src/HOL/List.thy
author nipkow
Wed Apr 16 22:14:08 2003 +0200 (2003-04-16)
changeset 13913 b3ed67af04b8
parent 13883 0451e0fb3f22
child 14025 d9b155757dc8
permissions -rw-r--r--
Added take/dropWhile thms
wenzelm@13462
     1
(*  Title:      HOL/List.thy
wenzelm@13462
     2
    ID:         $Id$
wenzelm@13462
     3
    Author:     Tobias Nipkow
wenzelm@13462
     4
    License:    GPL (GNU GENERAL PUBLIC LICENSE)
clasohm@923
     5
*)
clasohm@923
     6
wenzelm@13114
     7
header {* The datatype of finite lists *}
wenzelm@13122
     8
wenzelm@13122
     9
theory List = PreList:
clasohm@923
    10
wenzelm@13142
    11
datatype 'a list =
wenzelm@13366
    12
    Nil    ("[]")
wenzelm@13366
    13
  | Cons 'a  "'a list"    (infixr "#" 65)
clasohm@923
    14
clasohm@923
    15
consts
wenzelm@13366
    16
  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
wenzelm@13366
    17
  filter:: "('a => bool) => 'a list => 'a list"
wenzelm@13366
    18
  concat:: "'a list list => 'a list"
wenzelm@13366
    19
  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
wenzelm@13366
    20
  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
wenzelm@13366
    21
  hd:: "'a list => 'a"
wenzelm@13366
    22
  tl:: "'a list => 'a list"
wenzelm@13366
    23
  last:: "'a list => 'a"
wenzelm@13366
    24
  butlast :: "'a list => 'a list"
wenzelm@13366
    25
  set :: "'a list => 'a set"
wenzelm@13366
    26
  list_all:: "('a => bool) => ('a list => bool)"
wenzelm@13366
    27
  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
wenzelm@13366
    28
  map :: "('a=>'b) => ('a list => 'b list)"
wenzelm@13366
    29
  mem :: "'a => 'a list => bool"    (infixl 55)
wenzelm@13366
    30
  nth :: "'a list => nat => 'a"    (infixl "!" 100)
wenzelm@13366
    31
  list_update :: "'a list => nat => 'a => 'a list"
wenzelm@13366
    32
  take:: "nat => 'a list => 'a list"
wenzelm@13366
    33
  drop:: "nat => 'a list => 'a list"
wenzelm@13366
    34
  takeWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366
    35
  dropWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366
    36
  rev :: "'a list => 'a list"
wenzelm@13366
    37
  zip :: "'a list => 'b list => ('a * 'b) list"
wenzelm@13366
    38
  upt :: "nat => nat => nat list" ("(1[_../_'(])")
wenzelm@13366
    39
  remdups :: "'a list => 'a list"
wenzelm@13366
    40
  null:: "'a list => bool"
wenzelm@13366
    41
  "distinct":: "'a list => bool"
wenzelm@13366
    42
  replicate :: "nat => 'a => 'a list"
clasohm@923
    43
nipkow@13146
    44
nonterminals lupdbinds lupdbind
nipkow@5077
    45
clasohm@923
    46
syntax
wenzelm@13366
    47
  -- {* list Enumeration *}
wenzelm@13366
    48
  "@list" :: "args => 'a list"    ("[(_)]")
clasohm@923
    49
wenzelm@13366
    50
  -- {* Special syntax for filter *}
wenzelm@13366
    51
  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
clasohm@923
    52
wenzelm@13366
    53
  -- {* list update *}
wenzelm@13366
    54
  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
wenzelm@13366
    55
  "" :: "lupdbind => lupdbinds"    ("_")
wenzelm@13366
    56
  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
wenzelm@13366
    57
  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
nipkow@5077
    58
wenzelm@13366
    59
  upto:: "nat => nat => nat list"    ("(1[_../_])")
nipkow@5427
    60
clasohm@923
    61
translations
wenzelm@13366
    62
  "[x, xs]" == "x#[xs]"
wenzelm@13366
    63
  "[x]" == "x#[]"
wenzelm@13366
    64
  "[x:xs . P]"== "filter (%x. P) xs"
clasohm@923
    65
wenzelm@13366
    66
  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
wenzelm@13366
    67
  "xs[i:=x]" == "list_update xs i x"
nipkow@5077
    68
wenzelm@13366
    69
  "[i..j]" == "[i..(Suc j)(]"
nipkow@5427
    70
nipkow@5427
    71
wenzelm@12114
    72
syntax (xsymbols)
wenzelm@13366
    73
  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
paulson@3342
    74
paulson@3342
    75
wenzelm@13142
    76
text {*
wenzelm@13366
    77
  Function @{text size} is overloaded for all datatypes.Users may
wenzelm@13366
    78
  refer to the list version as @{text length}. *}
wenzelm@13142
    79
wenzelm@13142
    80
syntax length :: "'a list => nat"
wenzelm@13142
    81
translations "length" => "size :: _ list => nat"
wenzelm@13114
    82
wenzelm@13142
    83
typed_print_translation {*
wenzelm@13366
    84
  let
wenzelm@13366
    85
    fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
wenzelm@13366
    86
          Syntax.const "length" $ t
wenzelm@13366
    87
      | size_tr' _ _ _ = raise Match;
wenzelm@13366
    88
  in [("size", size_tr')] end
wenzelm@13114
    89
*}
paulson@3437
    90
berghofe@5183
    91
primrec
nipkow@13145
    92
"hd(x#xs) = x"
berghofe@5183
    93
primrec
nipkow@13145
    94
"tl([]) = []"
nipkow@13145
    95
"tl(x#xs) = xs"
berghofe@5183
    96
primrec
nipkow@13145
    97
"null([]) = True"
nipkow@13145
    98
"null(x#xs) = False"
paulson@8972
    99
primrec
nipkow@13145
   100
"last(x#xs) = (if xs=[] then x else last xs)"
berghofe@5183
   101
primrec
nipkow@13145
   102
"butlast []= []"
nipkow@13145
   103
"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
berghofe@5183
   104
primrec
nipkow@13145
   105
"x mem [] = False"
nipkow@13145
   106
"x mem (y#ys) = (if y=x then True else x mem ys)"
oheimb@5518
   107
primrec
nipkow@13145
   108
"set [] = {}"
nipkow@13145
   109
"set (x#xs) = insert x (set xs)"
berghofe@5183
   110
primrec
nipkow@13145
   111
list_all_Nil:"list_all P [] = True"
nipkow@13145
   112
list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
oheimb@5518
   113
primrec
nipkow@13145
   114
"map f [] = []"
nipkow@13145
   115
"map f (x#xs) = f(x)#map f xs"
berghofe@5183
   116
primrec
nipkow@13145
   117
append_Nil:"[]@ys = ys"
nipkow@13145
   118
append_Cons: "(x#xs)@ys = x#(xs@ys)"
berghofe@5183
   119
primrec
nipkow@13145
   120
"rev([]) = []"
nipkow@13145
   121
"rev(x#xs) = rev(xs) @ [x]"
berghofe@5183
   122
primrec
nipkow@13145
   123
"filter P [] = []"
nipkow@13145
   124
"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
berghofe@5183
   125
primrec
nipkow@13145
   126
foldl_Nil:"foldl f a [] = a"
nipkow@13145
   127
foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
berghofe@5183
   128
primrec
nipkow@13145
   129
"foldr f [] a = a"
nipkow@13145
   130
"foldr f (x#xs) a = f x (foldr f xs a)"
paulson@8000
   131
primrec
nipkow@13145
   132
"concat([]) = []"
nipkow@13145
   133
"concat(x#xs) = x @ concat(xs)"
berghofe@5183
   134
primrec
nipkow@13145
   135
drop_Nil:"drop n [] = []"
nipkow@13145
   136
drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
nipkow@13145
   137
-- {* Warning: simpset does not contain this definition *}
nipkow@13145
   138
-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
berghofe@5183
   139
primrec
nipkow@13145
   140
take_Nil:"take n [] = []"
nipkow@13145
   141
take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
nipkow@13145
   142
-- {* Warning: simpset does not contain this definition *}
nipkow@13145
   143
-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
berghofe@5183
   144
primrec
nipkow@13145
   145
nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
nipkow@13145
   146
-- {* Warning: simpset does not contain this definition *}
nipkow@13145
   147
-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
wenzelm@13142
   148
primrec
nipkow@13145
   149
"[][i:=v] = []"
nipkow@13145
   150
"(x#xs)[i:=v] =
nipkow@13145
   151
(case i of 0 => v # xs
nipkow@13145
   152
| Suc j => x # xs[j:=v])"
berghofe@5183
   153
primrec
nipkow@13145
   154
"takeWhile P [] = []"
nipkow@13145
   155
"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
berghofe@5183
   156
primrec
nipkow@13145
   157
"dropWhile P [] = []"
nipkow@13145
   158
"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
berghofe@5183
   159
primrec
nipkow@13145
   160
"zip xs [] = []"
nipkow@13145
   161
zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
nipkow@13145
   162
-- {* Warning: simpset does not contain this definition *}
nipkow@13145
   163
-- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
nipkow@5427
   164
primrec
nipkow@13145
   165
upt_0: "[i..0(] = []"
nipkow@13145
   166
upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
berghofe@5183
   167
primrec
nipkow@13145
   168
"distinct [] = True"
nipkow@13145
   169
"distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
berghofe@5183
   170
primrec
nipkow@13145
   171
"remdups [] = []"
nipkow@13145
   172
"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
berghofe@5183
   173
primrec
nipkow@13147
   174
replicate_0: "replicate 0 x = []"
nipkow@13145
   175
replicate_Suc: "replicate (Suc n) x = x # replicate n x"
nipkow@8115
   176
defs
wenzelm@13114
   177
 list_all2_def:
wenzelm@13142
   178
 "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
nipkow@8115
   179
paulson@3196
   180
wenzelm@13142
   181
subsection {* Lexicographic orderings on lists *}
nipkow@5281
   182
nipkow@5281
   183
consts
nipkow@13145
   184
lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
nipkow@5281
   185
primrec
nipkow@13145
   186
"lexn r 0 = {}"
nipkow@13145
   187
"lexn r (Suc n) =
nipkow@13145
   188
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
nipkow@13145
   189
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@5281
   190
nipkow@5281
   191
constdefs
nipkow@13145
   192
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@13145
   193
"lex r == \<Union>n. lexn r n"
nipkow@5281
   194
nipkow@13145
   195
lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@13145
   196
"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@9336
   197
nipkow@13145
   198
sublist :: "'a list => nat set => 'a list"
nipkow@13145
   199
"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
nipkow@5281
   200
wenzelm@13114
   201
wenzelm@13142
   202
lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
nipkow@13145
   203
by (induct xs) auto
wenzelm@13114
   204
wenzelm@13142
   205
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   206
wenzelm@13142
   207
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   208
by (induct xs) auto
wenzelm@13114
   209
wenzelm@13142
   210
lemma length_induct:
nipkow@13145
   211
"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
nipkow@13145
   212
by (rule measure_induct [of length]) rules
wenzelm@13114
   213
wenzelm@13114
   214
wenzelm@13142
   215
subsection {* @{text lists}: the list-forming operator over sets *}
wenzelm@13114
   216
wenzelm@13142
   217
consts lists :: "'a set => 'a list set"
wenzelm@13142
   218
inductive "lists A"
nipkow@13145
   219
intros
nipkow@13145
   220
Nil [intro!]: "[]: lists A"
nipkow@13145
   221
Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
wenzelm@13114
   222
wenzelm@13142
   223
inductive_cases listsE [elim!]: "x#l : lists A"
wenzelm@13114
   224
wenzelm@13366
   225
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@13145
   226
by (unfold lists.defs) (blast intro!: lfp_mono)
wenzelm@13114
   227
berghofe@13883
   228
lemma lists_IntI:
berghofe@13883
   229
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
berghofe@13883
   230
  by induct blast+
wenzelm@13142
   231
wenzelm@13142
   232
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@13145
   233
apply (rule mono_Int [THEN equalityI])
nipkow@13145
   234
apply (simp add: mono_def lists_mono)
nipkow@13145
   235
apply (blast intro!: lists_IntI)
nipkow@13145
   236
done
wenzelm@13114
   237
wenzelm@13142
   238
lemma append_in_lists_conv [iff]:
nipkow@13145
   239
"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@13145
   240
by (induct xs) auto
wenzelm@13142
   241
wenzelm@13142
   242
wenzelm@13142
   243
subsection {* @{text length} *}
wenzelm@13114
   244
wenzelm@13142
   245
text {*
nipkow@13145
   246
Needs to come before @{text "@"} because of theorem @{text
nipkow@13145
   247
append_eq_append_conv}.
wenzelm@13142
   248
*}
wenzelm@13114
   249
wenzelm@13142
   250
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   251
by (induct xs) auto
wenzelm@13114
   252
wenzelm@13142
   253
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   254
by (induct xs) auto
wenzelm@13114
   255
wenzelm@13142
   256
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   257
by (induct xs) auto
wenzelm@13114
   258
wenzelm@13142
   259
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   260
by (cases xs) auto
wenzelm@13114
   261
wenzelm@13142
   262
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   263
by (induct xs) auto
wenzelm@13114
   264
wenzelm@13142
   265
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   266
by (induct xs) auto
wenzelm@13114
   267
wenzelm@13114
   268
lemma length_Suc_conv:
nipkow@13145
   269
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   270
by (induct xs) auto
wenzelm@13142
   271
wenzelm@13114
   272
wenzelm@13142
   273
subsection {* @{text "@"} -- append *}
wenzelm@13114
   274
wenzelm@13142
   275
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   276
by (induct xs) auto
wenzelm@13114
   277
wenzelm@13142
   278
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   279
by (induct xs) auto
nipkow@3507
   280
wenzelm@13142
   281
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   282
by (induct xs) auto
wenzelm@13114
   283
wenzelm@13142
   284
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   285
by (induct xs) auto
wenzelm@13114
   286
wenzelm@13142
   287
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   288
by (induct xs) auto
wenzelm@13114
   289
wenzelm@13142
   290
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   291
by (induct xs) auto
wenzelm@13114
   292
berghofe@13883
   293
lemma append_eq_append_conv [simp]:
berghofe@13883
   294
 "!!ys. length xs = length ys \<or> length us = length vs
berghofe@13883
   295
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
berghofe@13883
   296
apply (induct xs)
nipkow@13145
   297
 apply (case_tac ys)
nipkow@13145
   298
apply simp
nipkow@13145
   299
 apply force
nipkow@13145
   300
apply (case_tac ys)
nipkow@13145
   301
 apply force
nipkow@13145
   302
apply simp
nipkow@13145
   303
done
wenzelm@13142
   304
wenzelm@13142
   305
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   306
by simp
wenzelm@13142
   307
wenzelm@13142
   308
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   309
by simp
wenzelm@13114
   310
wenzelm@13142
   311
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   312
by simp
wenzelm@13114
   313
wenzelm@13142
   314
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   315
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   316
wenzelm@13142
   317
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   318
using append_same_eq [of "[]"] by auto
wenzelm@13114
   319
wenzelm@13142
   320
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   321
by (induct xs) auto
wenzelm@13114
   322
wenzelm@13142
   323
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   324
by (induct xs) auto
wenzelm@13114
   325
wenzelm@13142
   326
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   327
by (simp add: hd_append split: list.split)
wenzelm@13114
   328
wenzelm@13142
   329
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   330
by (simp split: list.split)
wenzelm@13114
   331
wenzelm@13142
   332
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   333
by (simp add: tl_append split: list.split)
wenzelm@13114
   334
wenzelm@13114
   335
wenzelm@13142
   336
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   337
wenzelm@13114
   338
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   339
by simp
wenzelm@13114
   340
wenzelm@13142
   341
lemma Cons_eq_appendI:
nipkow@13145
   342
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   343
by (drule sym) simp
wenzelm@13114
   344
wenzelm@13142
   345
lemma append_eq_appendI:
nipkow@13145
   346
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   347
by (drule sym) simp
wenzelm@13114
   348
wenzelm@13114
   349
wenzelm@13142
   350
text {*
nipkow@13145
   351
Simplification procedure for all list equalities.
