src/HOL/List.thy
author wenzelm
Thu May 24 17:25:53 2012 +0200 (2012-05-24)
changeset 47988 e4b69e10b990
parent 47841 179b5e7c9803
child 48619 558e4e77ce69
permissions -rw-r--r--
tuned proofs;
wenzelm@13462
     1
(*  Title:      HOL/List.thy
wenzelm@13462
     2
    Author:     Tobias Nipkow
clasohm@923
     3
*)
clasohm@923
     4
wenzelm@13114
     5
header {* The datatype of finite lists *}
wenzelm@13122
     6
nipkow@15131
     7
theory List
krauss@44013
     8
imports Plain Presburger Code_Numeral Quotient ATP
bulwahn@41463
     9
uses
bulwahn@41463
    10
  ("Tools/list_code.ML")
bulwahn@41463
    11
  ("Tools/list_to_set_comprehension.ML")
nipkow@15131
    12
begin
clasohm@923
    13
wenzelm@13142
    14
datatype 'a list =
wenzelm@13366
    15
    Nil    ("[]")
wenzelm@13366
    16
  | Cons 'a  "'a list"    (infixr "#" 65)
clasohm@923
    17
haftmann@34941
    18
syntax
haftmann@34941
    19
  -- {* list Enumeration *}
wenzelm@35115
    20
  "_list" :: "args => 'a list"    ("[(_)]")
haftmann@34941
    21
haftmann@34941
    22
translations
haftmann@34941
    23
  "[x, xs]" == "x#[xs]"
haftmann@34941
    24
  "[x]" == "x#[]"
haftmann@34941
    25
wenzelm@35115
    26
wenzelm@35115
    27
subsection {* Basic list processing functions *}
nipkow@15302
    28
haftmann@34941
    29
primrec
haftmann@34941
    30
  hd :: "'a list \<Rightarrow> 'a" where
haftmann@34941
    31
  "hd (x # xs) = x"
haftmann@34941
    32
haftmann@34941
    33
primrec
haftmann@34941
    34
  tl :: "'a list \<Rightarrow> 'a list" where
haftmann@34941
    35
    "tl [] = []"
haftmann@34941
    36
  | "tl (x # xs) = xs"
haftmann@34941
    37
haftmann@34941
    38
primrec
haftmann@34941
    39
  last :: "'a list \<Rightarrow> 'a" where
haftmann@34941
    40
  "last (x # xs) = (if xs = [] then x else last xs)"
haftmann@34941
    41
haftmann@34941
    42
primrec
haftmann@34941
    43
  butlast :: "'a list \<Rightarrow> 'a list" where
haftmann@34941
    44
    "butlast []= []"
haftmann@34941
    45
  | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
haftmann@34941
    46
haftmann@34941
    47
primrec
haftmann@34941
    48
  set :: "'a list \<Rightarrow> 'a set" where
haftmann@34941
    49
    "set [] = {}"
haftmann@34941
    50
  | "set (x # xs) = insert x (set xs)"
haftmann@34941
    51
haftmann@46133
    52
definition
haftmann@46133
    53
  coset :: "'a list \<Rightarrow> 'a set" where
haftmann@46133
    54
  [simp]: "coset xs = - set xs"
haftmann@46133
    55
haftmann@34941
    56
primrec
haftmann@34941
    57
  map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
haftmann@34941
    58
    "map f [] = []"
haftmann@34941
    59
  | "map f (x # xs) = f x # map f xs"
haftmann@34941
    60
haftmann@34941
    61
primrec
haftmann@34941
    62
  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
haftmann@34941
    63
    append_Nil:"[] @ ys = ys"
haftmann@34941
    64
  | append_Cons: "(x#xs) @ ys = x # xs @ ys"
haftmann@34941
    65
haftmann@34941
    66
primrec
haftmann@34941
    67
  rev :: "'a list \<Rightarrow> 'a list" where
haftmann@34941
    68
    "rev [] = []"
haftmann@34941
    69
  | "rev (x # xs) = rev xs @ [x]"
haftmann@34941
    70
haftmann@34941
    71
primrec
haftmann@34941
    72
  filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
    73
    "filter P [] = []"
haftmann@34941
    74
  | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
haftmann@34941
    75
haftmann@34941
    76
syntax
haftmann@34941
    77
  -- {* Special syntax for filter *}
wenzelm@35115
    78
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
haftmann@34941
    79
haftmann@34941
    80
translations
haftmann@34941
    81
  "[x<-xs . P]"== "CONST filter (%x. P) xs"
haftmann@34941
    82
haftmann@34941
    83
syntax (xsymbols)
wenzelm@35115
    84
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
haftmann@34941
    85
syntax (HTML output)
wenzelm@35115
    86
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
haftmann@34941
    87
haftmann@47397
    88
primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@47397
    89
where
haftmann@47397
    90
  fold_Nil:  "fold f [] = id"
haftmann@47397
    91
| fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x" -- {* natural argument order *}
haftmann@47397
    92
haftmann@47397
    93
primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@47397
    94
where
haftmann@47397
    95
  foldr_Nil:  "foldr f [] = id"
haftmann@47397
    96
| foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs" -- {* natural argument order *}
haftmann@47397
    97
haftmann@47397
    98
primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
haftmann@47397
    99
where
haftmann@47397
   100
  foldl_Nil:  "foldl f a [] = a"
haftmann@47397
   101
| foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
haftmann@34941
   102
haftmann@34941
   103
primrec
haftmann@34941
   104
  concat:: "'a list list \<Rightarrow> 'a list" where
haftmann@34941
   105
    "concat [] = []"
haftmann@34941
   106
  | "concat (x # xs) = x @ concat xs"
haftmann@34941
   107
haftmann@39774
   108
definition (in monoid_add)
haftmann@34941
   109
  listsum :: "'a list \<Rightarrow> 'a" where
haftmann@39774
   110
  "listsum xs = foldr plus xs 0"
haftmann@34941
   111
haftmann@34941
   112
primrec
haftmann@34941
   113
  drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   114
    drop_Nil: "drop n [] = []"
haftmann@34941
   115
  | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
haftmann@34941
   116
  -- {*Warning: simpset does not contain this definition, but separate
haftmann@34941
   117
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
haftmann@34941
   118
haftmann@34941
   119
primrec
haftmann@34941
   120
  take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   121
    take_Nil:"take n [] = []"
haftmann@34941
   122
  | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
haftmann@34941
   123
  -- {*Warning: simpset does not contain this definition, but separate
haftmann@34941
   124
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
haftmann@34941
   125
haftmann@34941
   126
primrec
haftmann@34941
   127
  nth :: "'a list => nat => 'a" (infixl "!" 100) where
haftmann@34941
   128
  nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
haftmann@34941
   129
  -- {*Warning: simpset does not contain this definition, but separate
haftmann@34941
   130
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
haftmann@34941
   131
haftmann@34941
   132
primrec
haftmann@34941
   133
  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
haftmann@34941
   134
    "list_update [] i v = []"
haftmann@34941
   135
  | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
clasohm@923
   136
wenzelm@41229
   137
nonterminal lupdbinds and lupdbind
nipkow@5077
   138
clasohm@923
   139
syntax
wenzelm@13366
   140
  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
wenzelm@13366
   141
  "" :: "lupdbind => lupdbinds"    ("_")
wenzelm@13366
   142
  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
wenzelm@13366
   143
  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
nipkow@5077
   144
clasohm@923
   145
translations
wenzelm@35115
   146
  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
haftmann@34941
   147
  "xs[i:=x]" == "CONST list_update xs i x"
haftmann@34941
   148
haftmann@34941
   149
primrec
haftmann@34941
   150
  takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   151
    "takeWhile P [] = []"
haftmann@34941
   152
  | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
haftmann@34941
   153
haftmann@34941
   154
primrec
haftmann@34941
   155
  dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   156
    "dropWhile P [] = []"
haftmann@34941
   157
  | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
haftmann@34941
   158
haftmann@34941
   159
primrec
haftmann@34941
   160
  zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
haftmann@34941
   161
    "zip xs [] = []"
haftmann@34941
   162
  | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
haftmann@34941
   163
  -- {*Warning: simpset does not contain this definition, but separate
haftmann@34941
   164
       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
haftmann@34941
   165
haftmann@34941
   166
primrec 
haftmann@34941
   167
  upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
haftmann@34941
   168
    upt_0: "[i..<0] = []"
haftmann@34941
   169
  | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
haftmann@34941
   170
haftmann@34978
   171
definition
haftmann@34978
   172
  insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34978
   173
  "insert x xs = (if x \<in> set xs then xs else x # xs)"
haftmann@34978
   174
wenzelm@36176
   175
hide_const (open) insert
wenzelm@36176
   176
hide_fact (open) insert_def
haftmann@34978
   177
nipkow@47122
   178
primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
nipkow@47122
   179
  "find _ [] = None"
nipkow@47122
   180
| "find P (x#xs) = (if P x then Some x else find P xs)"
nipkow@47122
   181
nipkow@47122
   182
hide_const (open) find
nipkow@47122
   183
haftmann@34941
   184
primrec
haftmann@34941
   185
  remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   186
    "remove1 x [] = []"
haftmann@34941
   187
  | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
haftmann@34941
   188
haftmann@34941
   189
