auto

   330

   331 lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"

   332 using append_same_eq [of "[]"] by auto

   333

   334 lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"

   335 by (induct xs) auto

   336

   337 lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"

   338 by (induct xs) auto

   339

   340 lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"

   341 by (simp add: hd_append split: list.split)

   342

   343 lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"

   344 by (simp split: list.split)

   345

   346 lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"

   347 by (simp add: tl_append split: list.split)

   348

   349

   350 lemma Cons_eq_append_conv: "x#xs = ys@zs =

   351  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"

   352 by(cases ys) auto

   353

   354 lemma append_eq_Cons_conv: "(ys@zs = x#xs) =

   355  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"

   356 by(cases ys) auto

   357

   358

   359 text {* Trivial rules for solving @{text "@"}-equations automatically. *}

   360

   361 lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"

   362 by simp

   363

   364 lemma Cons_eq_appendI:

   365 "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"

   366 by (drule sym) simp

   367

   368 lemma append_eq_appendI:

   369 "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"

   370 by (drule sym) simp

   371

   372

   373 text {*

   374 Simplification procedure for all list equalities.

   375 Currently only tries to rearrange @{text "@"} to see if

   376 - both lists end in a singleton list,

   377 - or both lists end in the same list.

   378 *}

   379

   380 ML_setup {*

   381 local

   382

   383 val append_assoc = thm "append_assoc";

   384 val append_Nil = thm "append_Nil";

   385 val append_Cons = thm "append_Cons";

   386 val append1_eq_conv = thm "append1_eq_conv";

   387 val append_same_eq = thm "append_same_eq";

   388

   389 fun last (cons as Const("List.list.Cons",_) $_$ xs) =

   390   (case xs of Const("List.list.Nil",_) => cons | _ => last xs)

   391   | last (Const("List.op @",_) $_$ ys) = last ys

   392   | last t = t;

   393

   394 fun list1 (Const("List.list.Cons",_) $_$ Const("List.list.Nil",_)) = true

   395   | list1 _ = false;

   396

   397 fun butlast ((cons as Const("List.list.Cons",_) $x)$ xs) =

   398   (case xs of Const("List.list.Nil",_) => xs | _ => cons $butlast xs)   399 | butlast ((app as Const("List.op @",_)$ xs) $ys) = app$ butlast ys

   400   | butlast xs = Const("List.list.Nil",fastype_of xs);

   401

   402 val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];

   403

   404 fun list_eq ss (F as (eq as Const(_,eqT)) $lhs$ rhs) =

   405   let

   406     val lastl = last lhs and lastr = last rhs;

   407     fun rearr conv =

   408       let

   409         val lhs1 = butlast lhs and rhs1 = butlast rhs;

   410         val Type(_,listT::_) = eqT

   411         val appT = [listT,listT] ---> listT

   412         val app = Const("List.op @",appT)

   413         val F2 = eq $(app$lhs1$lastl)$ (app$rhs1$lastr)

   414         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));

   415         val thm = Goal.prove (Simplifier.the_context ss) [] [] eq

   416           (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));

   417       in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;

   418

   419   in

   420     if list1 lastl andalso list1 lastr then rearr append1_eq_conv

   421     else if lastl aconv lastr then rearr append_same_eq

   422     else NONE

   423   end;

   424

   425 in

   426

   427 val list_eq_simproc =

   428   Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq);

   429

   430 end;

   431

   432 Addsimprocs [list_eq_simproc];

   433 *}

   434

   435

   436 subsubsection {* @{text map} *}

   437

   438 lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"

   439 by (induct xs) simp_all

   440

   441 lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"

   442 by (rule ext, induct_tac xs) auto

   443

   444 lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"

   445 by (induct xs) auto

   446

   447 lemma map_compose: "map (f o g) xs = map f (map g xs)"

   448 by (induct xs) (auto simp add: o_def)

   449

   450 lemma rev_map: "rev (map f xs) = map f (rev xs)"

   451 by (induct xs) auto

   452

   453 lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"

   454 by (induct xs) auto

   455

   456 lemma map_cong [fundef_cong, recdef_cong]:

   457 "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"

   458 -- {* a congruence rule for @{text map} *}

   459 by simp

   460

   461 lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"

   462 by (cases xs) auto

   463

   464 lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"

   465 by (cases xs) auto

   466

   467 lemma map_eq_Cons_conv:

   468  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"

   469 by (cases xs) auto

   470

   471 lemma Cons_eq_map_conv:

   472  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"

   473 by (cases ys) auto

   474

   475 lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]

   476 lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]

   477 declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]

   478

   479 lemma ex_map_conv:

   480   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"

   481 by(induct ys, auto simp add: Cons_eq_map_conv)

   482

   483 lemma map_eq_imp_length_eq:

   484   "!!xs. map f xs = map f ys ==> length xs = length ys"

   485 apply (induct ys)

   486  apply simp

   487 apply(simp (no_asm_use))

   488 apply clarify

   489 apply(simp (no_asm_use))

   490 apply fast

   491 done

   492

   493 lemma map_inj_on:

   494  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]

   495   ==> xs = ys"

   496 apply(frule map_eq_imp_length_eq)

   497 apply(rotate_tac -1)

   498 apply(induct rule:list_induct2)

   499  apply simp

   500 apply(simp)

   501 apply (blast intro:sym)

   502 done

   503

   504 lemma inj_on_map_eq_map:

   505  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"

   506 by(blast dest:map_inj_on)

   507

   508 lemma map_injective:

   509  "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"

   510 by (induct ys) (auto dest!:injD)

   511

   512 lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"

   513 by(blast dest:map_injective)

   514

   515 lemma inj_mapI: "inj f ==> inj (map f)"

   516 by (iprover dest: map_injective injD intro: inj_onI)

   517

   518 lemma inj_mapD: "inj (map f) ==> inj f"

   519 apply (unfold inj_on_def, clarify)

   520 apply (erule_tac x = "[x]" in ballE)

   521  apply (erule_tac x = "[y]" in ballE, simp, blast)

   522 apply blast

   523 done

   524

   525 lemma inj_map[iff]: "inj (map f) = inj f"

   526 by (blast dest: inj_mapD intro: inj_mapI)

   527

   528 lemma inj_on_mapI: "inj_on f (\<Union>(set  A)) \<Longrightarrow> inj_on (map f) A"

   529 apply(rule inj_onI)

   530 apply(erule map_inj_on)

   531 apply(blast intro:inj_onI dest:inj_onD)

   532 done

   533

   534 lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"

   535 by (induct xs, auto)

   536

   537 lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"

   538 by (induct xs) auto

   539

   540 lemma map_fst_zip[simp]:

   541   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"

   542 by (induct rule:list_induct2, simp_all)

   543

   544 lemma map_snd_zip[simp]:

   545   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"

   546 by (induct rule:list_induct2, simp_all)

   547

   548

   549 subsubsection {* @{text rev} *}

   550

   551 lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"

   552 by (induct xs) auto

   553

   554 lemma rev_rev_ident [simp]: "rev (rev xs) = xs"

   555 by (induct xs) auto

   556

   557 lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"

   558 by auto

   559

   560 lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"

   561 by (induct xs) auto

   562

   563 lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"

   564 by (induct xs) auto

   565

   566 lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"

   567 by (cases xs) auto

   568

   569 lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"

   570 by (cases xs) auto

   571

   572 lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"

   573 apply (induct xs arbitrary: ys, force)

   574 apply (case_tac ys, simp, force)

   575 done

   576

   577 lemma inj_on_rev[iff]: "inj_on rev A"

   578 by(simp add:inj_on_def)

   579

   580 lemma rev_induct [case_names Nil snoc]:

   581   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"

   582 apply(simplesubst rev_rev_ident[symmetric])

   583 apply(rule_tac list = "rev xs" in list.induct, simp_all)

   584 done

   585

   586 ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"

   587

   588 lemma rev_exhaust [case_names Nil snoc]:

   589   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"

   590 by (induct xs rule: rev_induct) auto

   591

   592 lemmas rev_cases = rev_exhaust

   593

   594 lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"

   595 by(rule rev_cases[of xs]) auto

   596

   597

   598 subsubsection {* @{text set} *}

   599

   600 lemma finite_set [iff]: "finite (set xs)"

   601 by (induct xs) auto

   602

   603 lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"

   604 by (induct xs) auto

   605

   606 lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"

   607 by(cases xs) auto

   608

   609 lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"

   610 by auto

   611

   612 lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs"

   613 by auto

   614

   615 lemma set_empty [iff]: "(set xs = {}) = (xs = [])"

   616 by (induct xs) auto

   617

   618 lemma set_empty2[iff]: "({} = set xs) = (xs = [])"

   619 by(induct xs) auto

   620

   621 lemma set_rev [simp]: "set (rev xs) = set xs"

   622 by (induct xs) auto

   623

   624 lemma set_map [simp]: "set (map f xs) = f(set xs)"

   625 by (induct xs) auto

   626

   627 lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"

   628 by (induct xs) auto

   629

   630 lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"

   631 apply (induct j, simp_all)

   632 apply (erule ssubst, auto)

   633 done

   634

   635 lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"

   636 proof (induct xs)

   637   case Nil show ?case by simp

   638   case (Cons a xs)

   639   show ?case

   640   proof

   641     assume "x \<in> set (a # xs)"

   642     with prems show "\<exists>ys zs. a # xs = ys @ x # zs"

   643       by (simp, blast intro: Cons_eq_appendI)

   644   next

   645     assume "\<exists>ys zs. a # xs = ys @ x # zs"

   646     then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast

   647     show "x \<in> set (a # xs)"

   648       by (cases ys, auto simp add: eq)