nipkow@13145
   352
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   353
- both lists end in a singleton list,
nipkow@13145
   354
- or both lists end in the same list.
wenzelm@13142
   355
*}
wenzelm@13142
   356
wenzelm@13142
   357
ML_setup {*
nipkow@3507
   358
local
nipkow@3507
   359
wenzelm@13122
   360
val append_assoc = thm "append_assoc";
wenzelm@13122
   361
val append_Nil = thm "append_Nil";
wenzelm@13122
   362
val append_Cons = thm "append_Cons";
wenzelm@13122
   363
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   364
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   365
wenzelm@13114
   366
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   367
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   368
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   369
  | last t = t;
wenzelm@13114
   370
wenzelm@13114
   371
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   372
  | list1 _ = false;
wenzelm@13114
   373
wenzelm@13114
   374
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   375
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   376
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   377
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   378
wenzelm@13114
   379
val rearr_tac =
wenzelm@13462
   380
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   381
wenzelm@13114
   382
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   383
  let
wenzelm@13462
   384
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   385
    fun rearr conv =
wenzelm@13462
   386
      let
wenzelm@13462
   387
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   388
        val Type(_,listT::_) = eqT
wenzelm@13462
   389
        val appT = [listT,listT] ---> listT
wenzelm@13462
   390
        val app = Const("List.op @",appT)
wenzelm@13462
   391
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   392
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   393
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   394
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   395
wenzelm@13462
   396
  in
wenzelm@13462
   397
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   398
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   399
    else None
wenzelm@13462
   400
  end;
wenzelm@13462
   401
wenzelm@13114
   402
in
wenzelm@13462
   403
wenzelm@13462
   404
val list_eq_simproc =
wenzelm@13462
   405
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   406
wenzelm@13114
   407
end;
wenzelm@13114
   408
wenzelm@13114
   409
Addsimprocs [list_eq_simproc];
wenzelm@13114
   410
*}
wenzelm@13114
   411
wenzelm@13114
   412
wenzelm@13142
   413
subsection {* @{text map} *}
wenzelm@13114
   414
wenzelm@13142
   415
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   416
by (induct xs) simp_all
wenzelm@13114
   417
wenzelm@13142
   418
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   419
by (rule ext, induct_tac xs) auto
wenzelm@13114
   420
wenzelm@13142
   421
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   422
by (induct xs) auto
wenzelm@13114
   423
wenzelm@13142
   424
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   425
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   426
wenzelm@13142
   427
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   428
by (induct xs) auto
wenzelm@13114
   429
nipkow@13737
   430
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   431
by (induct xs) auto
nipkow@13737
   432
wenzelm@13366
   433
lemma map_cong [recdef_cong]:
nipkow@13145
   434
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   435
-- {* a congruence rule for @{text map} *}
nipkow@13737
   436
by simp
wenzelm@13114
   437
wenzelm@13142
   438
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   439
by (cases xs) auto
wenzelm@13114
   440
wenzelm@13142
   441
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   442
by (cases xs) auto
wenzelm@13114
   443
wenzelm@13114
   444
lemma map_eq_Cons:
nipkow@13145
   445
"(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)"
nipkow@13145
   446
by (cases xs) auto
wenzelm@13114
   447
wenzelm@13114
   448
lemma map_injective:
nipkow@13145
   449
"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys"
nipkow@13145
   450
by (induct ys) (auto simp add: map_eq_Cons)
wenzelm@13114
   451
wenzelm@13114
   452
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   453
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   454
wenzelm@13114
   455
lemma inj_mapD: "inj (map f) ==> inj f"
nipkow@13145
   456
apply (unfold inj_on_def)
nipkow@13145
   457
apply clarify
nipkow@13145
   458
apply (erule_tac x = "[x]" in ballE)
nipkow@13145
   459
 apply (erule_tac x = "[y]" in ballE)
nipkow@13145
   460
apply simp
nipkow@13145
   461
 apply blast
nipkow@13145
   462
apply blast
nipkow@13145
   463
done
wenzelm@13114
   464
wenzelm@13114
   465
lemma inj_map: "inj (map f) = inj f"
nipkow@13145
   466
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   467
wenzelm@13114
   468
wenzelm@13142
   469
subsection {* @{text rev} *}
wenzelm@13114
   470
wenzelm@13142
   471
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   472
by (induct xs) auto
wenzelm@13114
   473
wenzelm@13142
   474
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   475
by (induct xs) auto
wenzelm@13114
   476
wenzelm@13142
   477
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   478
by (induct xs) auto
wenzelm@13114
   479
wenzelm@13142
   480
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   481
by (induct xs) auto
wenzelm@13114
   482
wenzelm@13142
   483
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
nipkow@13145
   484
apply (induct xs)
nipkow@13145
   485
 apply force
nipkow@13145
   486
apply (case_tac ys)
nipkow@13145
   487
 apply simp
nipkow@13145
   488
apply force
nipkow@13145
   489
done
wenzelm@13114
   490
wenzelm@13366
   491
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   492
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   493
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   494
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   495
done
wenzelm@13114
   496
nipkow@13145
   497
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   498
wenzelm@13366
   499
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   500
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   501
by (induct xs rule: rev_induct) auto
wenzelm@13114
   502
wenzelm@13366
   503
lemmas rev_cases = rev_exhaust
wenzelm@13366
   504
wenzelm@13114
   505
wenzelm@13142
   506
subsection {* @{text set} *}
wenzelm@13114
   507
wenzelm@13142
   508
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   509
by (induct xs) auto
wenzelm@13114
   510
wenzelm@13142
   511
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   512
by (induct xs) auto
wenzelm@13114
   513
wenzelm@13142
   514
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   515
by auto
wenzelm@13114
   516
wenzelm@13142
   517
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   518
by (induct xs) auto
wenzelm@13114
   519
wenzelm@13142
   520
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   521
by (induct xs) auto
wenzelm@13114
   522
wenzelm@13142
   523
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   524
by (induct xs) auto
wenzelm@13114
   525
wenzelm@13142
   526
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   527
by (induct xs) auto
wenzelm@13114
   528
wenzelm@13142
   529
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
nipkow@13145
   530
apply (induct j)
nipkow@13145
   531
 apply simp_all
nipkow@13145
   532
apply(erule ssubst)
nipkow@13145
   533
apply auto
nipkow@13145
   534
done
wenzelm@13114
   535
wenzelm@13142
   536
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
nipkow@13145
   537
apply (induct xs)
nipkow@13145
   538
 apply simp
nipkow@13145
   539
apply simp
nipkow@13145
   540
apply (rule iffI)
nipkow@13145
   541
 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
nipkow@13145
   542
apply (erule exE)+
nipkow@13145
   543
apply (case_tac ys)
nipkow@13145
   544
apply auto
nipkow@13145
   545
done
wenzelm@13142
   546
wenzelm@13142
   547
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   548