primrec
haftmann@34941
   190
  removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@34941
   191
    "removeAll x [] = []"
haftmann@34941
   192
  | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
haftmann@34941
   193
haftmann@40122
   194
primrec
haftmann@39915
   195
  distinct :: "'a list \<Rightarrow> bool" where
haftmann@40122
   196
    "distinct [] \<longleftrightarrow> True"
haftmann@40122
   197
  | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
haftmann@39915
   198
haftmann@39915
   199
primrec
haftmann@39915
   200
  remdups :: "'a list \<Rightarrow> 'a list" where
haftmann@39915
   201
    "remdups [] = []"
haftmann@39915
   202
  | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
haftmann@39915
   203
haftmann@34941
   204
primrec
haftmann@34941
   205
  replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
haftmann@34941
   206
    replicate_0: "replicate 0 x = []"
haftmann@34941
   207
  | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
paulson@3342
   208
wenzelm@13142
   209
text {*
wenzelm@14589
   210
  Function @{text size} is overloaded for all datatypes. Users may
wenzelm@13366
   211
  refer to the list version as @{text length}. *}
wenzelm@13142
   212
wenzelm@19363
   213
abbreviation
haftmann@34941
   214
  length :: "'a list \<Rightarrow> nat" where
haftmann@34941
   215
  "length \<equiv> size"
paulson@15307
   216
blanchet@46440
   217
primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
blanchet@46440
   218
  "rotate1 [] = []" |
blanchet@46440
   219
  "rotate1 (x # xs) = xs @ [x]"
wenzelm@21404
   220
wenzelm@21404
   221
definition
wenzelm@21404
   222
  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
haftmann@30971
   223
  "rotate n = rotate1 ^^ n"
wenzelm@21404
   224
wenzelm@21404
   225
definition
wenzelm@21404
   226
  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
haftmann@37767
   227
  "list_all2 P xs ys =
haftmann@21061
   228
    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
wenzelm@21404
   229
wenzelm@21404
   230
definition
wenzelm@21404
   231
  sublist :: "'a list => nat set => 'a list" where
wenzelm@21404
   232
  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
nipkow@17086
   233
nipkow@40593
   234
fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
nipkow@40593
   235
"splice [] ys = ys" |
nipkow@40593
   236
"splice xs [] = xs" |
nipkow@40593
   237
"splice (x#xs) (y#ys) = x # y # splice xs ys"
haftmann@21061
   238
nipkow@26771
   239
text{*
nipkow@26771
   240
\begin{figure}[htbp]
nipkow@26771
   241
\fbox{
nipkow@26771
   242
\begin{tabular}{l}
wenzelm@27381
   243
@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
wenzelm@27381
   244
@{lemma "length [a,b,c] = 3" by simp}\\
wenzelm@27381
   245
@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
wenzelm@27381
   246
@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
wenzelm@27381
   247
@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
wenzelm@27381
   248
@{lemma "hd [a,b,c,d] = a" by simp}\\
wenzelm@27381
   249
@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
wenzelm@27381
   250
@{lemma "last [a,b,c,d] = d" by simp}\\
wenzelm@27381
   251
@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
wenzelm@27381
   252
@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
wenzelm@27381
   253
@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
haftmann@46133
   254
@{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
haftmann@47397
   255
@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
haftmann@47397
   256
@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
wenzelm@27381
   257
@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
wenzelm@27381
   258
@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
wenzelm@27381
   259
@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
wenzelm@27381
   260
@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
wenzelm@27381
   261
@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
wenzelm@27381
   262
@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
wenzelm@27381
   263
@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
wenzelm@27381
   264
@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
wenzelm@27381
   265
@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
wenzelm@27381
   266
@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
wenzelm@27381
   267
@{lemma "distinct [2,0,1::nat]" by simp}\\
wenzelm@27381
   268
@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
haftmann@34978
   269
@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
haftmann@35295
   270
@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
nipkow@47122
   271
@{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\
nipkow@47122
   272
@{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
wenzelm@27381
   273
@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
nipkow@27693
   274
@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
wenzelm@27381
   275
@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
wenzelm@27381
   276
@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
wenzelm@27381
   277
@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
blanchet@46440
   278
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
blanchet@46440
   279
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
nipkow@40077
   280
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
nipkow@40077
   281
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
haftmann@47397
   282
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
nipkow@26771
   283
\end{tabular}}
nipkow@26771
   284
\caption{Characteristic examples}
nipkow@26771
   285
\label{fig:Characteristic}
nipkow@26771
   286
\end{figure}
blanchet@29927
   287
Figure~\ref{fig:Characteristic} shows characteristic examples
nipkow@26771
   288
that should give an intuitive understanding of the above functions.
nipkow@26771
   289
*}
nipkow@26771
   290
nipkow@24616
   291
text{* The following simple sort functions are intended for proofs,
nipkow@24616
   292
not for efficient implementations. *}
nipkow@24616
   293
wenzelm@25221
   294
context linorder
wenzelm@25221
   295
begin
wenzelm@25221
   296
haftmann@39915
   297
inductive sorted :: "'a list \<Rightarrow> bool" where
haftmann@39915
   298
  Nil [iff]: "sorted []"
haftmann@39915
   299
| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
haftmann@39915
   300
haftmann@39915
   301
lemma sorted_single [iff]:
haftmann@39915
   302
  "sorted [x]"
haftmann@39915
   303
  by (rule sorted.Cons) auto
haftmann@39915
   304
haftmann@39915
   305
lemma sorted_many:
haftmann@39915
   306
  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
haftmann@39915
   307
  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
haftmann@39915
   308
haftmann@39915
   309
lemma sorted_many_eq [simp, code]:
haftmann@39915
   310
  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
haftmann@39915
   311
  by (auto intro: sorted_many elim: sorted.cases)
haftmann@39915
   312
haftmann@39915
   313
lemma [code]:
haftmann@39915
   314
  "sorted [] \<longleftrightarrow> True"
haftmann@39915
   315
  "sorted [x] \<longleftrightarrow> True"
haftmann@39915
   316
  by simp_all
nipkow@24697
   317
hoelzl@33639
   318
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@46133
   319
  "insort_key f x [] = [x]" |
haftmann@46133
   320
  "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
hoelzl@33639
   321
haftmann@35195
   322
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@46133
   323
  "sort_key f xs = foldr (insort_key f) xs []"
hoelzl@33639
   324
haftmann@40210
   325
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
haftmann@40210
   326
  "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
haftmann@40210
   327
hoelzl@33639
   328
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
hoelzl@33639
   329
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
haftmann@40210
   330
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
haftmann@35608
   331
wenzelm@25221
   332
end
wenzelm@25221
   333
nipkow@24616
   334
wenzelm@23388
   335
subsubsection {* List comprehension *}
nipkow@23192
   336
nipkow@24349
   337
text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349
   338
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
nipkow@24349
   339
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
nipkow@24349
   340
The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349
   341
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
nipkow@24349
   342
\verb![e| x <- xs, ...]!.
nipkow@24349
   343
nipkow@24349
   344
The qualifiers after the dot are
nipkow@24349
   345
\begin{description}
nipkow@24349
   346
\item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476
   347
 where @{text p} is a pattern and @{text xs} an expression of list type, or
nipkow@24476
   348
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476
   349
%\item[local bindings] @ {text"let x = e"}.
nipkow@24349
   350
\end{description}
nipkow@23240
   351
nipkow@24476
   352
Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476
   353
misunderstandings, the translation into desugared form is not reversed
nipkow@24476
   354
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476
   355
optmized to @{term"map (%x. e) xs"}.