   649   qed

   650 qed

   651

   652 lemma in_set_conv_decomp_first:

   653  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"

   654 proof (induct xs)

   655   case Nil show ?case by simp

   656 next

   657   case (Cons a xs)

   658   show ?case

   659   proof cases

   660     assume "x = a" thus ?case using Cons by force

   661   next

   662     assume "x \<noteq> a"

   663     show ?case

   664     proof

   665       assume "x \<in> set (a # xs)"

   666       from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"

   667 	by(fastsimp intro!: Cons_eq_appendI)

   668     next

   669       assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"

   670       then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast

   671       show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)

   672     qed

   673   qed

   674 qed

   675

   676 lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]

   677 lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]

   678

   679

   680 lemma finite_list: "finite A ==> EX l. set l = A"

   681 apply (erule finite_induct, auto)

   682 apply (rule_tac x="x#l" in exI, auto)

   683 done

   684

   685 lemma card_length: "card (set xs) \<le> length xs"

   686 by (induct xs) (auto simp add: card_insert_if)

   687

   688

   689 subsubsection {* @{text filter} *}

   690

   691 lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"

   692 by (induct xs) auto

   693

   694 lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"

   695 by (induct xs) simp_all

   696

   697 lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"

   698 by (induct xs) auto

   699

   700 lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"

   701 by (induct xs) (auto simp add: le_SucI)

   702

   703 lemma sum_length_filter_compl:

   704   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"

   705 by(induct xs) simp_all

   706

   707 lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"

   708 by (induct xs) auto

   709

   710 lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"

   711 by (induct xs) auto

   712

   713 lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"

   714   by (induct xs) simp_all

   715

   716 lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"

   717 apply (induct xs)

   718  apply auto

   719 apply(cut_tac P=P and xs=xs in length_filter_le)

   720 apply simp

   721 done

   722

   723 lemma filter_map:

   724   "filter P (map f xs) = map f (filter (P o f) xs)"

   725 by (induct xs) simp_all

   726

   727 lemma length_filter_map[simp]:

   728   "length (filter P (map f xs)) = length(filter (P o f) xs)"

   729 by (simp add:filter_map)

   730

   731 lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"

   732 by auto

   733

   734 lemma length_filter_less:

   735   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"

   736 proof (induct xs)

   737   case Nil thus ?case by simp

   738 next

   739   case (Cons x xs) thus ?case

   740     apply (auto split:split_if_asm)

   741     using length_filter_le[of P xs] apply arith

   742   done

   743 qed

   744

   745 lemma length_filter_conv_card:

   746  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"

   747 proof (induct xs)

   748   case Nil thus ?case by simp

   749 next

   750   case (Cons x xs)

   751   let ?S = "{i. i < length xs & p(xs!i)}"

   752   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)

   753   show ?case (is "?l = card ?S'")

   754   proof (cases)

   755     assume "p x"

   756     hence eq: "?S' = insert 0 (Suc  ?S)"

   757       by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)

   758     have "length (filter p (x # xs)) = Suc(card ?S)"

   759       using Cons by simp

   760     also have "\<dots> = Suc(card(Suc  ?S))" using fin

   761       by (simp add: card_image inj_Suc)

   762     also have "\<dots> = card ?S'" using eq fin

   763       by (simp add:card_insert_if) (simp add:image_def)

   764     finally show ?thesis .

   765   next

   766     assume "\<not> p x"

   767     hence eq: "?S' = Suc  ?S"

   768       by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)

   769     have "length (filter p (x # xs)) = card ?S"

   770       using Cons by simp

   771     also have "\<dots> = card(Suc  ?S)" using fin

   772       by (simp add: card_image inj_Suc)

   773     also have "\<dots> = card ?S'" using eq fin

   774       by (simp add:card_insert_if)

   775     finally show ?thesis .

   776   qed

   777 qed

   778

   779 lemma Cons_eq_filterD:

   780  "x#xs = filter P ys \<Longrightarrow>

   781   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"

   782   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")

   783 proof(induct ys)

   784   case Nil thus ?case by simp

   785 next

   786   case (Cons y ys)

   787   show ?case (is "\<exists>x. ?Q x")

   788   proof cases

   789     assume Py: "P y"

   790     show ?thesis

   791     proof cases

   792       assume xy: "x = y"

   793       show ?thesis

   794       proof from Py xy Cons(2) show "?Q []" by simp qed

   795     next

   796       assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp

   797     qed

   798   next

   799     assume Py: "\<not> P y"

   800     with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp

   801     show ?thesis (is "? us. ?Q us")

   802     proof show "?Q (y#us)" using 1 by simp qed

   803   qed

   804 qed

   805

   806 lemma filter_eq_ConsD:

   807  "filter P ys = x#xs \<Longrightarrow>

   808   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"

   809 by(rule Cons_eq_filterD) simp

   810

   811 lemma filter_eq_Cons_iff:

   812  "(filter P ys = x#xs) =

   813   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"

   814 by(auto dest:filter_eq_ConsD)

   815

   816 lemma Cons_eq_filter_iff:

   817  "(x#xs = filter P ys) =

   818   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"

   819 by(auto dest:Cons_eq_filterD)

   820

   821 lemma filter_cong[fundef_cong, recdef_cong]:

   822  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"

   823 apply simp

   824 apply(erule thin_rl)

   825 by (induct ys) simp_all

   826

   827

   828 subsubsection {* @{text concat} *}

   829

   830 lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"

   831 by (induct xs) auto

   832

   833 lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"

   834 by (induct xss) auto

   835

   836 lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"

   837 by (induct xss) auto

   838

   839 lemma set_concat [simp]: "set (concat xs) = \<Union>(set  set xs)"

   840 by (induct xs) auto

   841

   842 lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"

   843 by (induct xs) auto

   844

   845 lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"

   846 by (induct xs) auto

   847

   848 lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"

   849 by (induct xs) auto

   850

   851

   852 subsubsection {* @{text nth} *}

   853

   854 lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"

   855 by auto

   856

   857 lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"

   858 by auto

   859

   860 declare nth.simps [simp del]

   861

   862 lemma nth_append:

   863 "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"

   864 apply (induct "xs", simp)

   865 apply (case_tac n, auto)

   866 done

   867

   868 lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"

   869 by (induct "xs") auto

   870

   871 lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"

   872 by (induct "xs") auto

   873

   874 lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"

   875 apply (induct xs, simp)

   876 apply (case_tac n, auto)

   877 done

   878

   879 lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"

   880 by(cases xs) simp_all

   881

   882

   883 lemma list_eq_iff_nth_eq:

   884  "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"

   885 apply(induct xs)

   886  apply simp apply blast

   887 apply(case_tac ys)

   888  apply simp

   889 apply(simp add:nth_Cons split:nat.split)apply blast

   890 done

   891

   892 lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"

   893 apply (induct xs, simp, simp)

   894 apply safe

   895 apply (rule_tac x = 0 in exI, simp)

   896  apply (rule_tac x = "Suc i" in exI, simp)

   897 apply (case_tac i, simp)

   898 apply (rename_tac j)

   899 apply (rule_tac x = j in exI, simp)

   900 done

   901

   902 lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"

   903 by(auto simp:set_conv_nth)

   904

   905 lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"

   906 by (auto simp add: set_conv_nth)

   907

   908 lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"

   909 by (auto simp add: set_conv_nth)

   910

   911 lemma all_nth_imp_all_set:

   912 "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"

   913 by (auto simp add: set_conv_nth)

   914

   915 lemma all_set_conv_all_nth:

   916 "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"

   917 by (auto simp add: set_conv_nth)

   918

   919

   920 subsubsection {* @{text list_update} *}

   921

   922 lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"

   923 by (induct xs) (auto split: nat.split)

   924

   925 lemma nth_list_update:

   926 "!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"

   927 by (induct xs) (auto simp add: nth_Cons split: nat.split)

   928

   929 lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"

   930 by (simp add: nth_list_update)

   931

   932 lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"

   933 by (induct xs) (auto simp add: nth_Cons split: nat.split)

   934

   935 lemma list_update_overwrite [simp]:

   936 "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"

   937 by (induct xs) (auto split: nat.split)

   938

   939 lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"

   940 apply (induct xs, simp)

   941 apply(simp split:nat.splits)

   942 done

   943

   944 lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"

   945 apply (induct xs)

   946  apply simp

   947 apply (case_tac i)

   948 apply simp_all

   949 done

   950

   951 lemma list_update_same_conv:

   952 "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"

   953 by (induct xs) (auto split: nat.split)

   954

   955 lemma list_update_append1:

   956  "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"

   957 apply (induct xs, simp)

   958 apply(simp split:nat.split)

   959 done

   960

   961 lemma list_update_append:

   962   "!!n. (xs @ ys) [n:= x] =

   963   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"

   964 by (induct xs) (auto split:nat.splits)

   965

   966 lemma list_update_length [simp]:

   967  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"

   968 by (induct xs, auto)

   969

   970 lemma update_zip:

   971 "!!i xy xs. length xs = length ys ==>

   972 (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"

   973 by (induct ys) (auto, case_tac xs, auto split: nat.split)

   974

   975 lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"

   976 by (induct xs) (auto split: nat.split)

   977

   978 lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"

   979 by (blast dest!: set_update_subset_insert [THEN subsetD])

   980

   981 lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"

   982 by (induct xs) (auto split:nat.splits)

   983

   984

   985 subsubsection {* @{text last} and @{text butlast} *}

   986

   987 lemma last_snoc [simp]: "last (xs @ [x]) = x"

   988 by (induct xs) auto

   989

   990 lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"

   991 by (induct xs) auto

   992

   993 lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"