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   549
by (induct xs) auto
wenzelm@13142
   550
wenzelm@13142
   551
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   552
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   553
wenzelm@13142
   554
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   555
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   556
paulson@13508
   557
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   558
apply (erule finite_induct, auto)
paulson@13508
   559
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   560
done
paulson@13508
   561
wenzelm@13114
   562
wenzelm@13142
   563
subsection {* @{text mem} *}
wenzelm@13114
   564
wenzelm@13114
   565
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   566
by (induct xs) auto
wenzelm@13114
   567
wenzelm@13114
   568
wenzelm@13142
   569
subsection {* @{text list_all} *}
wenzelm@13114
   570
wenzelm@13142
   571
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   572
by (induct xs) auto
wenzelm@13114
   573
wenzelm@13142
   574
lemma list_all_append [simp]:
nipkow@13145
   575
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   576
by (induct xs) auto
wenzelm@13114
   577
wenzelm@13114
   578
wenzelm@13142
   579
subsection {* @{text filter} *}
wenzelm@13114
   580
wenzelm@13142
   581
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   582
by (induct xs) auto
wenzelm@13114
   583
wenzelm@13142
   584
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   585
by (induct xs) auto
wenzelm@13114
   586
wenzelm@13142
   587
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   588
by (induct xs) auto
wenzelm@13114
   589
wenzelm@13142
   590
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   591
by (induct xs) auto
wenzelm@13114
   592
wenzelm@13142
   593
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   594
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   595
wenzelm@13142
   596
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   597
by auto
wenzelm@13114
   598
wenzelm@13114
   599
wenzelm@13142
   600
subsection {* @{text concat} *}
wenzelm@13114
   601
wenzelm@13142
   602
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   603
by (induct xs) auto
wenzelm@13114
   604
wenzelm@13142
   605
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   606
by (induct xss) auto
wenzelm@13114
   607
wenzelm@13142
   608
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   609
by (induct xss) auto
wenzelm@13114
   610
wenzelm@13142
   611
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   612
by (induct xs) auto
wenzelm@13114
   613
wenzelm@13142
   614
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   615
by (induct xs) auto
wenzelm@13114
   616
wenzelm@13142
   617
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   618
by (induct xs) auto
wenzelm@13114
   619
wenzelm@13142
   620
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   621
by (induct xs) auto
wenzelm@13114
   622
wenzelm@13114
   623
wenzelm@13142
   624
subsection {* @{text nth} *}
wenzelm@13114
   625
wenzelm@13142
   626
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   627
by auto
wenzelm@13114
   628
wenzelm@13142
   629
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   630
by auto
wenzelm@13114
   631
wenzelm@13142
   632
declare nth.simps [simp del]
wenzelm@13114
   633
wenzelm@13114
   634
lemma nth_append:
nipkow@13145
   635
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@13145
   636
apply(induct "xs")
nipkow@13145
   637
 apply simp
nipkow@13145
   638
apply (case_tac n)
nipkow@13145
   639
 apply auto
nipkow@13145
   640
done
wenzelm@13114
   641
wenzelm@13142
   642
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@13145
   643
apply(induct xs)
nipkow@13145
   644
 apply simp
nipkow@13145
   645
apply (case_tac n)
nipkow@13145
   646
 apply auto
nipkow@13145
   647
done
wenzelm@13114
   648
wenzelm@13142
   649
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
nipkow@13145
   650
apply (induct_tac xs)
nipkow@13145
   651
 apply simp
nipkow@13145
   652
apply simp
nipkow@13145
   653
apply safe
nipkow@13145
   654
apply (rule_tac x = 0 in exI)
nipkow@13145
   655
apply simp
nipkow@13145
   656
 apply (rule_tac x = "Suc i" in exI)
nipkow@13145
   657
 apply simp
nipkow@13145
   658
apply (case_tac i)
nipkow@13145
   659
 apply simp
nipkow@13145
   660
apply (rename_tac j)
nipkow@13145
   661
apply (rule_tac x = j in exI)
nipkow@13145
   662
apply simp
nipkow@13145
   663
done
wenzelm@13114
   664
nipkow@13145
   665
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   666
by (auto simp add: set_conv_nth)
wenzelm@13114
   667
wenzelm@13142
   668
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   669
by (auto simp add: set_conv_nth)
wenzelm@13114
   670
wenzelm@13114
   671
lemma all_nth_imp_all_set:
nipkow@13145
   672
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   673
by (auto simp add: set_conv_nth)
wenzelm@13114
   674
wenzelm@13114
   675
lemma all_set_conv_all_nth:
nipkow@13145
   676
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   677
by (auto simp add: set_conv_nth)
wenzelm@13114
   678
wenzelm@13114
   679
wenzelm@13142
   680
subsection {* @{text list_update} *}
wenzelm@13114
   681
wenzelm@13142
   682
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   683
by (induct xs) (auto split: nat.split)
wenzelm@13114
   684
wenzelm@13114
   685
lemma nth_list_update:
nipkow@13145
   686
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   687
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   688
wenzelm@13142
   689
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   690
by (simp add: nth_list_update)
wenzelm@13114
   691
wenzelm@13142
   692
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   693
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   694
wenzelm@13142
   695
lemma list_update_overwrite [simp]:
nipkow@13145
   696
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   697
by (induct xs) (auto split: nat.split)
wenzelm@13114
   698
wenzelm@13114
   699
lemma list_update_same_conv:
nipkow@13145
   700
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   701
by (induct xs) (auto split: nat.split)
wenzelm@13114
   702
wenzelm@13114
   703
lemma update_zip:
nipkow@13145
   704
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   705
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   706
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   707
wenzelm@13114
   708
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   709
by (induct xs) (auto split: nat.split)
wenzelm@13114
   710
wenzelm@13114
   711
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   712
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   713
wenzelm@13114
   714
wenzelm@13142
   715
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   716
wenzelm@13142
   717
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   718
by (induct xs) auto
wenzelm@13114
   719
wenzelm@13142
   720
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   721
by (induct xs) auto
wenzelm@13114
   722
wenzelm@13142
   723
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   724
by (induct xs rule: rev_induct) auto
wenzelm@13114
   725
wenzelm@13114
   726
lemma butlast_append:
nipkow@13145
   727
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   728
by (induct xs) auto
wenzelm@13114
   729
wenzelm@13142
   730
lemma append_butlast_last_id [simp]:
nipkow@13145
   731
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   732
by (induct xs) auto
wenzelm@13114
   733
wenzelm@13142
   734
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   735
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   736
wenzelm@13114
   737
lemma in_set_butlast_appendI:
nipkow@13145
   738
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   739
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   740
wenzelm@13142
   741
wenzelm@13142
   742
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   743
wenzelm@13142
   744
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   745
by (induct xs) auto
wenzelm@13114
   746
wenzelm@13142
   747
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   748
by (induct xs) auto
wenzelm@13114