nipkow@23240
   356
nipkow@24349
   357
It is easy to write short list comprehensions which stand for complex
nipkow@24349
   358
expressions. During proofs, they may become unreadable (and
nipkow@24349
   359
mangled). In such cases it can be advisable to introduce separate
nipkow@24349
   360
definitions for the list comprehensions in question.  *}
nipkow@24349
   361
wenzelm@46138
   362
nonterminal lc_qual and lc_quals
nipkow@23192
   363
nipkow@23192
   364
syntax
wenzelm@46138
   365
  "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
wenzelm@46138
   366
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
wenzelm@46138
   367
  "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
wenzelm@46138
   368
  (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
wenzelm@46138
   369
  "_lc_end" :: "lc_quals" ("]")
wenzelm@46138
   370
  "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
wenzelm@46138
   371
  "_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   372
nipkow@24476
   373
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   374
   translation of [e. p<-xs]
nipkow@23192
   375
translations
wenzelm@46138
   376
  "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
wenzelm@46138
   377
  "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
wenzelm@46138
   378
   => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
wenzelm@46138
   379
  "[e. P]" => "if P then [e] else []"
wenzelm@46138
   380
  "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
wenzelm@46138
   381
   => "if P then (_listcompr e Q Qs) else []"
wenzelm@46138
   382
  "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
wenzelm@46138
   383
   => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   384
*)
nipkow@23240
   385
nipkow@23279
   386
syntax (xsymbols)
wenzelm@46138
   387
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@23279
   388
syntax (HTML output)
wenzelm@46138
   389
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
nipkow@24349
   390
nipkow@24349
   391
parse_translation (advanced) {*
wenzelm@46138
   392
  let
wenzelm@46138
   393
    val NilC = Syntax.const @{const_syntax Nil};
wenzelm@46138
   394
    val ConsC = Syntax.const @{const_syntax Cons};
wenzelm@46138
   395
    val mapC = Syntax.const @{const_syntax map};
wenzelm@46138
   396
    val concatC = Syntax.const @{const_syntax concat};
wenzelm@46138
   397
    val IfC = Syntax.const @{const_syntax If};
wenzelm@46138
   398
wenzelm@46138
   399
    fun single x = ConsC $ x $ NilC;
wenzelm@46138
   400
wenzelm@46138
   401
    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
wenzelm@46138
   402
      let
wenzelm@46138
   403
        (* FIXME proper name context!? *)
wenzelm@46138
   404
        val x =
wenzelm@46138
   405
          Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
wenzelm@46138
   406
        val e = if opti then single e else e;
wenzelm@46138
   407
        val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
wenzelm@46138
   408
        val case2 =
wenzelm@46138
   409
          Syntax.const @{syntax_const "_case1"} $
wenzelm@46138
   410
            Syntax.const @{const_syntax dummy_pattern} $ NilC;
wenzelm@46138
   411
        val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
wenzelm@46138
   412
      in Syntax_Trans.abs_tr [x, Datatype_Case.case_tr false ctxt [x, cs]] end;
wenzelm@46138
   413
wenzelm@46138
   414
    fun abs_tr ctxt p e opti =
wenzelm@46138
   415
      (case Term_Position.strip_positions p of
wenzelm@46138
   416
        Free (s, T) =>
wenzelm@46138
   417
          let
wenzelm@46138
   418
            val thy = Proof_Context.theory_of ctxt;
wenzelm@46138
   419
            val s' = Proof_Context.intern_const ctxt s;
wenzelm@46138
   420
          in
wenzelm@46138
   421
            if Sign.declared_const thy s'
wenzelm@46138
   422
            then (pat_tr ctxt p e opti, false)
wenzelm@46138
   423
            else (Syntax_Trans.abs_tr [p, e], true)
wenzelm@46138
   424
          end
wenzelm@46138
   425
      | _ => (pat_tr ctxt p e opti, false));
wenzelm@46138
   426
wenzelm@46138
   427
    fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
wenzelm@46138
   428
          let
wenzelm@46138
   429
            val res =
wenzelm@46138
   430
              (case qs of
wenzelm@46138
   431
                Const (@{syntax_const "_lc_end"}, _) => single e
wenzelm@46138
   432
              | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
wenzelm@46138
   433
          in IfC $ b $ res $ NilC end
wenzelm@46138
   434
      | lc_tr ctxt
wenzelm@46138
   435
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   436
              Const(@{syntax_const "_lc_end"}, _)] =
wenzelm@46138
   437
          (case abs_tr ctxt p e true of
wenzelm@46138
   438
            (f, true) => mapC $ f $ es
wenzelm@46138
   439
          | (f, false) => concatC $ (mapC $ f $ es))
wenzelm@46138
   440
      | lc_tr ctxt
wenzelm@46138
   441
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
wenzelm@46138
   442
              Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
wenzelm@46138
   443
          let val e' = lc_tr ctxt [e, q, qs];
wenzelm@46138
   444
          in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
wenzelm@46138
   445
wenzelm@46138
   446
  in [(@{syntax_const "_listcompr"}, lc_tr)] end
nipkow@24349
   447
*}
nipkow@23279
   448
wenzelm@42167
   449
ML {*
wenzelm@42167
   450
  let
wenzelm@42167
   451
    val read = Syntax.read_term @{context};
wenzelm@42167
   452
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
wenzelm@42167
   453
  in
wenzelm@42167
   454
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
wenzelm@42167
   455
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
wenzelm@42167
   456
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
wenzelm@42167
   457
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
wenzelm@42167
   458
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
wenzelm@42167
   459
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
wenzelm@42167
   460
    check "[(x,y). Cons True x \<leftarrow> xs]"
wenzelm@42167
   461
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
wenzelm@42167
   462
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
wenzelm@42167
   463
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
wenzelm@42167
   464
    check "[(x,y,z). x<a, x>b, x=d]"
wenzelm@42167
   465
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
wenzelm@42167
   466
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
wenzelm@42167
   467
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
wenzelm@42167
   468
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
wenzelm@42167
   469
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
wenzelm@42167
   470
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
wenzelm@42167
   471
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
wenzelm@42167
   472
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
wenzelm@42167
   473
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
wenzelm@42167
   474
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
wenzelm@42167
   475
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
wenzelm@42167
   476
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
wenzelm@42167
   477
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
wenzelm@42167
   478
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
wenzelm@42167
   479
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
wenzelm@42167
   480
  end;
wenzelm@42167
   481
*}
wenzelm@42167
   482
wenzelm@35115
   483
(*
nipkow@24349
   484
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   485
*)
nipkow@23192
   486
wenzelm@42167
   487
bulwahn@41463
   488
use "Tools/list_to_set_comprehension.ML"
bulwahn@41463
   489
bulwahn@41463
   490
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
bulwahn@41463
   491
haftmann@46133
   492
code_datatype set coset
haftmann@46133
   493
haftmann@46133
   494
hide_const (open) coset
wenzelm@35115
   495
haftmann@21061
   496
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   497
haftmann@21061
   498
lemma not_Cons_self [simp]:
haftmann@21061
   499
  "xs \<noteq> x # xs"
nipkow@13145
   500
by (induct xs) auto
wenzelm@13114
   501
wenzelm@41697
   502
lemma not_Cons_self2 [simp]:
wenzelm@41697
   503
  "x # xs \<noteq> xs"
wenzelm@41697
   504
by (rule not_Cons_self [symmetric])
wenzelm@13114
   505
wenzelm@13142
   506
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   507
by (induct xs) auto
wenzelm@13114
   508
wenzelm@13142
   509
lemma length_induct:
haftmann@21061
   510
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   511
by (rule measure_induct [of length]) iprover
wenzelm@13114
   512
haftmann@37289
   513
lemma list_nonempty_induct [consumes 1, case_names single cons]:
haftmann@37289
   514
  assumes "xs \<noteq> []"
haftmann@37289
   515
  assumes single: "\<And>x. P [x]"
haftmann@37289
   516
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
haftmann@37289
   517
  shows "P xs"
haftmann@37289
   518
using `xs \<noteq> []` proof (induct xs)
haftmann@37289
   519
  case Nil then show ?case by simp
haftmann@37289
   520
next
haftmann@37289
   521
  case (Cons x xs) show ?case proof (cases xs)
haftmann@37289
   522
    case Nil with single show ?thesis by simp
haftmann@37289
   523
  next
haftmann@37289
   524
    case Cons then have "xs \<noteq> []" by simp
haftmann@37289
   525
    moreover with Cons.hyps have "P xs" .
haftmann@37289
   526
    ultimately show ?thesis by (rule cons)
haftmann@37289
   527
  qed
haftmann@37289
   528
qed
haftmann@37289
   529
hoelzl@45714
   530
lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
hoelzl@45714
   531
  by (auto intro!: inj_onI)
wenzelm@13114
   532
haftmann@21061
   533
subsubsection {* @{const length} *}
wenzelm@13114
   534
wenzelm@13142
   535
text {*
haftmann@21061
   536
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   537
  append_eq_append_conv}.
wenzelm@13142
   538
*}
wenzelm@13114
   539
wenzelm@13142
   540
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   541
by (induct xs) auto
wenzelm@13114
   542
wenzelm@13142
   543
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   544
by (induct xs) auto
wenzelm@13114
   545
wenzelm@13142
   546
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   547
by (induct xs) auto
wenzelm@13114
   548
wenzelm@13142
   549
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   550
by (cases xs) auto
wenzelm@13114
   551
wenzelm@13142
   552
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   553
by (induct xs) auto
wenzelm@13114
   554
wenzelm@13142
   555
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   556
by (induct xs) auto
wenzelm@13114
   557
nipkow@23479
   558
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   559
by auto
nipkow@23479
   560
wenzelm@13114
   561
lemma length_Suc_conv:
nipkow@13145
   562
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   563
by (induct xs) auto
wenzelm@13142
   564
nipkow@14025
   565
lemma Suc_length_conv:
nipkow@14025
   566
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   567
apply (induct xs, simp, simp)
nipkow@14025
   568
apply blast
nipkow@14025
   569
done
nipkow@14025
   570
wenzelm@25221
   571
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   572
  by (induct xs) auto
wenzelm@25221
   573
haftmann@26442
   574
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   575
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   576
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   577
   \<Longrightarrow> P xs ys"
haftmann@26442
   578
proof (induct xs arbitrary: ys)
haftmann@26442
   579
  case Nil then show ?case by simp
haftmann@26442
   580
next
haftmann@26442
   581
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   582
qed
haftmann@26442
   583
haftmann@26442
   584
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   585
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   586
   (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
haftmann@26442
   587
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   588
proof (induct xs arbitrary: ys zs)
haftmann@26442
   589
  case Nil then show ?case by simp
haftmann@26442
   590
next
haftmann@26442
   591
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   592
    (cases zs, simp_all)
haftmann@26442
   593
qed
wenzelm@13114
   594
kaliszyk@36154
   595
lemma list_induct4 [consumes 3, case_names Nil Cons]:
kaliszyk@36154
   596
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
kaliszyk@36154
   597
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
kaliszyk@36154
   598
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
kaliszyk@36154
   599
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
kaliszyk@36154
   600
proof (induct xs arbitrary: ys zs ws)
kaliszyk@36154
   601
  case Nil then show ?case by simp
kaliszyk@36154
   602
next
kaliszyk@36154
   603
  case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
kaliszyk@36154
   604
qed
kaliszyk@36154
   605
krauss@22493
   606
lemma list_induct2': 
krauss@22493
   607
  "\<lbrakk> P [] [];
krauss@22493
   608
  \<And>x xs. P (x#xs) [];