   994 by(simp add:last.simps)

   995

   996 lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"

   997 by(simp add:last.simps)

   998

   999 lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"

  1000 by (induct xs) (auto)

  1001

  1002 lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"

  1003 by(simp add:last_append)

  1004

  1005 lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"

  1006 by(simp add:last_append)

  1007

  1008 lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"

  1009 by(rule rev_exhaust[of xs]) simp_all

  1010

  1011 lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"

  1012 by(cases xs) simp_all

  1013

  1014 lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"

  1015 by (induct as) auto

  1016

  1017 lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"

  1018 by (induct xs rule: rev_induct) auto

  1019

  1020 lemma butlast_append:

  1021 "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"

  1022 by (induct xs) auto

  1023

  1024 lemma append_butlast_last_id [simp]:

  1025 "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"

  1026 by (induct xs) auto

  1027

  1028 lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"

  1029 by (induct xs) (auto split: split_if_asm)

  1030

  1031 lemma in_set_butlast_appendI:

  1032 "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"

  1033 by (auto dest: in_set_butlastD simp add: butlast_append)

  1034

  1035 lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"

  1036 apply (induct xs)

  1037  apply simp

  1038 apply (auto split:nat.split)

  1039 done

  1040

  1041 lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"

  1042 by(induct xs)(auto simp:neq_Nil_conv)

  1043

  1044 subsubsection {* @{text take} and @{text drop} *}

  1045

  1046 lemma take_0 [simp]: "take 0 xs = []"

  1047 by (induct xs) auto

  1048

  1049 lemma drop_0 [simp]: "drop 0 xs = xs"

  1050 by (induct xs) auto

  1051

  1052 lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"

  1053 by simp

  1054

  1055 lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"

  1056 by simp

  1057

  1058 declare take_Cons [simp del] and drop_Cons [simp del]

  1059

  1060 lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"

  1061 by(clarsimp simp add:neq_Nil_conv)

  1062

  1063 lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"

  1064 by(cases xs, simp_all)

  1065

  1066 lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"

  1067 by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)

  1068

  1069 lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"

  1070 apply (induct xs, simp)

  1071 apply(simp add:drop_Cons nth_Cons split:nat.splits)

  1072 done

  1073

  1074 lemma take_Suc_conv_app_nth:

  1075  "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"

  1076 apply (induct xs, simp)

  1077 apply (case_tac i, auto)

  1078 done

  1079

  1080 lemma drop_Suc_conv_tl:

  1081   "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"

  1082 apply (induct xs, simp)

  1083 apply (case_tac i, auto)

  1084 done

  1085

  1086 lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"

  1087 by (induct n) (auto, case_tac xs, auto)

  1088

  1089 lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"

  1090 by (induct n) (auto, case_tac xs, auto)

  1091

  1092 lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"

  1093 by (induct n) (auto, case_tac xs, auto)

  1094

  1095 lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"

  1096 by (induct n) (auto, case_tac xs, auto)

  1097

  1098 lemma take_append [simp]:

  1099 "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"

  1100 by (induct n) (auto, case_tac xs, auto)

  1101

  1102 lemma drop_append [simp]:

  1103 "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"

  1104 by (induct n) (auto, case_tac xs, auto)

  1105

  1106 lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"

  1107 apply (induct m, auto)

  1108 apply (case_tac xs, auto)

  1109 apply (case_tac n, auto)

  1110 done

  1111

  1112 lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"

  1113 apply (induct m, auto)

  1114 apply (case_tac xs, auto)

  1115 done

  1116

  1117 lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"

  1118 apply (induct m, auto)

  1119 apply (case_tac xs, auto)

  1120 done

  1121

  1122 lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"

  1123 apply(induct xs)

  1124  apply simp

  1125 apply(simp add: take_Cons drop_Cons split:nat.split)

  1126 done

  1127

  1128 lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"

  1129 apply (induct n, auto)

  1130 apply (case_tac xs, auto)

  1131 done

  1132

  1133 lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"

  1134 apply(induct xs)

  1135  apply simp

  1136 apply(simp add:take_Cons split:nat.split)

  1137 done

  1138

  1139 lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"

  1140 apply(induct xs)

  1141 apply simp

  1142 apply(simp add:drop_Cons split:nat.split)

  1143 done

  1144

  1145 lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"

  1146 apply (induct n, auto)

  1147 apply (case_tac xs, auto)

  1148 done

  1149

  1150 lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"

  1151 apply (induct n, auto)

  1152 apply (case_tac xs, auto)

  1153 done

  1154

  1155 lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"

  1156 apply (induct xs, auto)

  1157 apply (case_tac i, auto)

  1158 done

  1159

  1160 lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"

  1161 apply (induct xs, auto)

  1162 apply (case_tac i, auto)

  1163 done

  1164

  1165 lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"

  1166 apply (induct xs, auto)

  1167 apply (case_tac n, blast)

  1168 apply (case_tac i, auto)

  1169 done

  1170

  1171 lemma nth_drop [simp]:

  1172 "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"

  1173 apply (induct n, auto)

  1174 apply (case_tac xs, auto)

  1175 done

  1176

  1177 lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"

  1178 by(simp add: hd_conv_nth)

  1179

  1180 lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"

  1181 by(induct xs)(auto simp:take_Cons split:nat.split)

  1182

  1183 lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"

  1184 by(induct xs)(auto simp:drop_Cons split:nat.split)

  1185

  1186 lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"

  1187 using set_take_subset by fast

  1188

  1189 lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"

  1190 using set_drop_subset by fast

  1191

  1192 lemma append_eq_conv_conj:

  1193 "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"

  1194 apply (induct xs, simp, clarsimp)

  1195 apply (case_tac zs, auto)

  1196 done

  1197

  1198 lemma take_add [rule_format]:

  1199     "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"

  1200 apply (induct xs, auto)

  1201 apply (case_tac i, simp_all)

  1202 done

  1203

  1204 lemma append_eq_append_conv_if:

  1205  "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =

  1206   (if size xs\<^isub>1 \<le> size ys\<^isub>1

  1207    then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2

  1208    else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"

  1209 apply(induct xs\<^isub>1)

  1210  apply simp

  1211 apply(case_tac ys\<^isub>1)

  1212 apply simp_all

  1213 done

  1214

  1215 lemma take_hd_drop:

  1216   "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"

  1217 apply(induct xs)

  1218 apply simp

  1219 apply(simp add:drop_Cons split:nat.split)

  1220 done

  1221

  1222 lemma id_take_nth_drop:

  1223  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs"

  1224 proof -

  1225   assume si: "i < length xs"

  1226   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto

  1227   moreover

  1228   from si have "take (Suc i) xs = take i xs @ [xs!i]"

  1229     apply (rule_tac take_Suc_conv_app_nth) by arith

  1230   ultimately show ?thesis by auto

  1231 qed

  1232

  1233 lemma upd_conv_take_nth_drop:

  1234  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"

  1235 proof -

  1236   assume i: "i < length xs"

  1237   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"

  1238     by(rule arg_cong[OF id_take_nth_drop[OF i]])

  1239   also have "\<dots> = take i xs @ a # drop (Suc i) xs"

  1240     using i by (simp add: list_update_append)

  1241   finally show ?thesis .

  1242 qed

  1243

  1244

  1245 subsubsection {* @{text takeWhile} and @{text dropWhile} *}

  1246

  1247 lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"

  1248 by (induct xs) auto

  1249

  1250 lemma takeWhile_append1 [simp]:

  1251 "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"

  1252 by (induct xs) auto

  1253

  1254 lemma takeWhile_append2 [simp]:

  1255 "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"

  1256 by (induct xs) auto

  1257

  1258 lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"

  1259 by (induct xs) auto

  1260

  1261 lemma dropWhile_append1 [simp]:

  1262 "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"

  1263 by (induct xs) auto

  1264

  1265 lemma dropWhile_append2 [simp]:

  1266 "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"

  1267 by (induct xs) auto

  1268

  1269 lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"

  1270 by (induct xs) (auto split: split_if_asm)

  1271

  1272 lemma takeWhile_eq_all_conv[simp]:

  1273  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"

  1274 by(induct xs, auto)

  1275

  1276 lemma dropWhile_eq_Nil_conv[simp]:

  1277  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"

  1278 by(induct xs, auto)

  1279

  1280 lemma dropWhile_eq_Cons_conv:

  1281  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"

  1282 by(induct xs, auto)

  1283

  1284 text{* The following two lemmmas could be generalized to an arbitrary

  1285 property. *}

  1286

  1287 lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>

  1288  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"

  1289 by(induct xs) (auto simp: takeWhile_tail[where l="[]"])

  1290

  1291 lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>

  1292   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"

  1293 apply(induct xs)

  1294  apply simp

  1295 apply auto

  1296 apply(subst dropWhile_append2)

  1297 apply auto

  1298 done

  1299

  1300 lemma takeWhile_not_last:

  1301  "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"

  1302 apply(induct xs)

  1303  apply simp

  1304 apply(case_tac xs)

  1305 apply(auto)

  1306 done

  1307

  1308 lemma takeWhile_cong [fundef_cong, recdef_cong]:

  1309   "[| l = k; !!x. x : set l ==> P x = Q x |]

  1310   ==> takeWhile P l = takeWhile Q k"

  1311   by (induct k arbitrary: l) (simp_all)

  1312

  1313 lemma dropWhile_cong [fundef_cong, recdef_cong]:

  1314   "[| l = k; !!x. x : set l ==> P x = Q x |]

  1315   ==> dropWhile P l = dropWhile Q k"

  1316   by (induct k arbitrary: l, simp_all)