   749
wenzelm@13142
   750
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   751
by simp
wenzelm@13114
   752
wenzelm@13142
   753
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   754
by simp
wenzelm@13114
   755
wenzelm@13142
   756
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   757
nipkow@13913
   758
lemma take_Suc_conv_app_nth:
nipkow@13913
   759
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@13913
   760
apply(induct xs)
nipkow@13913
   761
 apply simp
nipkow@13913
   762
apply(case_tac i)
nipkow@13913
   763
apply auto
nipkow@13913
   764
done
nipkow@13913
   765
wenzelm@13142
   766
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   767
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   768
wenzelm@13142
   769
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   770
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   771
wenzelm@13142
   772
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   773
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   774
wenzelm@13142
   775
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   776
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   777
wenzelm@13142
   778
lemma take_append [simp]:
nipkow@13145
   779
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   780
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   781
wenzelm@13142
   782
lemma drop_append [simp]:
nipkow@13145
   783
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   784
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   785
wenzelm@13142
   786
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
nipkow@13145
   787
apply (induct m)
nipkow@13145
   788
 apply auto
nipkow@13145
   789
apply (case_tac xs)
nipkow@13145
   790
 apply auto
nipkow@13145
   791
apply (case_tac na)
nipkow@13145
   792
 apply auto
nipkow@13145
   793
done
wenzelm@13114
   794
wenzelm@13142
   795
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
nipkow@13145
   796
apply (induct m)
nipkow@13145
   797
 apply auto
nipkow@13145
   798
apply (case_tac xs)
nipkow@13145
   799
 apply auto
nipkow@13145
   800
done
wenzelm@13114
   801
wenzelm@13114
   802
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@13145
   803
apply (induct m)
nipkow@13145
   804
 apply auto
nipkow@13145
   805
apply (case_tac xs)
nipkow@13145
   806
 apply auto
nipkow@13145
   807
done
wenzelm@13114
   808
wenzelm@13142
   809
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
nipkow@13145
   810
apply (induct n)
nipkow@13145
   811
 apply auto
nipkow@13145
   812
apply (case_tac xs)
nipkow@13145
   813
 apply auto
nipkow@13145
   814
done
wenzelm@13114
   815
wenzelm@13114
   816
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
nipkow@13145
   817
apply (induct n)
nipkow@13145
   818
 apply auto
nipkow@13145
   819
apply (case_tac xs)
nipkow@13145
   820
 apply auto
nipkow@13145
   821
done
wenzelm@13114
   822
wenzelm@13142
   823
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
nipkow@13145
   824
apply (induct n)
nipkow@13145
   825
 apply auto
nipkow@13145
   826
apply (case_tac xs)
nipkow@13145
   827
 apply auto
nipkow@13145
   828
done
wenzelm@13114
   829
wenzelm@13114
   830
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@13145
   831
apply (induct xs)
nipkow@13145
   832
 apply auto
nipkow@13145
   833
apply (case_tac i)
nipkow@13145
   834
 apply auto
nipkow@13145
   835
done
wenzelm@13114
   836
wenzelm@13114
   837
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@13145
   838
apply (induct xs)
nipkow@13145
   839
 apply auto
nipkow@13145
   840
apply (case_tac i)
nipkow@13145
   841
 apply auto
nipkow@13145
   842
done
wenzelm@13114
   843
wenzelm@13142
   844
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
nipkow@13145
   845
apply (induct xs)
nipkow@13145
   846
 apply auto
nipkow@13145
   847
apply (case_tac n)
nipkow@13145
   848
 apply(blast )
nipkow@13145
   849
apply (case_tac i)
nipkow@13145
   850
 apply auto
nipkow@13145
   851
done
wenzelm@13114
   852
wenzelm@13142
   853
lemma nth_drop [simp]:
nipkow@13145
   854
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@13145
   855
apply (induct n)
nipkow@13145
   856
 apply auto
nipkow@13145
   857
apply (case_tac xs)
nipkow@13145
   858
 apply auto
nipkow@13145
   859
done
nipkow@3507
   860
wenzelm@13114
   861
lemma append_eq_conv_conj:
nipkow@13145
   862
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@13145
   863
apply(induct xs)
nipkow@13145
   864
 apply simp
nipkow@13145
   865
apply clarsimp
nipkow@13145
   866
apply (case_tac zs)
nipkow@13145
   867
apply auto
nipkow@13145
   868
done
wenzelm@13142
   869
wenzelm@13114
   870
wenzelm@13142
   871
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
   872
wenzelm@13142
   873
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
   874
by (induct xs) auto
wenzelm@13114
   875
wenzelm@13142
   876
lemma takeWhile_append1 [simp]:
nipkow@13145
   877
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
   878
by (induct xs) auto
wenzelm@13114
   879
wenzelm@13142
   880
lemma takeWhile_append2 [simp]:
nipkow@13145
   881
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
   882
by (induct xs) auto
wenzelm@13114
   883
wenzelm@13142
   884
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
   885
by (induct xs) auto
wenzelm@13114
   886
wenzelm@13142
   887
lemma dropWhile_append1 [simp]:
nipkow@13145
   888
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
   889
by (induct xs) auto
wenzelm@13114
   890
wenzelm@13142
   891
lemma dropWhile_append2 [simp]:
nipkow@13145
   892
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
   893
by (induct xs) auto
wenzelm@13114
   894
wenzelm@13142
   895
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
   896
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   897
nipkow@13913
   898
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
   899
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   900
by(induct xs, auto)
nipkow@13913
   901
nipkow@13913
   902
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
   903
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   904
by(induct xs, auto)
nipkow@13913
   905
nipkow@13913
   906
lemma dropWhile_eq_Cons_conv:
nipkow@13913
   907
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
   908
by(induct xs, auto)
nipkow@13913
   909
wenzelm@13114
   910
wenzelm@13142
   911
subsection {* @{text zip} *}
wenzelm@13114
   912
wenzelm@13142
   913
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
   914
by (induct ys) auto
wenzelm@13114
   915
wenzelm@13142
   916
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
   917
by simp
wenzelm@13114
   918
wenzelm@13142
   919
declare zip_Cons [simp del]
wenzelm@13114
   920
wenzelm@13142
   921
lemma length_zip [simp]:
nipkow@13145
   922
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
nipkow@13145
   923
apply(induct ys)
nipkow@13145
   924
 apply simp
nipkow@13145
   925
apply (case_tac xs)
nipkow@13145
   926
 apply auto
nipkow@13145
   927
done
wenzelm@13114
   928
wenzelm@13114
   929
lemma zip_append1:
nipkow@13145
   930
"!!xs. zip (xs @ ys) zs =
nipkow@13145
   931
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
nipkow@13145
   932
apply (induct zs)
nipkow@13145
   933
 apply simp
nipkow@13145
   934
apply (case_tac xs)
nipkow@13145
   935
 apply simp_all
nipkow@13145
   936
done
wenzelm@13114
   937
wenzelm@13114
   938
lemma zip_append2:
nipkow@13145
   939
"!!ys. zip xs (ys @ zs) =
nipkow@13145
   940
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
nipkow@13145
   941
apply (induct xs)
nipkow@13145
   942
 apply simp
nipkow@13145
   943
apply (case_tac ys)
nipkow@13145
   944
 apply simp_all
nipkow@13145
   945
done
wenzelm@13114
   946
wenzelm@13142
   947
lemma zip_append [simp]:
wenzelm@13142
   948
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
   949
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
   950
by (simp add: zip_append1)
wenzelm@13114
   951
wenzelm@13114
   952
lemma zip_rev:
nipkow@13145
   953
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@13145
   954
apply(induct ys)
nipkow@13145
   955
 apply simp
nipkow@13145
   956
apply (case_tac xs)
nipkow@13145