krauss@22493
   609
  \<And>y ys. P [] (y#ys);
krauss@22493
   610
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   611
 \<Longrightarrow> P xs ys"
krauss@22493
   612
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   613
nipkow@22143
   614
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   615
by (rule Eq_FalseI) auto
wenzelm@24037
   616
wenzelm@24037
   617
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   618
(*
nipkow@22143
   619
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   620
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   621
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   622
*)
wenzelm@24037
   623
wenzelm@24037
   624
let
nipkow@22143
   625
huffman@29856
   626
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   627
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   628
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   629
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   630
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   631
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   632
wenzelm@24037
   633
fun list_neq _ ss ct =
nipkow@22143
   634
  let
wenzelm@24037
   635
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   636
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   637
    fun prove_neq() =
nipkow@22143
   638
      let
nipkow@22143
   639
        val Type(_,listT::_) = eqT;
haftmann@22994
   640
        val size = HOLogic.size_const listT;
nipkow@22143
   641
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   642
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   643
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   644
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   645
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   646
  in
wenzelm@23214
   647
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   648
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   649
    then prove_neq() else NONE
nipkow@22143
   650
  end;
wenzelm@24037
   651
in list_neq end;
nipkow@22143
   652
*}
nipkow@22143
   653
nipkow@22143
   654
nipkow@15392
   655
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   656
wenzelm@13142
   657
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   658
by (induct xs) auto
wenzelm@13114
   659
wenzelm@13142
   660
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   661
by (induct xs) auto
nipkow@3507
   662
wenzelm@13142
   663
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   664
by (induct xs) auto
wenzelm@13114
   665
wenzelm@13142
   666
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   667
by (induct xs) auto
wenzelm@13114
   668
wenzelm@13142
   669
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   670
by (induct xs) auto
wenzelm@13114
   671
wenzelm@13142
   672
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   673
by (induct xs) auto
wenzelm@13114
   674
blanchet@35828
   675
lemma append_eq_append_conv [simp, no_atp]:
nipkow@24526
   676
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   677
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   678
apply (induct xs arbitrary: ys)
paulson@14208
   679
 apply (case_tac ys, simp, force)
paulson@14208
   680
apply (case_tac ys, force, simp)
nipkow@13145
   681
done
wenzelm@13142
   682
nipkow@24526
   683
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   684
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   685
apply (induct xs arbitrary: ys zs ts)
nipkow@44890
   686
 apply fastforce
nipkow@14495
   687
apply(case_tac zs)
nipkow@14495
   688
 apply simp
nipkow@44890
   689
apply fastforce
nipkow@14495
   690
done
nipkow@14495
   691
berghofe@34910
   692
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   693
by simp
wenzelm@13142
   694
wenzelm@13142
   695
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   696
by simp
wenzelm@13114
   697
berghofe@34910
   698
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   699
by simp
wenzelm@13114
   700
wenzelm@13142
   701
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   702
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   703
wenzelm@13142
   704
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   705
using append_same_eq [of "[]"] by auto
wenzelm@13114
   706
blanchet@35828
   707
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   708
by (induct xs) auto
wenzelm@13114
   709
wenzelm@13142
   710
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   711
by (induct xs) auto
wenzelm@13114
   712
wenzelm@13142
   713
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   714
by (simp add: hd_append split: list.split)
wenzelm@13114
   715
wenzelm@13142
   716
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   717
by (simp split: list.split)
wenzelm@13114
   718
wenzelm@13142
   719
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   720
by (simp add: tl_append split: list.split)
wenzelm@13114
   721
wenzelm@13114
   722
nipkow@14300
   723
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   724
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   725
by(cases ys) auto
nipkow@14300
   726
nipkow@15281
   727
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   728
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   729
by(cases ys) auto
nipkow@15281
   730
nipkow@14300
   731
wenzelm@13142
   732
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   733
wenzelm@13114
   734
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   735
by simp
wenzelm@13114
   736
wenzelm@13142
   737
lemma Cons_eq_appendI:
nipkow@13145
   738
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   739
by (drule sym) simp
wenzelm@13114
   740
wenzelm@13142
   741
lemma append_eq_appendI:
nipkow@13145
   742
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   743
by (drule sym) simp
wenzelm@13114
   744
wenzelm@13114
   745
wenzelm@13142
   746
text {*
nipkow@13145
   747
Simplification procedure for all list equalities.
nipkow@13145
   748
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   749
- both lists end in a singleton list,
nipkow@13145
   750
- or both lists end in the same list.
wenzelm@13142
   751
*}
wenzelm@13142
   752
wenzelm@43594
   753
simproc_setup list_eq ("(xs::'a list) = ys")  = {*
wenzelm@13462
   754
  let
wenzelm@43594
   755
    fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
wenzelm@43594
   756
          (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
wenzelm@43594
   757
      | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@43594
   758
      | last t = t;
wenzelm@43594
   759
    
wenzelm@43594
   760
    fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@43594
   761
      | list1 _ = false;
wenzelm@43594
   762
    
wenzelm@43594
   763
    fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
wenzelm@43594
   764
          (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
wenzelm@43594
   765
      | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
wenzelm@43594
   766
      | butlast xs = Const(@{const_name Nil}, fastype_of xs);
wenzelm@43594
   767
    
wenzelm@43594
   768
    val rearr_ss =
wenzelm@43594
   769
      HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
wenzelm@43594
   770
    
wenzelm@43594
   771
    fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   772
      let
wenzelm@43594
   773
        val lastl = last lhs and lastr = last rhs;
wenzelm@43594
   774
        fun rearr conv =
wenzelm@43594
   775
          let
wenzelm@43594
   776
            val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@43594
   777
            val Type(_,listT::_) = eqT
wenzelm@43594
   778
            val appT = [listT,listT] ---> listT
wenzelm@43594
   779
            val app = Const(@{const_name append},appT)
wenzelm@43594
   780
            val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@43594
   781
            val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@43594
   782
            val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@43594
   783
              (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
wenzelm@43594
   784
          in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@43594
   785
      in
wenzelm@43594
   786
        if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
wenzelm@43594
   787
        else if lastl aconv lastr then rearr @{thm append_same_eq}
wenzelm@43594
   788
        else NONE
wenzelm@43594
   789
      end;
wenzelm@43594
   790
  in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
wenzelm@13114
   791
*}
wenzelm@13114
   792
wenzelm@13114
   793
nipkow@15392
   794
subsubsection {* @{text map} *}
wenzelm@13114
   795
haftmann@40210
   796
lemma hd_map:
haftmann@40210
   797
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
haftmann@40210
   798
  by (cases xs) simp_all
haftmann@40210
   799
haftmann@40210
   800
lemma map_tl:
haftmann@40210
   801
  "map f (tl xs) = tl (map f xs)"
haftmann@40210
   802
  by (cases xs) simp_all
haftmann@40210
   803
wenzelm@13142
   804
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   805
by (induct xs) simp_all
wenzelm@13114
   806
wenzelm@13142
   807
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   808
by (rule ext, induct_tac xs) auto
wenzelm@13114
   809
wenzelm@13142
   810
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   811
by (induct xs) auto
wenzelm@13114
   812
hoelzl@33639
   813
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
hoelzl@33639
   814
by (induct xs) auto
hoelzl@33639
   815
nipkow@35208
   816
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
nipkow@35208
   817
apply(rule ext)
nipkow@35208
   818
apply(simp)
nipkow@35208
   819
done
nipkow@35208
   820
wenzelm@13142
   821
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   822
by (induct xs) auto
wenzelm@13114
   823
nipkow@13737
   824
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   825
by (induct xs) auto
nipkow@13737
   826
krauss@44013
   827
lemma map_cong [fundef_cong]:
haftmann@40122
   828
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
haftmann@40122
   829
  by simp
wenzelm@13114
   830
wenzelm@13142
   831
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   832
by (cases xs) auto
wenzelm@13114
   833
wenzelm@13142
   834
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   835
by (cases xs) auto
wenzelm@13114
   836
paulson@18447
   837
lemma map_eq_Cons_conv:
nipkow@14025
   838
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   839
by (cases xs) auto
wenzelm@13114
   840
paulson@18447
   841
lemma Cons_eq_map_conv:
nipkow@14025
   842
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   843
by (cases ys) auto
nipkow@14025
   844
paulson@18447
   845
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   846
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   847
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   848
nipkow@14111
   849
lemma ex_map_conv:
nipkow@14111
   850
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   851
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   852
nipkow@15110
   853
lemma map_eq_imp_length_eq:
paulson@35510
   854
  assumes "map f xs = map g ys"
haftmann@26734
   855
  shows "length xs = length ys"
haftmann@26734
   856
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   857
  case Nil then show ?case by simp
haftmann@26734
   858
next
haftmann@26734
   859
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
paulson@35510
   860
  from Cons xs have "map f zs = map g ys" by simp
haftmann@26734
   861
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   862
  with xs show ?case by simp
haftmann@26734
   863
qed
haftmann@26734
   864
  
nipkow@15110
   865
lemma map_inj_on:
nipkow@15110
   866
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   867
  ==> xs = ys"
nipkow@15110
   868
apply(frule map_eq_imp_length_eq)
nipkow@15110
   869
apply(rotate_tac -1)
nipkow@15110
   870
apply(induct rule:list_induct2)
nipkow@15110
   871
 apply simp
nipkow@15110
   872
apply(simp)
nipkow@15110
   873
apply (blast intro:sym)
nipkow@15110
   874
done
nipkow@15110
   875
nipkow@15110
   876
lemma inj_on_map_eq_map:
nipkow@15110
   877
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   878
by(blast dest:map_inj_on)
nipkow@15110
   879
wenzelm@13114
   880
lemma map_injective:
nipkow@24526
   881
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   882
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   883
nipkow@14339
   884
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   885
by(blast dest:map_injective)
nipkow@14339
   886
wenzelm@13114
   887
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   888
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   889
wenzelm@13114
   890
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   891
apply (unfold inj_on_def, clarify)
nipkow@13145
   892
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   893
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   894
apply blast
nipkow@13145
   895
done
wenzelm@13114
   896
nipkow@14339
   897
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   898
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   899
nipkow@15303
   900