  1317

  1318

  1319 subsubsection {* @{text zip} *}

  1320

  1321 lemma zip_Nil [simp]: "zip [] ys = []"

  1322 by (induct ys) auto

  1323

  1324 lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"

  1325 by simp

  1326

  1327 declare zip_Cons [simp del]

  1328

  1329 lemma zip_Cons1:

  1330  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"

  1331 by(auto split:list.split)

  1332

  1333 lemma length_zip [simp]:

  1334 "!!xs. length (zip xs ys) = min (length xs) (length ys)"

  1335 apply (induct ys, simp)

  1336 apply (case_tac xs, auto)

  1337 done

  1338

  1339 lemma zip_append1:

  1340 "!!xs. zip (xs @ ys) zs =

  1341 zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"

  1342 apply (induct zs, simp)

  1343 apply (case_tac xs, simp_all)

  1344 done

  1345

  1346 lemma zip_append2:

  1347 "!!ys. zip xs (ys @ zs) =

  1348 zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"

  1349 apply (induct xs, simp)

  1350 apply (case_tac ys, simp_all)

  1351 done

  1352

  1353 lemma zip_append [simp]:

  1354  "[| length xs = length us; length ys = length vs |] ==>

  1355 zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"

  1356 by (simp add: zip_append1)

  1357

  1358 lemma zip_rev:

  1359 "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"

  1360 by (induct rule:list_induct2, simp_all)

  1361

  1362 lemma nth_zip [simp]:

  1363 "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"

  1364 apply (induct ys, simp)

  1365 apply (case_tac xs)

  1366  apply (simp_all add: nth.simps split: nat.split)

  1367 done

  1368

  1369 lemma set_zip:

  1370 "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"

  1371 by (simp add: set_conv_nth cong: rev_conj_cong)

  1372

  1373 lemma zip_update:

  1374 "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"

  1375 by (rule sym, simp add: update_zip)

  1376

  1377 lemma zip_replicate [simp]:

  1378 "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"

  1379 apply (induct i, auto)

  1380 apply (case_tac j, auto)

  1381 done

  1382

  1383 lemma take_zip:

  1384  "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"

  1385 apply (induct n)

  1386  apply simp

  1387 apply (case_tac xs, simp)

  1388 apply (case_tac ys, simp_all)

  1389 done

  1390

  1391 lemma drop_zip:

  1392  "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"

  1393 apply (induct n)

  1394  apply simp

  1395 apply (case_tac xs, simp)

  1396 apply (case_tac ys, simp_all)

  1397 done

  1398

  1399

  1400 subsubsection {* @{text list_all2} *}

  1401

  1402 lemma list_all2_lengthD [intro?]:

  1403   "list_all2 P xs ys ==> length xs = length ys"

  1404   by (simp add: list_all2_def)

  1405

  1406 lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"

  1407   by (simp add: list_all2_def)

  1408

  1409 lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"

  1410   by (simp add: list_all2_def)

  1411

  1412 lemma list_all2_Cons [iff, code]:

  1413   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"

  1414   by (auto simp add: list_all2_def)

  1415

  1416 lemma list_all2_Cons1:

  1417 "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"

  1418 by (cases ys) auto

  1419

  1420 lemma list_all2_Cons2:

  1421 "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"

  1422 by (cases xs) auto

  1423

  1424 lemma list_all2_rev [iff]:

  1425 "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"

  1426 by (simp add: list_all2_def zip_rev cong: conj_cong)

  1427

  1428 lemma list_all2_rev1:

  1429 "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"

  1430 by (subst list_all2_rev [symmetric]) simp

  1431

  1432 lemma list_all2_append1:

  1433 "list_all2 P (xs @ ys) zs =

  1434 (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>

  1435 list_all2 P xs us \<and> list_all2 P ys vs)"

  1436 apply (simp add: list_all2_def zip_append1)

  1437 apply (rule iffI)

  1438  apply (rule_tac x = "take (length xs) zs" in exI)

  1439  apply (rule_tac x = "drop (length xs) zs" in exI)

  1440  apply (force split: nat_diff_split simp add: min_def, clarify)

  1441 apply (simp add: ball_Un)

  1442 done

  1443

  1444 lemma list_all2_append2:

  1445 "list_all2 P xs (ys @ zs) =

  1446 (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>

  1447 list_all2 P us ys \<and> list_all2 P vs zs)"

  1448 apply (simp add: list_all2_def zip_append2)

  1449 apply (rule iffI)

  1450  apply (rule_tac x = "take (length ys) xs" in exI)

  1451  apply (rule_tac x = "drop (length ys) xs" in exI)

  1452  apply (force split: nat_diff_split simp add: min_def, clarify)

  1453 apply (simp add: ball_Un)

  1454 done

  1455

  1456 lemma list_all2_append:

  1457   "length xs = length ys \<Longrightarrow>

  1458   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"

  1459 by (induct rule:list_induct2, simp_all)

  1460

  1461 lemma list_all2_appendI [intro?, trans]:

  1462   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"

  1463   by (simp add: list_all2_append list_all2_lengthD)

  1464

  1465 lemma list_all2_conv_all_nth:

  1466 "list_all2 P xs ys =

  1467 (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"

  1468 by (force simp add: list_all2_def set_zip)

  1469

  1470 lemma list_all2_trans:

  1471   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"

  1472   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"

  1473         (is "!!bs cs. PROP ?Q as bs cs")

  1474 proof (induct as)

  1475   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"

  1476   show "!!cs. PROP ?Q (x # xs) bs cs"

  1477   proof (induct bs)

  1478     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"

  1479     show "PROP ?Q (x # xs) (y # ys) cs"

  1480       by (induct cs) (auto intro: tr I1 I2)

  1481   qed simp

  1482 qed simp

  1483

  1484 lemma list_all2_all_nthI [intro?]:

  1485   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"

  1486   by (simp add: list_all2_conv_all_nth)

  1487

  1488 lemma list_all2I:

  1489   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"

  1490   by (simp add: list_all2_def)

  1491

  1492 lemma list_all2_nthD:

  1493   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"

  1494   by (simp add: list_all2_conv_all_nth)

  1495

  1496 lemma list_all2_nthD2:

  1497   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"

  1498   by (frule list_all2_lengthD) (auto intro: list_all2_nthD)

  1499

  1500 lemma list_all2_map1:

  1501   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"

  1502   by (simp add: list_all2_conv_all_nth)

  1503

  1504 lemma list_all2_map2:

  1505   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"

  1506   by (auto simp add: list_all2_conv_all_nth)

  1507

  1508 lemma list_all2_refl [intro?]:

  1509   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"

  1510   by (simp add: list_all2_conv_all_nth)

  1511

  1512 lemma list_all2_update_cong:

  1513   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"

  1514   by (simp add: list_all2_conv_all_nth nth_list_update)

  1515

  1516 lemma list_all2_update_cong2:

  1517   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"

  1518   by (simp add: list_all2_lengthD list_all2_update_cong)

  1519

  1520 lemma list_all2_takeI [simp,intro?]:

  1521   "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"

  1522   apply (induct xs)

  1523    apply simp

  1524   apply (clarsimp simp add: list_all2_Cons1)

  1525   apply (case_tac n)

  1526   apply auto

  1527   done

  1528

  1529 lemma list_all2_dropI [simp,intro?]:

  1530   "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"

  1531   apply (induct as, simp)

  1532   apply (clarsimp simp add: list_all2_Cons1)

  1533   apply (case_tac n, simp, simp)

  1534   done

  1535

  1536 lemma list_all2_mono [intro?]:

  1537   "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"

  1538   apply (induct x, simp)

  1539   apply (case_tac y, auto)

  1540   done

  1541

  1542

  1543 subsubsection {* @{text foldl} and @{text foldr} *}

  1544

  1545 lemma foldl_append [simp]:

  1546 "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"

  1547 by (induct xs) auto

  1548

  1549 lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"

  1550 by (induct xs) auto

  1551

  1552 lemma foldl_cong [fundef_cong, recdef_cong]:

  1553   "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |]

  1554   ==> foldl f a l = foldl g b k"

  1555   by (induct k arbitrary: a b l) simp_all

  1556

  1557 lemma foldr_cong [fundef_cong, recdef_cong]:

  1558   "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |]

  1559   ==> foldr f l a = foldr g k b"

  1560   by (induct k arbitrary: a b l) simp_all

  1561

  1562 lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"

  1563 by (induct xs) auto

  1564

  1565 lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"

  1566 by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])

  1567

  1568 text {*

  1569 Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more

  1570 difficult to use because it requires an additional transitivity step.