   957
 apply simp_all
nipkow@13145
   958
done
wenzelm@13114
   959
wenzelm@13142
   960
lemma nth_zip [simp]:
nipkow@13145
   961
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@13145
   962
apply (induct ys)
nipkow@13145
   963
 apply simp
nipkow@13145
   964
apply (case_tac xs)
nipkow@13145
   965
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
   966
done
wenzelm@13114
   967
wenzelm@13114
   968
lemma set_zip:
nipkow@13145
   969
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
   970
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
   971
wenzelm@13114
   972
lemma zip_update:
nipkow@13145
   973
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
   974
by (rule sym, simp add: update_zip)
wenzelm@13114
   975
wenzelm@13142
   976
lemma zip_replicate [simp]:
nipkow@13145
   977
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@13145
   978
apply (induct i)
nipkow@13145
   979
 apply auto
nipkow@13145
   980
apply (case_tac j)
nipkow@13145
   981
 apply auto
nipkow@13145
   982
done
wenzelm@13114
   983
wenzelm@13142
   984
wenzelm@13142
   985
subsection {* @{text list_all2} *}
wenzelm@13114
   986
wenzelm@13114
   987
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
   988
by (simp add: list_all2_def)
wenzelm@13114
   989
wenzelm@13142
   990
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
   991
by (simp add: list_all2_def)
wenzelm@13114
   992
wenzelm@13142
   993
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
   994
by (simp add: list_all2_def)
wenzelm@13114
   995
wenzelm@13142
   996
lemma list_all2_Cons [iff]:
nipkow@13145
   997
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
   998
by (auto simp add: list_all2_def)
wenzelm@13114
   999
wenzelm@13114
  1000
lemma list_all2_Cons1:
nipkow@13145
  1001
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1002
by (cases ys) auto
wenzelm@13114
  1003
wenzelm@13114
  1004
lemma list_all2_Cons2:
nipkow@13145
  1005
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1006
by (cases xs) auto
wenzelm@13114
  1007
wenzelm@13142
  1008
lemma list_all2_rev [iff]:
nipkow@13145
  1009
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1010
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1011
kleing@13863
  1012
lemma list_all2_rev1:
kleing@13863
  1013
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1014
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1015
wenzelm@13114
  1016
lemma list_all2_append1:
nipkow@13145
  1017
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1018
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1019
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1020
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1021
apply (rule iffI)
nipkow@13145
  1022
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1023
 apply (rule_tac x = "drop (length xs) zs" in exI)
nipkow@13145
  1024
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
  1025
apply clarify
nipkow@13145
  1026
apply (simp add: ball_Un)
nipkow@13145
  1027
done
wenzelm@13114
  1028
wenzelm@13114
  1029
lemma list_all2_append2:
nipkow@13145
  1030
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1031
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1032
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1033
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1034
apply (rule iffI)
nipkow@13145
  1035
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1036
 apply (rule_tac x = "drop (length ys) xs" in exI)
nipkow@13145
  1037
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
  1038
apply clarify
nipkow@13145
  1039
apply (simp add: ball_Un)
nipkow@13145
  1040
done
wenzelm@13114
  1041
kleing@13863
  1042
lemma list_all2_append:
kleing@13863
  1043
  "\<And>b. length a = length b \<Longrightarrow>
kleing@13863
  1044
  list_all2 P (a@c) (b@d) = (list_all2 P a b \<and> list_all2 P c d)"
kleing@13863
  1045
  apply (induct a)
kleing@13863
  1046
   apply simp
kleing@13863
  1047
  apply (case_tac b)
kleing@13863
  1048
  apply auto
kleing@13863
  1049
  done
kleing@13863
  1050
kleing@13863
  1051
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1052
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1053
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1054
wenzelm@13114
  1055
lemma list_all2_conv_all_nth:
nipkow@13145
  1056
"list_all2 P xs ys =
nipkow@13145
  1057
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1058
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1059
berghofe@13883
  1060
lemma list_all2_trans:
berghofe@13883
  1061
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1062
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1063
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1064
proof (induct as)
berghofe@13883
  1065
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1066
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1067
  proof (induct bs)
berghofe@13883
  1068
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1069
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1070
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1071
  qed simp
berghofe@13883
  1072
qed simp
berghofe@13883
  1073
kleing@13863
  1074
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1075
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1076
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1077
kleing@13863
  1078
lemma list_all2_nthD [dest?]:
kleing@13863
  1079
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1080
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1081
kleing@13863
  1082
lemma list_all2_map1: 
kleing@13863
  1083
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1084
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1085
kleing@13863
  1086
lemma list_all2_map2: 
kleing@13863
  1087
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1088
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1089
kleing@13863
  1090
lemma list_all2_refl:
kleing@13863
  1091
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1092
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1093
kleing@13863
  1094
lemma list_all2_update_cong:
kleing@13863
  1095
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1096
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1097
kleing@13863
  1098
lemma list_all2_update_cong2:
kleing@13863
  1099
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1100
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1101
kleing@13863
  1102
lemma list_all2_dropI [intro?]:
kleing@13863
  1103
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
kleing@13863
  1104
  apply (induct as)
kleing@13863
  1105
   apply simp
kleing@13863
  1106
  apply (clarsimp simp add: list_all2_Cons1)
kleing@13863
  1107
  apply (case_tac n)
kleing@13863
  1108
   apply simp
kleing@13863
  1109
  apply simp
kleing@13863
  1110
  done
kleing@13863
  1111
kleing@13863
  1112
lemma list_all2_mono [intro?]:
kleing@13863
  1113
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
kleing@13863
  1114
  apply (induct x)
kleing@13863
  1115
   apply simp
kleing@13863
  1116
  apply (case_tac y)
kleing@13863
  1117
  apply auto
kleing@13863
  1118
  done
kleing@13863
  1119
wenzelm@13142
  1120
wenzelm@13142
  1121
subsection {* @{text foldl} *}
wenzelm@13142
  1122
wenzelm@13142
  1123
lemma foldl_append [simp]:
nipkow@13145
  1124
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1125
by (induct xs) auto
wenzelm@13142
  1126
wenzelm@13142
  1127
text {*
nipkow@13145
  1128
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1129
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1130
*}
wenzelm@13142
  1131
wenzelm@13142
  1132
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1133
by (induct ns) auto
wenzelm@13142
  1134
wenzelm@13142
  1135
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1136
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1137
wenzelm@13142
  1138
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1139
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1140
by (induct ns) auto
wenzelm@13114
  1141
wenzelm@13114
  1142
wenzelm@13142
  1143
subsection {* @{text upto} *}
wenzelm@13114
  1144
wenzelm@13142
  1145
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1146