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   901
apply(rule inj_onI)
nipkow@15303
   902
apply(erule map_inj_on)
nipkow@15303
   903
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   904
done
nipkow@15303
   905
kleing@14343
   906
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   907
by (induct xs, auto)
wenzelm@13114
   908
nipkow@14402
   909
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   910
by (induct xs) auto
nipkow@14402
   911
nipkow@15110
   912
lemma map_fst_zip[simp]:
nipkow@15110
   913
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   914
by (induct rule:list_induct2, simp_all)
nipkow@15110
   915
nipkow@15110
   916
lemma map_snd_zip[simp]:
nipkow@15110
   917
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   918
by (induct rule:list_induct2, simp_all)
nipkow@15110
   919
haftmann@41505
   920
enriched_type map: map
nipkow@47122
   921
by (simp_all add: id_def)
nipkow@47122
   922
nipkow@47122
   923
declare map.id[simp]
nipkow@15110
   924
nipkow@15392
   925
subsubsection {* @{text rev} *}
wenzelm@13114
   926
wenzelm@13142
   927
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   928
by (induct xs) auto
wenzelm@13114
   929
wenzelm@13142
   930
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   931
by (induct xs) auto
wenzelm@13114
   932
kleing@15870
   933
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   934
by auto
kleing@15870
   935
wenzelm@13142
   936
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   937
by (induct xs) auto
wenzelm@13114
   938
wenzelm@13142
   939
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   940
by (induct xs) auto
wenzelm@13114
   941
kleing@15870
   942
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   943
by (cases xs) auto
kleing@15870
   944
kleing@15870
   945
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   946
by (cases xs) auto
kleing@15870
   947
blanchet@46439
   948
lemma rev_is_rev_conv [iff, no_atp]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   949
apply (induct xs arbitrary: ys, force)
paulson@14208
   950
apply (case_tac ys, simp, force)
nipkow@13145
   951
done
wenzelm@13114
   952
nipkow@15439
   953
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   954
by(simp add:inj_on_def)
nipkow@15439
   955
wenzelm@13366
   956
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   957
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   958
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   959
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   960
done
wenzelm@13114
   961
wenzelm@13366
   962
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   963
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   964
by (induct xs rule: rev_induct) auto
wenzelm@13114
   965
wenzelm@13366
   966
lemmas rev_cases = rev_exhaust
wenzelm@13366
   967
nipkow@18423
   968
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   969
by(rule rev_cases[of xs]) auto
nipkow@18423
   970
wenzelm@13114
   971
nipkow@15392
   972
subsubsection {* @{text set} *}
wenzelm@13114
   973
nipkow@46698
   974
declare set.simps [code_post]  --"pretty output"
nipkow@46698
   975
wenzelm@13142
   976
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   977
by (induct xs) auto
wenzelm@13114
   978
wenzelm@13142
   979
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   980
by (induct xs) auto
wenzelm@13114
   981
nipkow@17830
   982
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   983
by(cases xs) auto
oheimb@14099
   984
wenzelm@13142
   985
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   986
by auto
wenzelm@13114
   987
oheimb@14099
   988
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   989
by auto
oheimb@14099
   990
wenzelm@13142
   991
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   992
by (induct xs) auto
wenzelm@13114
   993
nipkow@15245
   994
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   995
by(induct xs) auto
nipkow@15245
   996
wenzelm@13142
   997
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   998
by (induct xs) auto
wenzelm@13114
   999
wenzelm@13142
  1000
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
  1001
by (induct xs) auto
wenzelm@13114
  1002
wenzelm@13142
  1003
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
  1004
by (induct xs) auto
wenzelm@13114
  1005
nipkow@32417
  1006
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
bulwahn@41463
  1007
by (induct j) auto
wenzelm@13114
  1008
wenzelm@13142
  1009
wenzelm@25221
  1010
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
  1011
proof (induct xs)
nipkow@26073
  1012
  case Nil thus ?case by simp
nipkow@26073
  1013
next
nipkow@26073
  1014
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
  1015
qed
nipkow@26073
  1016
haftmann@26734
  1017
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
  1018
  by (auto elim: split_list)
nipkow@26073
  1019
nipkow@26073
  1020
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
  1021
proof (induct xs)
nipkow@26073
  1022
  case Nil thus ?case by simp
nipkow@18049
  1023
next
nipkow@18049
  1024
  case (Cons a xs)
nipkow@18049
  1025
  show ?case
nipkow@18049
  1026
  proof cases
nipkow@44890
  1027
    assume "x = a" thus ?case using Cons by fastforce
nipkow@18049
  1028
  next
nipkow@44890
  1029
    assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
nipkow@26073
  1030
  qed
nipkow@26073
  1031
qed
nipkow@26073
  1032
nipkow@26073
  1033
lemma in_set_conv_decomp_first:
nipkow@26073
  1034
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
  1035
  by (auto dest!: split_list_first)
nipkow@26073
  1036
haftmann@40122
  1037
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
haftmann@40122
  1038
proof (induct xs rule: rev_induct)
nipkow@26073
  1039
  case Nil thus ?case by simp
nipkow@26073
  1040
next
nipkow@26073
  1041
  case (snoc a xs)
nipkow@26073
  1042
  show ?case
nipkow@26073
  1043
  proof cases
haftmann@40122
  1044
    assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
nipkow@26073
  1045
  next
nipkow@44890
  1046
    assume "x \<noteq> a" thus ?case using snoc by fastforce
nipkow@18049
  1047
  qed
nipkow@18049
  1048
qed
nipkow@18049
  1049
nipkow@26073
  1050
lemma in_set_conv_decomp_last:
nipkow@26073
  1051
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
  1052
  by (auto dest!: split_list_last)
nipkow@26073
  1053
nipkow@26073
  1054
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
  1055
proof (induct xs)
nipkow@26073
  1056
  case Nil thus ?case by simp
nipkow@26073
  1057
next
nipkow@26073
  1058
  case Cons thus ?case
nipkow@26073
  1059
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
  1060
qed
nipkow@26073
  1061
nipkow@26073
  1062
lemma split_list_propE:
haftmann@26734
  1063
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1064
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
  1065
using split_list_prop [OF assms] by blast
nipkow@26073
  1066
nipkow@26073
  1067
lemma split_list_first_prop:
nipkow@26073
  1068
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1069
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
  1070
proof (induct xs)
nipkow@26073
  1071
  case Nil thus ?case by simp
nipkow@26073
  1072
next
nipkow@26073
  1073
  case (Cons x xs)
nipkow@26073
  1074
  show ?case
nipkow@26073
  1075
  proof cases
nipkow@26073
  1076
    assume "P x"
haftmann@40122
  1077
    thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
  1078
  next
nipkow@26073
  1079
    assume "\<not> P x"
nipkow@26073
  1080
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
  1081
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
  1082
  qed
nipkow@26073
  1083
qed
nipkow@26073
  1084
nipkow@26073
  1085
lemma split_list_first_propE:
haftmann@26734
  1086
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1087
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
  1088
using split_list_first_prop [OF assms] by blast
nipkow@26073
  1089
nipkow@26073
  1090
lemma split_list_first_prop_iff:
nipkow@26073
  1091
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1092
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
  1093
by (rule, erule split_list_first_prop) auto
nipkow@26073
  1094
nipkow@26073
  1095
lemma split_list_last_prop:
nipkow@26073
  1096
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
  1097
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
  1098
proof(induct xs rule:rev_induct)
nipkow@26073
  1099
  case Nil thus ?case by simp
nipkow@26073
  1100
next
nipkow@26073
  1101
  case (snoc x xs)
nipkow@26073
  1102
  show ?case
nipkow@26073
  1103
  proof cases
nipkow@26073
  1104
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
  1105
  next
nipkow@26073
  1106
    assume "\<not> P x"
nipkow@26073
  1107
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@44890
  1108
    thus ?thesis using `\<not> P x` snoc(1) by fastforce
nipkow@26073
  1109
  qed
nipkow@26073
  1110
qed
nipkow@26073
  1111
nipkow@26073
  1112
lemma split_list_last_propE:
haftmann@26734
  1113
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
  1114
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
  1115
using split_list_last_prop [OF assms] by blast
nipkow@26073
  1116
nipkow@26073
  1117
lemma split_list_last_prop_iff:
nipkow@26073
  1118
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1119
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1120
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1121
nipkow@26073
  1122
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1123
  by (erule finite_induct)
haftmann@26734
  1124
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1125
kleing@14388
  1126
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1127
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1128
haftmann@26442
  1129
lemma set_minus_filter_out:
haftmann@26442
  1130
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1131
  by (induct xs) auto
paulson@15168
  1132
wenzelm@35115
  1133
nipkow@15392
  1134
subsubsection {* @{text filter} *}
wenzelm@13114
  1135
wenzelm@13142
  1136
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1137
by (induct xs) auto
wenzelm@13114
  1138
nipkow@15305
  1139
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1140
by (induct xs) simp_all
nipkow@15305
  1141
wenzelm@13142
  1142
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1143
by (induct xs) auto
wenzelm@13114
  1144
nipkow@16998
  1145
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1146
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1147
nipkow@18423
  1148
lemma sum_length_filter_compl:
nipkow@18423
  1149
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1150
by(induct xs) simp_all
nipkow@18423
  1151
wenzelm@13142
  1152
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1153
by (induct xs) auto
wenzelm@13114
  1154
wenzelm@13142
  1155
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1156
by (induct xs) auto
wenzelm@13114
  1157
nipkow@16998
  1158
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1159
by (induct xs) simp_all
nipkow@16998
  1160
nipkow@16998
  1161
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1162
apply (induct xs)
nipkow@16998
  1163
 apply auto
nipkow@16998
  1164
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1165
apply simp
nipkow@16998
  1166
done
wenzelm@13114
  1167
nipkow@16965
  1168
lemma filter_map:
nipkow@16965
  1169
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1170
by (induct xs) simp_all
nipkow@16965
  1171
nipkow@16965
  1172
lemma length_filter_map[simp]:
nipkow@16965
  1173
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1174
by (simp add:filter_map)
nipkow@16965
  1175
wenzelm@13142
  1176
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1177
by auto
wenzelm@13114
  1178
nipkow@15246
  1179
lemma length_filter_less:
nipkow@15246
  1180
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1181
proof (induct xs)
nipkow@15246
  1182
  case Nil thus ?case by simp
nipkow@15246
  1183
next
nipkow@15246
  1184
  case (Cons x xs) thus ?case
nipkow@15246
  1185
    apply (auto split:split_if_asm)
nipkow@15246
  1186
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1187
  done
nipkow@15246
  1188
qed
wenzelm@13114
  1189
nipkow@15281
  1190
lemma length_filter_conv_card:
nipkow@15281
  1191
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1192
proof (induct xs)
nipkow@15281
  1193
  case Nil thus ?case by simp
nipkow@15281
  1194
next
nipkow@15281
  1195
  case (Cons x xs)
nipkow@15281
  1196
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1197
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1198
  show ?case (is "?l = card ?S'")
nipkow@15281
  1199
  proof (cases)
nipkow@15281
  1200
    assume "p x"
nipkow@15281
  1201
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1202
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1203
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1204
      using Cons `p x` by simp
nipkow@15281
  1205
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
huffman@44921
  1206
      by (simp add: card_image)
nipkow@15281
  1207
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1208
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1209
    finally show ?thesis .