  1571 *}

  1572

  1573 lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"

  1574 by (induct ns) auto

  1575

  1576 lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"

  1577 by (force intro: start_le_sum simp add: in_set_conv_decomp)

  1578

  1579 lemma sum_eq_0_conv [iff]:

  1580 "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"

  1581 by (induct ns) auto

  1582

  1583

  1584 subsubsection {* @{text upto} *}

  1585

  1586 lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"

  1587 -- {* simp does not terminate! *}

  1588 by (induct j) auto

  1589

  1590 lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"

  1591 by (subst upt_rec) simp

  1592

  1593 lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"

  1594 by(induct j)simp_all

  1595

  1596 lemma upt_eq_Cons_conv:

  1597  "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"

  1598 apply(induct j)

  1599  apply simp

  1600 apply(clarsimp simp add: append_eq_Cons_conv)

  1601 apply arith

  1602 done

  1603

  1604 lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"

  1605 -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}

  1606 by simp

  1607

  1608 lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"

  1609 apply(rule trans)

  1610 apply(subst upt_rec)

  1611  prefer 2 apply (rule refl, simp)

  1612 done

  1613

  1614 lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"

  1615 -- {* LOOPS as a simprule, since @{text "j <= j"}. *}

  1616 by (induct k) auto

  1617

  1618 lemma length_upt [simp]: "length [i..<j] = j - i"

  1619 by (induct j) (auto simp add: Suc_diff_le)

  1620

  1621 lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"

  1622 apply (induct j)

  1623 apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)

  1624 done

  1625

  1626

  1627 lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"

  1628 by(simp add:upt_conv_Cons)

  1629

  1630 lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"

  1631 apply(cases j)

  1632  apply simp

  1633 by(simp add:upt_Suc_append)

  1634

  1635 lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"

  1636 apply (induct m, simp)

  1637 apply (subst upt_rec)

  1638 apply (rule sym)

  1639 apply (subst upt_rec)

  1640 apply (simp del: upt.simps)

  1641 done

  1642

  1643 lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"

  1644 apply(induct j)

  1645 apply auto

  1646 done

  1647

  1648 lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"

  1649 by (induct n) auto

  1650

  1651 lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"

  1652 apply (induct n m rule: diff_induct)

  1653 prefer 3 apply (subst map_Suc_upt[symmetric])

  1654 apply (auto simp add: less_diff_conv nth_upt)

  1655 done

  1656

  1657 lemma nth_take_lemma:

  1658   "!!xs ys. k <= length xs ==> k <= length ys ==>

  1659      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"

  1660 apply (atomize, induct k)

  1661 apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)

  1662 txt {* Both lists must be non-empty *}

  1663 apply (case_tac xs, simp)

  1664 apply (case_tac ys, clarify)

  1665  apply (simp (no_asm_use))

  1666 apply clarify

  1667 txt {* prenexing's needed, not miniscoping *}

  1668 apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)

  1669 apply blast

  1670 done

  1671

  1672 lemma nth_equalityI:

  1673  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"

  1674 apply (frule nth_take_lemma [OF le_refl eq_imp_le])

  1675 apply (simp_all add: take_all)

  1676 done

  1677

  1678 (* needs nth_equalityI *)

  1679 lemma list_all2_antisym:

  1680   "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk>

  1681   \<Longrightarrow> xs = ys"

  1682   apply (simp add: list_all2_conv_all_nth)

  1683   apply (rule nth_equalityI, blast, simp)

  1684   done

  1685

  1686 lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"

  1687 -- {* The famous take-lemma. *}

  1688 apply (drule_tac x = "max (length xs) (length ys)" in spec)

  1689 apply (simp add: le_max_iff_disj take_all)

  1690 done

  1691

  1692

  1693 lemma take_Cons':

  1694      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"

  1695 by (cases n) simp_all

  1696

  1697 lemma drop_Cons':

  1698      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"

  1699 by (cases n) simp_all

  1700

  1701 lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"

  1702 by (cases n) simp_all

  1703

  1704 lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]

  1705 lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]

  1706 lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]

  1707

  1708 declare take_Cons_number_of [simp]

  1709         drop_Cons_number_of [simp]

  1710         nth_Cons_number_of [simp]

  1711

  1712

  1713 subsubsection {* @{text "distinct"} and @{text remdups} *}

  1714

  1715 lemma distinct_append [simp]:

  1716 "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"

  1717 by (induct xs) auto

  1718

  1719 lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"

  1720 by(induct xs) auto

  1721

  1722 lemma set_remdups [simp]: "set (remdups xs) = set xs"

  1723 by (induct xs) (auto simp add: insert_absorb)

  1724

  1725 lemma distinct_remdups [iff]: "distinct (remdups xs)"

  1726 by (induct xs) auto

  1727

  1728 lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"

  1729   by (induct x, auto)

  1730

  1731 lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"

  1732   by (induct x, auto)

  1733

  1734 lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"

  1735 by (induct xs) auto

  1736

  1737 lemma length_remdups_eq[iff]:

  1738   "(length (remdups xs) = length xs) = (remdups xs = xs)"

  1739 apply(induct xs)

  1740  apply auto

  1741 apply(subgoal_tac "length (remdups xs) <= length xs")

  1742  apply arith

  1743 apply(rule length_remdups_leq)

  1744 done

  1745

  1746

  1747 lemma distinct_map:

  1748   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"

  1749 by (induct xs) auto

  1750

  1751

  1752 lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"

  1753 by (induct xs) auto

  1754

  1755 lemma distinct_upt[simp]: "distinct[i..<j]"

  1756 by (induct j) auto

  1757

  1758 lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"

  1759 apply(induct xs)

  1760  apply simp

  1761 apply (case_tac i)

  1762  apply simp_all

  1763 apply(blast dest:in_set_takeD)

  1764 done

  1765

  1766 lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"

  1767 apply(induct xs)

  1768  apply simp

  1769 apply (case_tac i)

  1770  apply simp_all

  1771 done

  1772

  1773 lemma distinct_list_update:

  1774 assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"

  1775 shows "distinct (xs[i:=a])"

  1776 proof (cases "i < length xs")

  1777   case True

  1778   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"

  1779     apply (drule_tac id_take_nth_drop) by simp

  1780   with d True show ?thesis

  1781     apply (simp add: upd_conv_take_nth_drop)

  1782     apply (drule subst [OF id_take_nth_drop]) apply assumption

  1783     apply simp apply (cases "a = xs!i") apply simp by blast

  1784 next

  1785   case False with d show ?thesis by auto

  1786 qed

  1787

  1788

  1789 text {* It is best to avoid this indexed version of distinct, but

  1790 sometimes it is useful. *}

  1791

  1792 lemma distinct_conv_nth:

  1793 "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"

  1794 apply (induct xs, simp, simp)

  1795 apply (rule iffI, clarsimp)

  1796  apply (case_tac i)

  1797 apply (case_tac j, simp)

  1798 apply (simp add: set_conv_nth)

  1799  apply (case_tac j)

  1800 apply (clarsimp simp add: set_conv_nth, simp)

  1801 apply (rule conjI)

  1802  apply (clarsimp simp add: set_conv_nth)

  1803  apply (erule_tac x = 0 in allE, simp)

  1804  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)

  1805 apply (erule_tac x = "Suc i" in allE, simp)

  1806 apply (erule_tac x = "Suc j" in allE, simp)

  1807 done

  1808

  1809 lemma nth_eq_iff_index_eq:

  1810  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"

  1811 by(auto simp: distinct_conv_nth)

  1812

  1813 lemma distinct_card: "distinct xs ==> card (set xs) = size xs"

  1814   by (induct xs) auto

  1815

  1816 lemma card_distinct: "card (set xs) = size xs ==> distinct xs"

  1817 proof (induct xs)

  1818   case Nil thus ?case by simp

  1819 next

  1820   case (Cons x xs)

  1821   show ?case

  1822   proof (cases "x \<in> set xs")

  1823     case False with Cons show ?thesis by simp

  1824   next

  1825     case True with Cons.prems

  1826     have "card (set xs) = Suc (length xs)"

  1827       by (simp add: card_insert_if split: split_if_asm)

  1828     moreover have "card (set xs) \<le> length xs" by (rule card_length)

  1829     ultimately have False by simp

  1830     thus ?thesis ..

  1831   qed

  1832 qed

  1833

  1834

  1835 lemma length_remdups_concat:

  1836  "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"

  1837 by(simp add: distinct_card[symmetric])

  1838

  1839

  1840 subsubsection {* @{text remove1} *}

  1841

  1842 lemma remove1_append:

  1843   "remove1 x (xs @ ys) =

  1844   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"

  1845 by (induct xs) auto

  1846

  1847 lemma set_remove1_subset: "set(remove1 x xs) <= set xs"

  1848 apply(induct xs)

  1849  apply simp

  1850 apply simp

  1851 apply blast

  1852 done

  1853

  1854 lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"

  1855 apply(induct xs)

  1856  apply simp

  1857 apply simp

  1858 apply blast

  1859 done

  1860

  1861 lemma remove1_filter_not[simp]:

  1862   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"

  1863 by(induct xs) auto

  1864

  1865 lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"

  1866 apply(insert set_remove1_subset)

  1867 apply fast

  1868 done

  1869

  1870 lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"

  1871 by (induct xs) simp_all

  1872

  1873

  1874 subsubsection {* @{text replicate} *}

  1875

  1876 lemma length_replicate [simp]: "length (replicate n x) = n"

  1877 by (induct n) auto

  1878

  1879 lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"

  1880 by (induct n) auto

  1881

  1882 lemma replicate_app_Cons_same:

  1883 "(replicate n x) @ (x # xs) = x # replicate n x @ xs"

  1884 by (induct n) auto

  1885

  1886 lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"

  1887 apply (induct n, simp)

  1888 apply (simp add: replicate_app_Cons_same)

  1889 done

  1890

  1891 lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"

  1892 by (induct n) auto

  1893

  1894 text{* Courtesy of Matthias Daum: *}

  1895 lemma append_replicate_commute:

  1896   "replicate n x @ replicate k x = replicate k x @ replicate n x"

  1897 apply (simp add: replicate_add [THEN sym])

  1898 apply (simp add: add_commute)

  1899 done

  1900

  1901 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"

  1902 by (induct n) auto

  1903

  1904 lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"

  1905 by (induct n) auto

  1906

  1907 lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"

  1908 by (atomize (full), induct n) auto

  1909

  1910 lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"

  1911 apply (induct n, simp)

  1912 apply (simp add: nth_Cons split: nat.split)

  1913 done

  1914

  1915 text{* Courtesy of Matthias Daum (2 lemmas): *}

  1916 lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"

  1917 apply (case_tac "k \<le> i")

  1918  apply  (simp add: min_def)