-- {* Does not terminate! *}
nipkow@13145
  1147
by (induct j) auto
wenzelm@13142
  1148
wenzelm@13142
  1149
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1150
by (subst upt_rec) simp
wenzelm@13114
  1151
wenzelm@13142
  1152
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1153
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1154
by simp
wenzelm@13114
  1155
wenzelm@13142
  1156
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1157
apply(rule trans)
nipkow@13145
  1158
apply(subst upt_rec)
nipkow@13145
  1159
 prefer 2 apply(rule refl)
nipkow@13145
  1160
apply simp
nipkow@13145
  1161
done
wenzelm@13114
  1162
wenzelm@13142
  1163
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1164
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1165
by (induct k) auto
wenzelm@13114
  1166
wenzelm@13142
  1167
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1168
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1169
wenzelm@13142
  1170
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1171
apply (induct j)
nipkow@13145
  1172
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1173
done
wenzelm@13114
  1174
wenzelm@13142
  1175
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
nipkow@13145
  1176
apply (induct m)
nipkow@13145
  1177
 apply simp
nipkow@13145
  1178
apply (subst upt_rec)
nipkow@13145
  1179
apply (rule sym)
nipkow@13145
  1180
apply (subst upt_rec)
nipkow@13145
  1181
apply (simp del: upt.simps)
nipkow@13145
  1182
done
nipkow@3507
  1183
wenzelm@13114
  1184
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1185
by (induct n) auto
wenzelm@13114
  1186
wenzelm@13114
  1187
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1188
apply (induct n m rule: diff_induct)
nipkow@13145
  1189
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1190
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1191
done
wenzelm@13114
  1192
berghofe@13883
  1193
lemma nth_take_lemma:
berghofe@13883
  1194
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1195
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1196
apply (atomize, induct k)
nipkow@13145
  1197
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib)
nipkow@13145
  1198
apply clarify
nipkow@13145
  1199
txt {* Both lists must be non-empty *}
nipkow@13145
  1200
apply (case_tac xs)
nipkow@13145
  1201
 apply simp
nipkow@13145
  1202
apply (case_tac ys)
nipkow@13145
  1203
 apply clarify
nipkow@13145
  1204
 apply (simp (no_asm_use))
nipkow@13145
  1205
apply clarify
nipkow@13145
  1206
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1207
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1208
apply blast
nipkow@13145
  1209
done
wenzelm@13114
  1210
wenzelm@13114
  1211
lemma nth_equalityI:
wenzelm@13114
  1212
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1213
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1214
apply (simp_all add: take_all)
nipkow@13145
  1215
done
wenzelm@13142
  1216
kleing@13863
  1217
(* needs nth_equalityI *)
kleing@13863
  1218
lemma list_all2_antisym:
kleing@13863
  1219
  "\<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> 
kleing@13863
  1220
  \<Longrightarrow> xs = ys"
kleing@13863
  1221
  apply (simp add: list_all2_conv_all_nth) 
kleing@13863
  1222
  apply (rule nth_equalityI)
kleing@13863
  1223
   apply blast
kleing@13863
  1224
  apply simp
kleing@13863
  1225
  done
kleing@13863
  1226
wenzelm@13142
  1227
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1228
-- {* The famous take-lemma. *}
nipkow@13145
  1229
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1230
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1231
done
wenzelm@13142
  1232
wenzelm@13142
  1233
wenzelm@13142
  1234
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1235
wenzelm@13142
  1236
lemma distinct_append [simp]:
nipkow@13145
  1237
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1238
by (induct xs) auto
wenzelm@13142
  1239
wenzelm@13142
  1240
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1241
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1242
wenzelm@13142
  1243
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1244
by (induct xs) auto
wenzelm@13142
  1245
wenzelm@13142
  1246
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1247
by (induct xs) auto
wenzelm@13114
  1248
wenzelm@13142
  1249
text {*
nipkow@13145
  1250
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1251
it is useful. *}
wenzelm@13142
  1252
lemma distinct_conv_nth:
nipkow@13145
  1253
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
nipkow@13145
  1254
apply (induct_tac xs)
nipkow@13145
  1255
 apply simp
nipkow@13145
  1256
apply simp
nipkow@13145
  1257
apply (rule iffI)
nipkow@13145
  1258
 apply clarsimp
nipkow@13145
  1259
 apply (case_tac i)
nipkow@13145
  1260
apply (case_tac j)
nipkow@13145
  1261
 apply simp
nipkow@13145
  1262
apply (simp add: set_conv_nth)
nipkow@13145
  1263
 apply (case_tac j)
nipkow@13145
  1264
apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1265
 apply simp
nipkow@13145
  1266
apply (rule conjI)
nipkow@13145
  1267
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1268
 apply (erule_tac x = 0 in allE)
nipkow@13145
  1269
 apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1270
 apply simp
nipkow@13145
  1271
apply clarsimp
nipkow@13145
  1272
apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1273
apply (erule_tac x = "Suc j" in allE)
nipkow@13145
  1274
apply simp
nipkow@13145
  1275
done
wenzelm@13114
  1276
wenzelm@13114
  1277
wenzelm@13142
  1278
subsection {* @{text replicate} *}
wenzelm@13114
  1279
wenzelm@13142
  1280
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1281
by (induct n) auto
nipkow@13124
  1282
wenzelm@13142
  1283
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1284
by (induct n) auto
wenzelm@13114
  1285
wenzelm@13114
  1286
lemma replicate_app_Cons_same:
nipkow@13145
  1287
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1288
by (induct n) auto
wenzelm@13114
  1289
wenzelm@13142
  1290
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
nipkow@13145
  1291
apply(induct n)
nipkow@13145
  1292
 apply simp
nipkow@13145
  1293
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1294
done
wenzelm@13114
  1295
wenzelm@13142
  1296
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1297
by (induct n) auto
wenzelm@13114
  1298
wenzelm@13142
  1299
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1300
by (induct n) auto
wenzelm@13114
  1301
wenzelm@13142
  1302
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1303
by (induct n) auto
wenzelm@13114
  1304
wenzelm@13142
  1305
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1306
by (atomize (full), induct n) auto
wenzelm@13114
  1307
wenzelm@13142
  1308
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
nipkow@13145
  1309
apply(induct n)
nipkow@13145
  1310
 apply simp
nipkow@13145
  1311
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1312
done
wenzelm@13114
  1313
wenzelm@13142
  1314
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1315
by (induct n) auto
wenzelm@13114
  1316
wenzelm@13142
  1317
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1318
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1319
wenzelm@13142
  1320
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1321
by auto
wenzelm@13114
  1322
wenzelm@13142
  1323
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1324
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1325
wenzelm@13114
  1326
wenzelm@13142
  1327
subsection {* Lexcicographic orderings on lists *}
nipkow@3507
  1328
wenzelm@13142
  1329
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@13145
  1330
apply (induct_tac n)
nipkow@13145
  1331
 apply simp
nipkow@13145
  1332
apply simp
nipkow@13145
  1333
apply(rule wf_subset)
nipkow@13145
  1334
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1335
apply(rule wf_prod_fun_image)
paulson@13585
  1336
 prefer 2 apply (rule inj_onI)
nipkow@13145
  1337
apply auto
nipkow@13145
  1338
done
wenzelm@13114
  1339
wenzelm@13114
  1340
lemma lexn_length:
nipkow@13145
  1341
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1342
by (induct n) auto
wenzelm@13114
  1343
wenzelm@13142
  1344