nipkow@15281
  1210
  next
nipkow@15281
  1211
    assume "\<not> p x"
nipkow@15281
  1212
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1213
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1214
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1215
      using Cons `\<not> p x` by simp
nipkow@15281
  1216
    also have "\<dots> = card(Suc ` ?S)" using fin
huffman@44921
  1217
      by (simp add: card_image)
nipkow@15281
  1218
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1219
      by (simp add:card_insert_if)
nipkow@15281
  1220
    finally show ?thesis .
nipkow@15281
  1221
  qed
nipkow@15281
  1222
qed
nipkow@15281
  1223
nipkow@17629
  1224
lemma Cons_eq_filterD:
nipkow@17629
  1225
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1226
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585
  1227
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1228
proof(induct ys)
nipkow@17629
  1229
  case Nil thus ?case by simp
nipkow@17629
  1230
next
nipkow@17629
  1231
  case (Cons y ys)
nipkow@17629
  1232
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1233
  proof cases
nipkow@17629
  1234
    assume Py: "P y"
nipkow@17629
  1235
    show ?thesis
nipkow@17629
  1236
    proof cases
wenzelm@25221
  1237
      assume "x = y"
wenzelm@25221
  1238
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1239
      then show ?thesis ..
nipkow@17629
  1240
    next
wenzelm@25221
  1241
      assume "x \<noteq> y"
wenzelm@25221
  1242
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1243
    qed
nipkow@17629
  1244
  next
wenzelm@25221
  1245
    assume "\<not> P y"
nipkow@44890
  1246
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
wenzelm@25221
  1247
    then have "?Q (y#us)" by simp
wenzelm@25221
  1248
    then show ?thesis ..
nipkow@17629
  1249
  qed
nipkow@17629
  1250
qed
nipkow@17629
  1251
nipkow@17629
  1252
lemma filter_eq_ConsD:
nipkow@17629
  1253
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1254
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629
  1255
by(rule Cons_eq_filterD) simp
nipkow@17629
  1256
nipkow@17629
  1257
lemma filter_eq_Cons_iff:
nipkow@17629
  1258
 "(filter P ys = x#xs) =
nipkow@17629
  1259
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1260
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1261
nipkow@17629
  1262
lemma Cons_eq_filter_iff:
nipkow@17629
  1263
 "(x#xs = filter P ys) =
nipkow@17629
  1264
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1265
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1266
krauss@44013
  1267
lemma filter_cong[fundef_cong]:
nipkow@17501
  1268
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1269
apply simp
nipkow@17501
  1270
apply(erule thin_rl)
nipkow@17501
  1271
by (induct ys) simp_all
nipkow@17501
  1272
nipkow@15281
  1273
haftmann@26442
  1274
subsubsection {* List partitioning *}
haftmann@26442
  1275
haftmann@26442
  1276
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1277
  "partition P [] = ([], [])"
haftmann@26442
  1278
  | "partition P (x # xs) = 
haftmann@26442
  1279
      (let (yes, no) = partition P xs
haftmann@26442
  1280
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1281
haftmann@26442
  1282
lemma partition_filter1:
haftmann@26442
  1283
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1284
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1285
haftmann@26442
  1286
lemma partition_filter2:
haftmann@26442
  1287
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1288
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1289
haftmann@26442
  1290
lemma partition_P:
haftmann@26442
  1291
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1292
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1293
proof -
haftmann@26442
  1294
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1295
    by simp_all
haftmann@26442
  1296
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1297
qed
haftmann@26442
  1298
haftmann@26442
  1299
lemma partition_set:
haftmann@26442
  1300
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1301
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1302
proof -
haftmann@26442
  1303
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1304
    by simp_all
haftmann@26442
  1305
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1306
qed
haftmann@26442
  1307
hoelzl@33639
  1308
lemma partition_filter_conv[simp]:
hoelzl@33639
  1309
  "partition f xs = (filter f xs,filter (Not o f) xs)"
hoelzl@33639
  1310
unfolding partition_filter2[symmetric]
hoelzl@33639
  1311
unfolding partition_filter1[symmetric] by simp
hoelzl@33639
  1312
hoelzl@33639
  1313
declare partition.simps[simp del]
haftmann@26442
  1314
wenzelm@35115
  1315
nipkow@15392
  1316
subsubsection {* @{text concat} *}
wenzelm@13114
  1317
wenzelm@13142
  1318
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1319
by (induct xs) auto
wenzelm@13114
  1320
paulson@18447
  1321
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1322
by (induct xss) auto
wenzelm@13114
  1323
paulson@18447
  1324
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1325
by (induct xss) auto
wenzelm@13114
  1326
nipkow@24308
  1327
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1328
by (induct xs) auto
wenzelm@13114
  1329
nipkow@24476
  1330
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1331
by (induct xs) auto
nipkow@24349
  1332
wenzelm@13142
  1333
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1334
by (induct xs) auto
wenzelm@13114
  1335
wenzelm@13142
  1336
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1337
by (induct xs) auto
wenzelm@13114
  1338
wenzelm@13142
  1339
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1340
by (induct xs) auto
wenzelm@13114
  1341
bulwahn@40365
  1342
lemma concat_eq_concat_iff: "\<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> length xs = length ys ==> (concat xs = concat ys) = (xs = ys)"
bulwahn@40365
  1343
proof (induct xs arbitrary: ys)
bulwahn@40365
  1344
  case (Cons x xs ys)
bulwahn@40365
  1345
  thus ?case by (cases ys) auto
bulwahn@40365
  1346
qed (auto)
bulwahn@40365
  1347
bulwahn@40365
  1348
lemma concat_injective: "concat xs = concat ys ==> length xs = length ys ==> \<forall>(x, y) \<in> set (zip xs ys). length x = length y ==> xs = ys"
bulwahn@40365
  1349
by (simp add: concat_eq_concat_iff)
bulwahn@40365
  1350
wenzelm@13114
  1351
nipkow@15392
  1352
subsubsection {* @{text nth} *}
wenzelm@13114
  1353
haftmann@29827
  1354
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1355
by auto
wenzelm@13114
  1356
haftmann@29827
  1357
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1358
by auto
wenzelm@13114
  1359
wenzelm@13142
  1360
declare nth.simps [simp del]
wenzelm@13114
  1361
nipkow@41842
  1362
lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
nipkow@41842
  1363
by(auto simp: Nat.gr0_conv_Suc)
nipkow@41842
  1364
wenzelm@13114
  1365
lemma nth_append:
nipkow@24526
  1366
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1367
apply (induct xs arbitrary: n, simp)
paulson@14208
  1368
apply (case_tac n, auto)
nipkow@13145
  1369
done
wenzelm@13114
  1370
nipkow@14402
  1371
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1372
by (induct xs) auto
nipkow@14402
  1373
nipkow@14402
  1374
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1375
by (induct xs) auto
nipkow@14402
  1376
nipkow@24526
  1377
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1378
apply (induct xs arbitrary: n, simp)
paulson@14208
  1379
apply (case_tac n, auto)
nipkow@13145
  1380
done
wenzelm@13114
  1381
noschinl@45841
  1382
lemma nth_tl:
noschinl@45841
  1383
  assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n"
noschinl@45841
  1384
using assms by (induct x) auto
noschinl@45841
  1385
nipkow@18423
  1386
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1387
by(cases xs) simp_all
nipkow@18423
  1388
nipkow@18049
  1389
nipkow@18049
  1390
lemma list_eq_iff_nth_eq:
nipkow@24526
  1391
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1392
apply(induct xs arbitrary: ys)
paulson@24632
  1393
 apply force
nipkow@18049
  1394
apply(case_tac ys)
nipkow@18049
  1395
 apply simp
nipkow@18049
  1396
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1397
done
nipkow@18049
  1398
wenzelm@13142
  1399
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1400
apply (induct xs, simp, simp)
nipkow@13145
  1401
apply safe
paulson@24632
  1402
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1403
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1404
apply (case_tac i, simp)
paulson@24632
  1405
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1406
done
wenzelm@13114
  1407
nipkow@17501
  1408
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1409
by(auto simp:set_conv_nth)
nipkow@17501
  1410
nipkow@13145
  1411
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1412
by (auto simp add: set_conv_nth)
wenzelm@13114
  1413
wenzelm@13142
  1414
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1415
by (auto simp add: set_conv_nth)
wenzelm@13114
  1416
wenzelm@13114
  1417
lemma all_nth_imp_all_set:
nipkow@13145
  1418
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1419
by (auto simp add: set_conv_nth)
wenzelm@13114
  1420
wenzelm@13114
  1421
lemma all_set_conv_all_nth:
nipkow@13145
  1422
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1423
by (auto simp add: set_conv_nth)
wenzelm@13114
  1424
kleing@25296
  1425
lemma rev_nth:
kleing@25296
  1426
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1427
proof (induct xs arbitrary: n)
kleing@25296
  1428
  case Nil thus ?case by simp
kleing@25296
  1429
next
kleing@25296
  1430
  case (Cons x xs)
kleing@25296
  1431
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1432
  moreover
kleing@25296
  1433
  { assume "n < length xs"
kleing@25296
  1434
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1435
      by (cases "length xs - n", auto)
kleing@25296
  1436
    moreover
kleing@25296
  1437
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1438
    ultimately
kleing@25296
  1439
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1440
  }
kleing@25296
  1441
  ultimately
kleing@25296
  1442
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1443
qed
wenzelm@13114
  1444
nipkow@31159
  1445
lemma Skolem_list_nth:
nipkow@31159
  1446
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1447
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1448
proof(induct k)
nipkow@31159
  1449
  case 0 show ?case by simp
nipkow@31159
  1450
next
nipkow@31159
  1451
  case (Suc k)
nipkow@31159
  1452
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1453
  proof
nipkow@31159
  1454
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1455
  next
nipkow@31159
  1456
    assume "?L"
nipkow@31159
  1457
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1458
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1459
    thus "?R" ..