  1919 apply (drule not_leE)

  1920 apply (simp add: min_def)

  1921 apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")

  1922  apply  simp

  1923 apply (simp add: replicate_add [symmetric])

  1924 done

  1925

  1926 lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"

  1927 apply (induct k)

  1928  apply simp

  1929 apply clarsimp

  1930 apply (case_tac i)

  1931  apply simp

  1932 apply clarsimp

  1933 done

  1934

  1935

  1936 lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"

  1937 by (induct n) auto

  1938

  1939 lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"

  1940 by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)

  1941

  1942 lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"

  1943 by auto

  1944

  1945 lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"

  1946 by (simp add: set_replicate_conv_if split: split_if_asm)

  1947

  1948

  1949 subsubsection{*@{text rotate1} and @{text rotate}*}

  1950

  1951 lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"

  1952 by(simp add:rotate1_def)

  1953

  1954 lemma rotate0[simp]: "rotate 0 = id"

  1955 by(simp add:rotate_def)

  1956

  1957 lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"

  1958 by(simp add:rotate_def)

  1959

  1960 lemma rotate_add:

  1961   "rotate (m+n) = rotate m o rotate n"

  1962 by(simp add:rotate_def funpow_add)

  1963

  1964 lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"

  1965 by(simp add:rotate_add)

  1966

  1967 lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"

  1968 by(simp add:rotate_def funpow_swap1)

  1969

  1970 lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"

  1971 by(cases xs) simp_all

  1972

  1973 lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"

  1974 apply(induct n)

  1975  apply simp

  1976 apply (simp add:rotate_def)

  1977 done

  1978

  1979 lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"

  1980 by(simp add:rotate1_def split:list.split)

  1981

  1982 lemma rotate_drop_take:

  1983   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"

  1984 apply(induct n)

  1985  apply simp

  1986 apply(simp add:rotate_def)

  1987 apply(cases "xs = []")

  1988  apply (simp)

  1989 apply(case_tac "n mod length xs = 0")

  1990  apply(simp add:mod_Suc)

  1991  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)

  1992 apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]

  1993                 take_hd_drop linorder_not_le)

  1994 done

  1995

  1996 lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"

  1997 by(simp add:rotate_drop_take)

  1998

  1999 lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"

  2000 by(simp add:rotate_drop_take)

  2001

  2002 lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"

  2003 by(simp add:rotate1_def split:list.split)

  2004

  2005 lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"

  2006 by (induct n) (simp_all add:rotate_def)

  2007

  2008 lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"

  2009 by(simp add:rotate1_def split:list.split) blast

  2010

  2011 lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"

  2012 by (induct n) (simp_all add:rotate_def)

  2013

  2014 lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"

  2015 by(simp add:rotate_drop_take take_map drop_map)

  2016

  2017 lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"

  2018 by(simp add:rotate1_def split:list.split)

  2019

  2020 lemma set_rotate[simp]: "set(rotate n xs) = set xs"

  2021 by (induct n) (simp_all add:rotate_def)

  2022

  2023 lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"

  2024 by(simp add:rotate1_def split:list.split)

  2025

  2026 lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"

  2027 by (induct n) (simp_all add:rotate_def)

  2028

  2029 lemma rotate_rev:

  2030   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"

  2031 apply(simp add:rotate_drop_take rev_drop rev_take)

  2032 apply(cases "length xs = 0")

  2033  apply simp

  2034 apply(cases "n mod length xs = 0")

  2035  apply simp

  2036 apply(simp add:rotate_drop_take rev_drop rev_take)

  2037 done

  2038

  2039 lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"

  2040 apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)

  2041 apply(subgoal_tac "length xs \<noteq> 0")

  2042  prefer 2 apply simp

  2043 using mod_less_divisor[of "length xs" n] by arith

  2044

  2045

  2046 subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}

  2047

  2048 lemma sublist_empty [simp]: "sublist xs {} = []"

  2049 by (auto simp add: sublist_def)

  2050

  2051 lemma sublist_nil [simp]: "sublist [] A = []"

  2052 by (auto simp add: sublist_def)

  2053

  2054 lemma length_sublist:

  2055   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"

  2056 by(simp add: sublist_def length_filter_conv_card cong:conj_cong)

  2057

  2058 lemma sublist_shift_lemma_Suc:

  2059   "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =

  2060          map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"

  2061 apply(induct xs)

  2062  apply simp

  2063 apply (case_tac "is")

  2064  apply simp

  2065 apply simp

  2066 done

  2067

  2068 lemma sublist_shift_lemma:

  2069      "map fst [p:zip xs [i..<i + length xs] . snd p : A] =

  2070       map fst [p:zip xs [0..<length xs] . snd p + i : A]"

  2071 by (induct xs rule: rev_induct) (simp_all add: add_commute)

  2072

  2073 lemma sublist_append:

  2074      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"

  2075 apply (unfold sublist_def)

  2076 apply (induct l' rule: rev_induct, simp)

  2077 apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)

  2078 apply (simp add: add_commute)

  2079 done

  2080

  2081 lemma sublist_Cons:

  2082 "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"

  2083 apply (induct l rule: rev_induct)

  2084  apply (simp add: sublist_def)

  2085 apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)

  2086 done

  2087

  2088 lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"

  2089 apply(induct xs)

  2090  apply simp

  2091 apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)

  2092  apply(erule lessE)

  2093   apply auto

  2094 apply(erule lessE)

  2095 apply auto

  2096 done

  2097

  2098 lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"

  2099 by(auto simp add:set_sublist)

  2100

  2101 lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"

  2102 by(auto simp add:set_sublist)

  2103

  2104 lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"

  2105 by(auto simp add:set_sublist)

  2106

  2107 lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"

  2108 by (simp add: sublist_Cons)

  2109

  2110

  2111 lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"

  2112 apply(induct xs)

  2113  apply simp

  2114 apply(auto simp add:sublist_Cons)

  2115 done

  2116

  2117

  2118 lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"

  2119 apply (induct l rule: rev_induct, simp)

  2120 apply (simp split: nat_diff_split add: sublist_append)

  2121 done

  2122

  2123 lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>

  2124   filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"

  2125 proof (induct xs)

  2126   case Nil thus ?case by simp

  2127 next

  2128   case (Cons a xs)

  2129   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto

  2130   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)

  2131 qed

  2132

  2133

  2134 subsubsection {* @{const splice} *}

  2135

  2136 lemma splice_Nil2 [simp, code]:

  2137  "splice xs [] = xs"

  2138 by (cases xs) simp_all

  2139

  2140 lemma splice_Cons_Cons [simp, code]:

  2141  "splice (x#xs) (y#ys) = x # y # splice xs ys"

  2142 by simp

  2143

  2144 declare splice.simps(2) [simp del, code del]

  2145

  2146 subsubsection{*Sets of Lists*}

  2147

  2148 subsubsection {* @{text lists}: the list-forming operator over sets *}

  2149

  2150 consts lists :: "'a set => 'a list set"

  2151 inductive "lists A"

  2152  intros

  2153   Nil [intro!]: "[]: lists A"

  2154   Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"

  2155

  2156 inductive_cases listsE [elim!]: "x#l : lists A"

  2157

  2158 lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"

  2159 by (unfold lists.defs) (blast intro!: lfp_mono)

  2160

  2161 lemma lists_IntI:

  2162   assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l

  2163   by induct blast+

  2164

  2165 lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"

  2166 proof (rule mono_Int [THEN equalityI])

  2167   show "mono lists" by (simp add: mono_def lists_mono)

  2168   show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)

  2169 qed

  2170

  2171 lemma append_in_lists_conv [iff]:

  2172      "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"

  2173 by (induct xs) auto

  2174

  2175 lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"

  2176 -- {* eliminate @{text lists} in favour of @{text set} *}

  2177 by (induct xs) auto

  2178

  2179 lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"

  2180 by (rule in_lists_conv_set [THEN iffD1])

  2181

  2182 lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"

  2183 by (rule in_lists_conv_set [THEN iffD2])

  2184

  2185 lemma lists_UNIV [simp]: "lists UNIV = UNIV"

  2186 by auto

  2187

  2188

  2189

  2190 subsubsection{* Inductive definition for membership *}

  2191

  2192 consts ListMem :: "('a \<times> 'a list)set"

  2193 inductive ListMem

  2194 intros

  2195  elem:  "(x,x#xs) \<in> ListMem"

  2196  insert:  "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem"

  2197

  2198 lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)"

  2199 apply (rule iffI)

  2200  apply (induct set: ListMem)

  2201   apply auto

  2202 apply (induct xs)

  2203  apply (auto intro: ListMem.intros)

  2204 done

  2205

  2206

  2207

  2208 subsubsection{*Lists as Cartesian products*}

  2209

  2210 text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from

  2211 @{term A} and tail drawn from @{term Xs}.*}

  2212

  2213 constdefs

  2214   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"

  2215   "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"

  2216

  2217 lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])A"

  2218 by (auto simp add: set_Cons_def)

  2219

  2220 text{*Yields the set of lists, all of the same length as the argument and

  2221 with elements drawn from the corresponding element of the argument.*}

  2222

  2223 consts  listset :: "'a set list \<Rightarrow> 'a list set"

  2224 primrec

  2225    "listset []    = {[]}"

  2226    "listset(A#As) = set_Cons A (listset As)"

  2227

  2228

  2229 subsection{*Relations on Lists*}

  2230

  2231 subsubsection {* Length Lexicographic Ordering *}

  2232

  2233 text{*These orderings preserve well-foundedness: shorter lists

  2234   precede longer lists. These ordering are not used in dictionaries.*}

  2235

  2236 consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"