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1345
apply (unfold lex_def)
nipkow@13145
  1346
apply (rule wf_UN)
nipkow@13145
  1347
apply (blast intro: wf_lexn)
nipkow@13145
  1348
apply clarify
nipkow@13145
  1349
apply (rename_tac m n)
nipkow@13145
  1350
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1351
 prefer 2 apply blast
nipkow@13145
  1352
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1353
done
wenzelm@13114
  1354
wenzelm@13114
  1355
lemma lexn_conv:
nipkow@13145
  1356
"lexn r n =
nipkow@13145
  1357
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1358
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1359
apply (induct_tac n)
nipkow@13145
  1360
 apply simp
nipkow@13145
  1361
 apply blast
nipkow@13145
  1362
apply (simp add: image_Collect lex_prod_def)
berghofe@13601
  1363
apply safe
nipkow@13145
  1364
apply blast
berghofe@13601
  1365
 apply (rule_tac x = "ab # xys" in exI)
nipkow@13145
  1366
 apply simp
nipkow@13145
  1367
apply (case_tac xys)
nipkow@13145
  1368
 apply simp_all
nipkow@13145
  1369
apply blast
nipkow@13145
  1370
done
wenzelm@13114
  1371
wenzelm@13114
  1372
lemma lex_conv:
nipkow@13145
  1373
"lex r =
nipkow@13145
  1374
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1375
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1376
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1377
wenzelm@13142
  1378
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1379
by (unfold lexico_def) blast
wenzelm@13114
  1380
wenzelm@13114
  1381
lemma lexico_conv:
nipkow@13145
  1382
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1383
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1384
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1385
wenzelm@13142
  1386
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1387
by (simp add: lex_conv)
wenzelm@13114
  1388
wenzelm@13142
  1389
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1390
by (simp add:lex_conv)
wenzelm@13114
  1391
wenzelm@13142
  1392
lemma Cons_in_lex [iff]:
nipkow@13145
  1393
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1394
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1395
apply (simp add: lex_conv)
nipkow@13145
  1396
apply (rule iffI)
nipkow@13145
  1397
 prefer 2 apply (blast intro: Cons_eq_appendI)
nipkow@13145
  1398
apply clarify
nipkow@13145
  1399
apply (case_tac xys)
nipkow@13145
  1400
 apply simp
nipkow@13145
  1401
apply simp
nipkow@13145
  1402
apply blast
nipkow@13145
  1403
done
wenzelm@13114
  1404
wenzelm@13114
  1405
wenzelm@13142
  1406
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1407
wenzelm@13142
  1408
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1409
by (auto simp add: sublist_def)
wenzelm@13114
  1410
wenzelm@13142
  1411
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1412
by (auto simp add: sublist_def)
wenzelm@13114
  1413
wenzelm@13114
  1414
lemma sublist_shift_lemma:
nipkow@13145
  1415
"map fst [p:zip xs [i..i + length xs(] . snd p : A] =
nipkow@13145
  1416
map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1417
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1418
wenzelm@13114
  1419
lemma sublist_append:
nipkow@13145
  1420
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1421
apply (unfold sublist_def)
nipkow@13145
  1422
apply (induct l' rule: rev_induct)
nipkow@13145
  1423
 apply simp
nipkow@13145
  1424
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1425
apply (simp add: add_commute)
nipkow@13145
  1426
done
wenzelm@13114
  1427
wenzelm@13114
  1428
lemma sublist_Cons:
nipkow@13145
  1429
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1430
apply (induct l rule: rev_induct)
nipkow@13145
  1431
 apply (simp add: sublist_def)
nipkow@13145
  1432
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1433
done
wenzelm@13114
  1434
wenzelm@13142
  1435
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1436
by (simp add: sublist_Cons)
wenzelm@13114
  1437
wenzelm@13142
  1438
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
nipkow@13145
  1439
apply (induct l rule: rev_induct)
nipkow@13145
  1440
 apply simp
nipkow@13145
  1441
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1442
done
wenzelm@13114
  1443
wenzelm@13114
  1444
wenzelm@13142
  1445
lemma take_Cons':
nipkow@13145
  1446
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1447
by (cases n) simp_all
wenzelm@13114
  1448
wenzelm@13142
  1449
lemma drop_Cons':
nipkow@13145
  1450
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1451
by (cases n) simp_all
wenzelm@13114
  1452
wenzelm@13142
  1453
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1454
by (cases n) simp_all
wenzelm@13142
  1455
nipkow@13145
  1456
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1457
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1458
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1459
wenzelm@13462
  1460
wenzelm@13366
  1461
subsection {* Characters and strings *}
wenzelm@13366
  1462
wenzelm@13366
  1463
datatype nibble =
wenzelm@13366
  1464
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1465
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1466
wenzelm@13366
  1467
datatype char = Char nibble nibble
wenzelm@13366
  1468
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1469
wenzelm@13366
  1470
types string = "char list"
wenzelm@13366
  1471
wenzelm@13366
  1472
syntax
wenzelm@13366
  1473
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1474
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1475
wenzelm@13366
  1476
parse_ast_translation {*
wenzelm@13366
  1477
  let
wenzelm@13366
  1478
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1479
wenzelm@13366
  1480
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1481
    fun mk_char c =
wenzelm@13366
  1482
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1483
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1484
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1485
wenzelm@13366
  1486
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1487
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1488
wenzelm@13366
  1489
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1490
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1491
          [c] => mk_char c
wenzelm@13366
  1492
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1493
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1494
wenzelm@13366
  1495
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1496
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1497
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  1498
        | cs => mk_string cs)
wenzelm@13366
  1499
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  1500
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  1501
*}
wenzelm@13366
  1502
wenzelm@13366
  1503
print_ast_translation {*
wenzelm@13366
  1504
  let
wenzelm@13366
  1505
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  1506
        (case explode c of
wenzelm@13366
  1507
          ["N", "i", "b", "b", "l", "e", h] =>
wenzelm@13366
  1508
            if "0" <= h andalso h <= "9" then ord h - ord "0"
wenzelm@13366
  1509
            else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
wenzelm@13366
  1510
            else raise Match
wenzelm@13366
  1511
        | _ => raise Match)
wenzelm@13366
  1512
      | dest_nib _ = raise Match;
wenzelm@13366
  1513
wenzelm@13366
  1514
    fun dest_chr c1 c2 =
wenzelm@13366
  1515
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  1516
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  1517
wenzelm@13366
  1518
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  1519
      | dest_char _ = raise Match;
wenzelm@13366
  1520
wenzelm@13366
  1521
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  1522
wenzelm@13366
  1523
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  1524
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  1525
wenzelm@13366
  1526
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  1527
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  1528
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  1529
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  1530
*}
wenzelm@13366
  1531
wenzelm@13122
  1532
end