nipkow@31159
  1460
  qed
nipkow@31159
  1461
qed
nipkow@31159
  1462
nipkow@31159
  1463
nipkow@15392
  1464
subsubsection {* @{text list_update} *}
wenzelm@13114
  1465
nipkow@24526
  1466
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1467
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1468
wenzelm@13114
  1469
lemma nth_list_update:
nipkow@24526
  1470
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1471
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1472
wenzelm@13142
  1473
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1474
by (simp add: nth_list_update)
wenzelm@13114
  1475
nipkow@24526
  1476
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1477
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1478
nipkow@24526
  1479
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1480
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1481
nipkow@24526
  1482
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1483
apply (induct xs arbitrary: i)
nipkow@17501
  1484
 apply simp
nipkow@17501
  1485
apply (case_tac i)
nipkow@17501
  1486
apply simp_all
nipkow@17501
  1487
done
nipkow@17501
  1488
nipkow@31077
  1489
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1490
by(metis length_0_conv length_list_update)
nipkow@31077
  1491
wenzelm@13114
  1492
lemma list_update_same_conv:
nipkow@24526
  1493
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1494
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1495
nipkow@14187
  1496
lemma list_update_append1:
nipkow@24526
  1497
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1498
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1499
apply(simp split:nat.split)
nipkow@14187
  1500
done
nipkow@14187
  1501
kleing@15868
  1502
lemma list_update_append:
nipkow@24526
  1503
  "(xs @ ys) [n:= x] = 
kleing@15868
  1504
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1505
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1506
nipkow@14402
  1507
lemma list_update_length [simp]:
nipkow@14402
  1508
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1509
by (induct xs, auto)
nipkow@14402
  1510
nipkow@31264
  1511
lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]"
nipkow@31264
  1512
by(induct xs arbitrary: k)(auto split:nat.splits)
nipkow@31264
  1513
nipkow@31264
  1514
lemma rev_update:
nipkow@31264
  1515
  "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]"
nipkow@31264
  1516
by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits)
nipkow@31264
  1517
wenzelm@13114
  1518
lemma update_zip:
nipkow@31080
  1519
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1520
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1521
nipkow@24526
  1522
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1523
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1524
wenzelm@13114
  1525
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1526
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1527
nipkow@24526
  1528
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1529
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1530
nipkow@31077
  1531
lemma list_update_overwrite[simp]:
haftmann@24796
  1532
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1533
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1534
apply (case_tac i, simp_all)
haftmann@24796
  1535
done
haftmann@24796
  1536
haftmann@24796
  1537
lemma list_update_swap:
haftmann@24796
  1538
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1539
apply (induct xs arbitrary: i i')
haftmann@24796
  1540
apply simp
haftmann@24796
  1541
apply (case_tac i, case_tac i')
haftmann@24796
  1542
apply auto
haftmann@24796
  1543
apply (case_tac i')
haftmann@24796
  1544
apply auto
haftmann@24796
  1545
done
haftmann@24796
  1546
haftmann@29827
  1547
lemma list_update_code [code]:
haftmann@29827
  1548
  "[][i := y] = []"
haftmann@29827
  1549
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1550
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1551
  by simp_all
haftmann@29827
  1552
wenzelm@13114
  1553
nipkow@15392
  1554
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1555
wenzelm@13142
  1556
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1557
by (induct xs) auto
wenzelm@13114
  1558
wenzelm@13142
  1559
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1560
by (induct xs) auto
wenzelm@13114
  1561
nipkow@14302
  1562
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
huffman@44921
  1563
  by simp
nipkow@14302
  1564
nipkow@14302
  1565
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
huffman@44921
  1566
  by simp
nipkow@14302
  1567
nipkow@14302
  1568
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1569
by (induct xs) (auto)
nipkow@14302
  1570
nipkow@14302
  1571
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1572
by(simp add:last_append)
nipkow@14302
  1573
nipkow@14302
  1574
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1575
by(simp add:last_append)
nipkow@14302
  1576
noschinl@45841
  1577
lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs"
noschinl@45841
  1578
by (induct xs) simp_all
noschinl@45841
  1579
noschinl@45841
  1580
lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)"
noschinl@45841
  1581
by (induct xs) simp_all
noschinl@45841
  1582
nipkow@17762
  1583
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1584
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1585
nipkow@17762
  1586
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1587
by(cases xs) simp_all
nipkow@17762
  1588
nipkow@17765
  1589
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1590
by (induct as) auto
nipkow@17762
  1591
wenzelm@13142
  1592
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1593
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1594
wenzelm@13114
  1595
lemma butlast_append:
nipkow@24526
  1596
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1597
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1598
wenzelm@13142
  1599
lemma append_butlast_last_id [simp]:
nipkow@13145
  1600
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1601
by (induct xs) auto
wenzelm@13114
  1602
wenzelm@13142
  1603
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1604
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1605
wenzelm@13114
  1606
lemma in_set_butlast_appendI:
nipkow@13145
  1607
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1608
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1609
nipkow@24526
  1610
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1611
apply (induct xs arbitrary: n)
nipkow@17501
  1612
 apply simp
nipkow@17501
  1613
apply (auto split:nat.split)
nipkow@17501
  1614
done
nipkow@17501
  1615
noschinl@45841
  1616
lemma nth_butlast:
noschinl@45841
  1617
  assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
noschinl@45841
  1618
proof (cases xs)
noschinl@45841
  1619
  case (Cons y ys)
noschinl@45841
  1620
  moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n"
noschinl@45841
  1621
    by (simp add: nth_append)
noschinl@45841
  1622
  ultimately show ?thesis using append_butlast_last_id by simp
noschinl@45841
  1623
qed simp
noschinl@45841
  1624
huffman@30128
  1625
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1626
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1627
huffman@30128
  1628
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1629
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1630
nipkow@31077
  1631
lemma last_list_update:
nipkow@31077
  1632
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1633
by (auto simp: last_conv_nth)
nipkow@31077
  1634
nipkow@31077
  1635
lemma butlast_list_update:
nipkow@31077
  1636
  "butlast(xs[k:=x]) =
nipkow@31077
  1637
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1638
apply(cases xs rule:rev_cases)
nipkow@31077
  1639
apply simp
nipkow@31077
  1640
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1641
done
nipkow@31077
  1642
haftmann@36851
  1643
lemma last_map:
haftmann@36851
  1644
  "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)"
haftmann@36851
  1645
  by (cases xs rule: rev_cases) simp_all
haftmann@36851
  1646
haftmann@36851
  1647
lemma map_butlast:
haftmann@36851
  1648
  "map f (butlast xs) = butlast (map f xs)"
haftmann@36851
  1649
  by (induct xs) simp_all
haftmann@36851
  1650
nipkow@40230
  1651
lemma snoc_eq_iff_butlast:
nipkow@40230
  1652
  "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] & butlast ys = xs & last ys = x)"
nipkow@40230
  1653
by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self)
nipkow@40230
  1654
haftmann@24796
  1655
nipkow@15392
  1656
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1657
wenzelm@13142
  1658
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1659
by (induct xs) auto
wenzelm@13114
  1660
wenzelm@13142
  1661
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1662
by (induct xs) auto
wenzelm@13114
  1663
wenzelm@13142
  1664
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1665
by simp
wenzelm@13114
  1666
wenzelm@13142
  1667
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1668
by simp
wenzelm@13114
  1669
wenzelm@13142
  1670
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1671
huffman@30128
  1672
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1673
  unfolding One_nat_def by simp
huffman@30128
  1674
huffman@30128
  1675
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1676
  unfolding One_nat_def by simp
huffman@30128
  1677
nipkow@15110
  1678
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1679
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1680
nipkow@14187
  1681
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1682
by(cases xs, simp_all)
nipkow@14187
  1683
huffman@26584
  1684
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1685
by (induct xs arbitrary: n) simp_all
huffman@26584
  1686
nipkow@24526
  1687
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1688
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1689
huffman@26584
  1690
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1691
by (cases n, simp, cases xs, auto)
huffman@26584
  1692
huffman@26584
  1693
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1694
by (simp only: drop_tl)
huffman@26584
  1695
nipkow@24526
  1696
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1697
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1698
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1699
done
nipkow@14187
  1700
nipkow@13913
  1701
lemma take_Suc_conv_app_nth:
nipkow@24526
  1702
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1703
apply (induct xs arbitrary: i, simp)
paulson@14208
  1704
apply (case_tac i, auto)
nipkow@13913
  1705
done
nipkow@13913
  1706
mehta@14591
  1707
lemma drop_Suc_conv_tl:
nipkow@24526
  1708
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1709
apply (induct xs arbitrary: i, simp)
mehta@14591
  1710
apply (case_tac i, auto)
mehta@14591
  1711
done
mehta@14591
  1712
nipkow@24526
  1713
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1714
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1715