  2237         --{*The lexicographic ordering for lists of the specified length*}

  2238 primrec

  2239   "lexn r 0 = {}"

  2240   "lexn r (Suc n) =

  2241     (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs)  (r <*lex*> lexn r n)) Int

  2242     {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"

  2243

  2244 constdefs

  2245   lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"

  2246     "lex r == \<Union>n. lexn r n"

  2247         --{*Holds only between lists of the same length*}

  2248

  2249   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"

  2250     "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"

  2251         --{*Compares lists by their length and then lexicographically*}

  2252

  2253

  2254 lemma wf_lexn: "wf r ==> wf (lexn r n)"

  2255 apply (induct n, simp, simp)

  2256 apply(rule wf_subset)

  2257  prefer 2 apply (rule Int_lower1)

  2258 apply(rule wf_prod_fun_image)

  2259  prefer 2 apply (rule inj_onI, auto)

  2260 done

  2261

  2262 lemma lexn_length:

  2263      "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"

  2264 by (induct n) auto

  2265

  2266 lemma wf_lex [intro!]: "wf r ==> wf (lex r)"

  2267 apply (unfold lex_def)

  2268 apply (rule wf_UN)

  2269 apply (blast intro: wf_lexn, clarify)

  2270 apply (rename_tac m n)

  2271 apply (subgoal_tac "m \<noteq> n")

  2272  prefer 2 apply blast

  2273 apply (blast dest: lexn_length not_sym)

  2274 done

  2275

  2276 lemma lexn_conv:

  2277   "lexn r n =

  2278     {(xs,ys). length xs = n \<and> length ys = n \<and>

  2279     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"

  2280 apply (induct n, simp)

  2281 apply (simp add: image_Collect lex_prod_def, safe, blast)

  2282  apply (rule_tac x = "ab # xys" in exI, simp)

  2283 apply (case_tac xys, simp_all, blast)

  2284 done

  2285

  2286 lemma lex_conv:

  2287   "lex r =

  2288     {(xs,ys). length xs = length ys \<and>

  2289     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"

  2290 by (force simp add: lex_def lexn_conv)

  2291

  2292 lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"

  2293 by (unfold lenlex_def) blast

  2294

  2295 lemma lenlex_conv:

  2296     "lenlex r = {(xs,ys). length xs < length ys |

  2297                  length xs = length ys \<and> (xs, ys) : lex r}"

  2298 by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)

  2299

  2300 lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"

  2301 by (simp add: lex_conv)

  2302

  2303 lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"

  2304 by (simp add:lex_conv)

  2305

  2306 lemma Cons_in_lex [simp]:

  2307     "((x # xs, y # ys) : lex r) =

  2308       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"

  2309 apply (simp add: lex_conv)

  2310 apply (rule iffI)

  2311  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)

  2312 apply (case_tac xys, simp, simp)

  2313 apply blast

  2314 done

  2315

  2316

  2317 subsubsection {* Lexicographic Ordering *}

  2318

  2319 text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".

  2320     This ordering does \emph{not} preserve well-foundedness.

  2321      Author: N. Voelker, March 2005. *}

  2322

  2323 constdefs

  2324   lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set"

  2325   "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or>

  2326             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"

  2327

  2328 lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"

  2329   by (unfold lexord_def, induct_tac y, auto)

  2330

  2331 lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"

  2332   by (unfold lexord_def, induct_tac x, auto)

  2333

  2334 lemma lexord_cons_cons[simp]:

  2335      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"

  2336   apply (unfold lexord_def, safe, simp_all)

  2337   apply (case_tac u, simp, simp)

  2338   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)

  2339   apply (erule_tac x="b # u" in allE)

  2340   by force

  2341

  2342 lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons

  2343

  2344 lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"

  2345   by (induct_tac x, auto)

  2346

  2347 lemma lexord_append_left_rightI:

  2348      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"

  2349   by (induct_tac u, auto)

  2350

  2351 lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"

  2352   by (induct x, auto)

  2353

  2354 lemma lexord_append_leftD:

  2355      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"

  2356   by (erule rev_mp, induct_tac x, auto)

  2357

  2358 lemma lexord_take_index_conv:

  2359    "((x,y) : lexord r) =

  2360     ((length x < length y \<and> take (length x) y = x) \<or>

  2361      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"

  2362   apply (unfold lexord_def Let_def, clarsimp)

  2363   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)

  2364   apply auto

  2365   apply (rule_tac x="hd (drop (length x) y)" in exI)

  2366   apply (rule_tac x="tl (drop (length x) y)" in exI)

  2367   apply (erule subst, simp add: min_def)

  2368   apply (rule_tac x ="length u" in exI, simp)

  2369   apply (rule_tac x ="take i x" in exI)

  2370   apply (rule_tac x ="x ! i" in exI)

  2371   apply (rule_tac x ="y ! i" in exI, safe)

  2372   apply (rule_tac x="drop (Suc i) x" in exI)

  2373   apply (drule sym, simp add: drop_Suc_conv_tl)

  2374   apply (rule_tac x="drop (Suc i) y" in exI)

  2375   by (simp add: drop_Suc_conv_tl)

  2376

  2377 -- {* lexord is extension of partial ordering List.lex *}

  2378 lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"

  2379   apply (rule_tac x = y in spec)

  2380   apply (induct_tac x, clarsimp)

  2381   by (clarify, case_tac x, simp, force)

  2382

  2383 lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"

  2384   by (induct y, auto)

  2385

  2386 lemma lexord_trans:

  2387     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"

  2388    apply (erule rev_mp)+

  2389    apply (rule_tac x = x in spec)

  2390   apply (rule_tac x = z in spec)

  2391   apply ( induct_tac y, simp, clarify)

  2392   apply (case_tac xa, erule ssubst)

  2393   apply (erule allE, erule allE) -- {* avoid simp recursion *}

  2394   apply (case_tac x, simp, simp)

  2395   apply (case_tac x, erule allE, erule allE, simp)

  2396   apply (erule_tac x = listb in allE)

  2397   apply (erule_tac x = lista in allE, simp)

  2398   apply (unfold trans_def)

  2399   by blast

  2400

  2401 lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"

  2402   by (rule transI, drule lexord_trans, blast)

  2403

  2404 lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"

  2405   apply (rule_tac x = y in spec)

  2406   apply (induct_tac x, rule allI)

  2407   apply (case_tac x, simp, simp)

  2408   apply (rule allI, case_tac x, simp, simp)

  2409   by blast

  2410

  2411

  2412 subsection {* Lexicographic combination of measure functions *}

  2413

  2414 text {* These are useful for termination proofs *}

  2415

  2416 definition

  2417   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"

  2418

  2419 lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"

  2420   unfolding measures_def

  2421   by blast

  2422

  2423 lemma in_measures[simp]:

  2424   "(x, y) \<in> measures [] = False"

  2425   "(x, y) \<in> measures (f # fs)

  2426          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"

  2427   unfolding measures_def

  2428   by auto

  2429

  2430 lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"

  2431   by simp

  2432

  2433 lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"

  2434   by auto

  2435

  2436 (* install the lexicographic_order method and the "fun" command *)

  2437 use "Tools/function_package/lexicographic_order.ML"

  2438 use "Tools/function_package/fundef_datatype.ML"

  2439 setup LexicographicOrder.setup

  2440 setup FundefDatatype.setup

  2441

  2442

  2443 subsubsection{*Lifting a Relation on List Elements to the Lists*}

  2444

  2445 consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"

  2446

  2447 inductive "listrel(r)"

  2448  intros

  2449    Nil:  "([],[]) \<in> listrel r"

  2450    Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"

  2451

  2452 inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"

  2453 inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"

  2454 inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"

  2455 inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"

  2456

  2457

  2458 lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"

  2459 apply clarify

  2460 apply (erule listrel.induct)

  2461 apply (blast intro: listrel.intros)+

  2462 done

  2463

  2464 lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"

  2465 apply clarify

  2466 apply (erule listrel.induct, auto)

  2467 done

  2468

  2469 lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)"

  2470 apply (simp add: refl_def listrel_subset Ball_def)

  2471 apply (rule allI)

  2472 apply (induct_tac x)

  2473 apply (auto intro: listrel.intros)

  2474 done

  2475

  2476 lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)"

  2477 apply (auto simp add: sym_def)

  2478 apply (erule listrel.induct)

  2479 apply (blast intro: listrel.intros)+

  2480 done

  2481

  2482 lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)"

  2483 apply (simp add: trans_def)

  2484 apply (intro allI)

  2485 apply (rule impI)

  2486 apply (erule listrel.induct)

  2487 apply (blast intro: listrel.intros)+

  2488 done

  2489

  2490 theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"

  2491 by (simp add: equiv_def listrel_refl listrel_sym listrel_trans)

  2492

  2493 lemma listrel_Nil [simp]: "listrel r  {[]} = {[]}"

  2494 by (blast intro: listrel.intros)

  2495

  2496 lemma listrel_Cons:

  2497      "listrel r  {x#xs} = set_Cons (r{x}) (listrel r  {xs})";

  2498 by (auto simp add: set_Cons_def intro: listrel.intros)

  2499

  2500

  2501 subsection{*Miscellany*}

  2502

  2503 subsubsection {* Characters and strings *}

  2504

  2505 datatype nibble =

  2506     Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7

  2507   | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF

  2508

  2509 datatype char = Char nibble nibble

  2510   -- "Note: canonical order of character encoding coincides with standard term ordering"

  2511

  2512 types string = "char list"

  2513

  2514 syntax

  2515   "_Char" :: "xstr => char"    ("CHR _")

  2516   "_String" :: "xstr => string"    ("_")

  2517

  2518 parse_ast_translation {*

  2519   let

  2520     val constants = Syntax.Appl o map Syntax.Constant;

  2521

  2522     fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));

  2523     fun mk_char c =

  2524       if Symbol.is_ascii c andalso Symbol.is_printable c then

  2525         constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]

  2526       else error ("Printable ASCII character expected: " ^ quote c);

  2527

  2528     fun mk_string [] = Syntax.Constant "Nil"

  2529       | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];

  2530

  2531     fun char_ast_tr [Syntax.Variable xstr] =

  2532         (case Syntax.explode_xstr xstr of

  2533           [c] => mk_char c

  2534         | _ => error ("Single character expected: " ^ xstr))

  2535       | char_ast_tr asts = raise AST ("char_ast_tr", asts);

  2536

  2537     fun string_ast_tr [Syntax.Variable xstr] =

  2538         (case Syntax.explode_xstr xstr of

  2539           [] => constants [Syntax.constrainC, "Nil", "string"]

  2540         | cs => mk_string cs)

  2541       | string_ast_tr asts = raise AST ("string_tr", asts);

  2542   in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;

  2543 *}

  2544

  2545 print_ast_translation {*

  2546   let

  2547     fun dest_nib (Syntax.Constant c) =

  2548         (case explode c of

  2549           ["N", "i", "b", "b", "l", "e", h] => HOLogic.int_of_nibble h

  2550         | _ => raise Match)

  2551       | dest_nib _ = raise Match;

  2552

  2553     fun dest_chr c1 c2 =

  2554       let val c = chr (dest_nib c1 * 16 + dest_nib c2)

  2555       in if Symbol.is_printable c then c else raise Match end;

  2556

  2557     fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2

  2558       | dest_char _ = raise Match;

  2559

  2560     fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];

  2561

  2562     fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]

  2563       | char_ast_tr' _ = raise Match;

  2564

  2565     fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",

  2566             xstr (map dest_char (Syntax.unfold_ast "_args" args))]

  2567       | list_ast_tr' ts = raise Match;

  2568   in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;

  2569 *}

  2570

  2571

  2572 subsection {* Code generator *}

  2573

  2574 subsubsection {* Setup *}

  2575

  2576 types_code

  2577   "list" ("_ list")

  2578 attach (term_of) {*

  2579 val term_of_list = HOLogic.mk_list;

  2580 *}

  2581 attach (test) {*

  2582 fun gen_list' aG i j = frequency

  2583   [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()

  2584 and gen_list aG i = gen_list' aG i i;

  2585 *}

  2586   "char" ("string")

  2587 attach (term_of) {*

  2588 val term_of_char = HOLogic.mk_char;

  2589 *}

  2590 attach (test) {*

  2591 fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));

  2592 *}

  2593

  2594 consts_code "Cons" ("(_ ::/ _)")

  2595

  2596 code_type list

  2597   (SML "_ list")

  2598   (Haskell "![_]")

  2599

  2600 code_const Nil

  2601   (SML "[]")

  2602   (Haskell "[]")

  2603

  2604 code_type char

  2605   (SML "char")

  2606   (Haskell "Char")

  2607

  2608 code_const Char

  2609   (SML "!((_),/ (_))")

  2610   (Haskell "!((_),/ (_))")

  2611

  2612 code_instance list :: eq and char :: eq

  2613   (Haskell - and -)

  2614

  2615 code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"

  2616   (Haskell infixl 4 "==")

  2617

  2618 code_const "op = \<Colon> char \<Rightarrow> char \<Rightarrow> bool"

  2619   (Haskell infixl 4 "==")

  2620

  2621 code_reserved SML

  2622   list char

  2623

  2624 code_reserved Haskell

  2625   Char

  2626

  2627 setup {*

  2628 let

  2629

  2630 fun list_codegen thy defs gr dep thyname b t =

  2631   let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)

  2632     (gr, HOLogic.dest_list t)

  2633   in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;

  2634

  2635 fun char_codegen thy defs gr dep thyname b t =

  2636   case (Option.map chr o HOLogic.dest_char) t

  2637    of SOME c =>

  2638         if Symbol.is_printable c

  2639         then SOME (gr, (Pretty.quote o Pretty.str) c)

  2640         else NONE

  2641     | NONE => NONE;

  2642

  2643 in

  2644

  2645   Codegen.add_codegen "list_codegen" list_codegen

  2646   #> Codegen.add_codegen "char_codegen" char_codegen

  2647   #> CodegenSerializer.add_pretty_list "SML" "List.list.Nil" "List.list.Cons"

  2648        (Pretty.enum "," "[" "]") NONE (7, "::")

  2649   #> CodegenSerializer.add_pretty_list "Haskell" "List.list.Nil" "List.list.Cons"

  2650        (Pretty.enum "," "[" "]") (SOME (HOLogic.print_char, HOLogic.print_string)) (5, ":")

  2651   #> CodegenPackage.add_appconst

  2652        ("List.char.Char", CodegenPackage.appgen_char HOLogic.dest_char)

  2653

  2654 end;

  2655 *}

  2656

  2657

  2658 subsubsection {* Generation of efficient code *}

  2659

  2660 consts

  2661   memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)

  2662   null:: "'a list \<Rightarrow> bool"

  2663   list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

  2664   list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"

  2665   list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"

  2666   itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

  2667   filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"

  2668   map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"

  2669

  2670 primrec

  2671   "x mem [] = False"

  2672   "x mem (y#ys) = (x = y \<or> x mem ys)"

  2673

  2674 primrec

  2675   "null [] = True"

  2676   "null (x#xs) = False"

  2677

  2678 primrec

  2679   "list_inter [] bs = []"

  2680   "list_inter (a#as) bs =

  2681      (if a \<in> set bs then a # list_inter as bs else list_inter as bs)"

  2682

  2683 primrec

  2684   "list_all P [] = True"

  2685   "list_all P (x#xs) = (P x \<and> list_all P xs)"

  2686

  2687 primrec

  2688   "list_ex P [] = False"

  2689   "list_ex P (x#xs) = (P x \<or> list_ex P xs)"

  2690

  2691 primrec

  2692   "filtermap f [] = []"

  2693   "filtermap f (x#xs) =

  2694      (case f x of None \<Rightarrow> filtermap f xs

  2695       | Some y \<Rightarrow> y # filtermap f xs)"

  2696

  2697 primrec

  2698   "map_filter f P [] = []"

  2699   "map_filter f P (x#xs) =

  2700      (if P x then f x # map_filter f P xs else map_filter f P xs)"

  2701

  2702 primrec

  2703   "itrev [] ys = ys"

  2704   "itrev (x#xs) ys = itrev xs (x#ys)"

  2705

  2706 text {*

  2707   Only use @{text mem} for generating executable code.  Otherwise

  2708   use @{prop "x : set xs"} instead --- it is much easier to reason about.

  2709   The same is true for @{const list_all} and @{const list_ex}: write

  2710   @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL

  2711   quantifiers are aleady known to the automatic provers. In fact,

  2712   the declarations in the code subsection make sure that @{text "\<in>"}, @{text "\<forall>x\<in>set xs"}

  2713   and @{text "\<exists>x\<in>set xs"} are implemented efficiently.

  2714

  2715   Efficient emptyness check is implemented by @{const null}.

  2716

  2717   The functions @{const itrev}, @{const filtermap} and @{const map_filter}

  2718   are just there to generate efficient code. Do not use them

  2719   for modelling and proving.

  2720 *}

  2721

  2722 lemma mem_iff [normal post]:

  2723   "(x mem xs) = (x \<in> set xs)"

  2724   by (induct xs) auto

  2725

  2726 lemmas in_set_code [code unfold] =

  2727   mem_iff [symmetric, THEN eq_reflection]

  2728

  2729 lemma empty_null [code inline]:

  2730   "(xs = []) = null xs"

  2731   by (cases xs) simp_all

  2732

  2733 lemmas null_empty [normal post] =

  2734   empty_null [symmetric]

  2735

  2736 lemma list_inter_conv:

  2737   "set (list_inter xs ys) = set xs \<inter> set ys"

  2738   by (induct xs) auto

  2739

  2740 lemma list_all_iff [normal post]:

  2741   "list_all P xs = (\<forall>x \<in> set xs. P x)"

  2742   by (induct xs) auto

  2743

  2744 lemmas list_ball_code [code unfold] =

  2745   list_all_iff [symmetric, THEN eq_reflection]

  2746

  2747 lemma list_all_append [simp]:

  2748   "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"

  2749   by (induct xs) auto

  2750

  2751 lemma list_all_rev [simp]:

  2752   "list_all P (rev xs) = list_all P xs"

  2753   by (simp add: list_all_iff)

  2754

  2755 lemma list_ex_iff [normal post]:

  2756   "list_ex P xs = (\<exists>x \<in> set xs. P x)"

  2757   by (induct xs) simp_all

  2758

  2759 lemmas list_bex_code [code unfold] =

  2760   list_ex_iff [symmetric, THEN eq_reflection]

  2761

  2762 lemma itrev [simp]:

  2763   "itrev xs ys = rev xs @ ys"

  2764   by (induct xs arbitrary: ys) simp_all

  2765

  2766 lemma filtermap_conv:

  2767    "filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)"

  2768   by (induct xs) (simp_all split: option.split)

  2769

  2770 lemma map_filter_conv [simp]:

  2771   "map_filter f P xs = map f (filter P xs)"

  2772   by (induct xs) auto

  2773

  2774 lemma rev_code [code func, code unfold, code noinline]:

  2775   "rev xs == itrev xs []"

  2776   by simp

  2777

  2778 end
`