nipkow@24526
  1716
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1717
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1718
nipkow@24526
  1719
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1720
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1721
nipkow@24526
  1722
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1723
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1724
wenzelm@13142
  1725
lemma take_append [simp]:
nipkow@24526
  1726
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1727
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1728
wenzelm@13142
  1729
lemma drop_append [simp]:
nipkow@24526
  1730
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1731
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1732
nipkow@24526
  1733
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1734
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1735
apply (case_tac xs, auto)
nipkow@15236
  1736
apply (case_tac n, auto)
nipkow@13145
  1737
done
wenzelm@13114
  1738
nipkow@24526
  1739
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1740
apply (induct m arbitrary: xs, auto)
paulson@14208
  1741
apply (case_tac xs, auto)
nipkow@13145
  1742
done
wenzelm@13114
  1743
nipkow@24526
  1744
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1745
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1746
apply (case_tac xs, auto)
nipkow@13145
  1747
done
wenzelm@13114
  1748
nipkow@24526
  1749
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1750
apply(induct xs arbitrary: m n)
nipkow@14802
  1751
 apply simp
nipkow@14802
  1752
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1753
done
nipkow@14802
  1754
nipkow@24526
  1755
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1756
apply (induct n arbitrary: xs, auto)
paulson@14208
  1757
apply (case_tac xs, auto)
nipkow@13145
  1758
done
wenzelm@13114
  1759
nipkow@24526
  1760
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1761
apply(induct xs arbitrary: n)
nipkow@15110
  1762
 apply simp
nipkow@15110
  1763
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1764
done
nipkow@15110
  1765
nipkow@24526
  1766
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1767
apply(induct xs arbitrary: n)
nipkow@15110
  1768
apply simp
nipkow@15110
  1769
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1770
done
nipkow@15110
  1771
nipkow@24526
  1772
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1773
apply (induct n arbitrary: xs, auto)
paulson@14208
  1774
apply (case_tac xs, auto)
nipkow@13145
  1775
done
wenzelm@13114
  1776
nipkow@24526
  1777
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1778
apply (induct n arbitrary: xs, auto)
paulson@14208
  1779
apply (case_tac xs, auto)
nipkow@13145
  1780
done
wenzelm@13114
  1781
nipkow@24526
  1782
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1783
apply (induct xs arbitrary: i, auto)
paulson@14208
  1784
apply (case_tac i, auto)
nipkow@13145
  1785
done
wenzelm@13114
  1786
nipkow@24526
  1787
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1788
apply (induct xs arbitrary: i, auto)
paulson@14208
  1789
apply (case_tac i, auto)
nipkow@13145
  1790
done
wenzelm@13114
  1791
nipkow@24526
  1792
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1793
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1794
apply (case_tac n, blast)
paulson@14208
  1795
apply (case_tac i, auto)
nipkow@13145
  1796
done
wenzelm@13114
  1797
wenzelm@13142
  1798
lemma nth_drop [simp]:
nipkow@24526
  1799
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1800
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1801
apply (case_tac xs, auto)
nipkow@13145
  1802
done
nipkow@3507
  1803
huffman@26584
  1804
lemma butlast_take:
huffman@30128
  1805
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1806
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1807
huffman@26584
  1808
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  1809
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1810
huffman@26584
  1811
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1812
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1813
huffman@26584
  1814
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  1815
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1816
bulwahn@46500
  1817
lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1818
by(simp add: hd_conv_nth)
nipkow@18423
  1819
nipkow@35248
  1820
lemma set_take_subset_set_take:
nipkow@35248
  1821
  "m <= n \<Longrightarrow> set(take m xs) <= set(take n xs)"
bulwahn@41463
  1822
apply (induct xs arbitrary: m n)
bulwahn@41463
  1823
apply simp
bulwahn@41463
  1824
apply (case_tac n)
bulwahn@41463
  1825
apply (auto simp: take_Cons)
bulwahn@41463
  1826
done
nipkow@35248
  1827
nipkow@24526
  1828
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1829
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1830
nipkow@24526
  1831
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1832
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1833
nipkow@35248
  1834
lemma set_drop_subset_set_drop:
nipkow@35248
  1835
  "m >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)"
nipkow@35248
  1836
apply(induct xs arbitrary: m n)
nipkow@35248
  1837
apply(auto simp:drop_Cons split:nat.split)
nipkow@35248
  1838
apply (metis set_drop_subset subset_iff)
nipkow@35248
  1839
done
nipkow@35248
  1840
nipkow@14187
  1841
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1842
using set_take_subset by fast
nipkow@14187
  1843
nipkow@14187
  1844
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1845
using set_drop_subset by fast
nipkow@14187
  1846
wenzelm@13114
  1847
lemma append_eq_conv_conj:
nipkow@24526
  1848
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1849
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1850
apply (case_tac zs, auto)
nipkow@13145
  1851
done
wenzelm@13142
  1852
nipkow@24526
  1853
lemma take_add: 
noschinl@42713
  1854
  "take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1855
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1856
apply (case_tac i, simp_all)
paulson@14050
  1857
done
paulson@14050
  1858
nipkow@14300
  1859
lemma append_eq_append_conv_if:
nipkow@24526
  1860
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1861
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1862
   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
nipkow@14300
  1863
   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)"
nipkow@24526
  1864
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1865
 apply simp
nipkow@14300
  1866
apply(case_tac ys\<^isub>1)
nipkow@14300
  1867
apply simp_all
nipkow@14300
  1868
done
nipkow@14300
  1869
nipkow@15110
  1870
lemma take_hd_drop:
huffman@30079
  1871
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  1872
apply(induct xs arbitrary: n)
nipkow@15110
  1873
apply simp
nipkow@15110
  1874
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1875
done
nipkow@15110
  1876
nipkow@17501
  1877
lemma id_take_nth_drop:
nipkow@17501
  1878
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1879
proof -
nipkow@17501
  1880
  assume si: "i < length xs"
nipkow@17501
  1881
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1882
  moreover
nipkow@17501
  1883
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1884
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1885
  ultimately show ?thesis by auto
nipkow@17501
  1886
qed
nipkow@17501
  1887
  
nipkow@17501
  1888
lemma upd_conv_take_nth_drop:
nipkow@17501
  1889
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1890
proof -
nipkow@17501
  1891
  assume i: "i < length xs"
nipkow@17501
  1892
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1893
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1894
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1895
    using i by (simp add: list_update_append)
nipkow@17501
  1896
  finally show ?thesis .
nipkow@17501
  1897
qed
nipkow@17501
  1898
haftmann@24796
  1899
lemma nth_drop':
haftmann@24796
  1900
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1901
apply (induct i arbitrary: xs)
haftmann@24796
  1902
apply (simp add: neq_Nil_conv)
haftmann@24796
  1903
apply (erule exE)+
haftmann@24796
  1904
apply simp
haftmann@24796
  1905
apply (case_tac xs)
haftmann@24796
  1906
apply simp_all
haftmann@24796
  1907
done
haftmann@24796
  1908
wenzelm@13114
  1909
nipkow@15392
  1910
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1911
hoelzl@33639
  1912
lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs"
hoelzl@33639
  1913
  by (induct xs) auto
hoelzl@33639
  1914
wenzelm@13142
  1915
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1916
by (induct xs) auto
wenzelm@13114
  1917
wenzelm@13142
  1918
lemma takeWhile_append1 [simp]:
nipkow@13145
  1919
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1920
by (induct xs) auto
wenzelm@13114
  1921
wenzelm@13142
  1922
lemma takeWhile_append2 [simp]:
nipkow@13145
  1923
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1924
by (induct xs) auto
wenzelm@13114
  1925
wenzelm@13142
  1926
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1927
by (induct xs) auto
wenzelm@13114
  1928
hoelzl@33639
  1929
lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j"
hoelzl@33639
  1930
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1931
hoelzl@33639
  1932
lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
hoelzl@33639
  1933
apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto
hoelzl@33639
  1934
hoelzl@33639
  1935
lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs"
hoelzl@33639
  1936
by (induct xs) auto
hoelzl@33639
  1937
wenzelm@13142
  1938
lemma dropWhile_append1 [simp]:
nipkow@13145
  1939
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1940
by (induct xs) auto
wenzelm@13114
  1941
wenzelm@13142
  1942
lemma dropWhile_append2 [simp]:
nipkow@13145
  1943
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1944
by (induct xs) auto
wenzelm@13114
  1945
noschinl@45841
  1946
lemma dropWhile_append3:
noschinl@45841
  1947
  "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
noschinl@45841
  1948
by (induct xs) auto
noschinl@45841
  1949
noschinl@45841
  1950
lemma dropWhile_last:
noschinl@45841
  1951
  "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs"
noschinl@45841
  1952
by (auto simp add: dropWhile_append3 in_set_conv_decomp)
noschinl@45841
  1953
noschinl@45841
  1954
lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs"
noschinl@45841
  1955
by (induct xs) (auto split: split_if_asm)
noschinl@45841
  1956
krauss@23971
  1957
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1958
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1959
nipkow@13913
  1960
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1961
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1962
by(induct xs, auto)
nipkow@13913
  1963
nipkow@13913
  1964
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1965
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1966
by(induct xs, auto)
nipkow@13913
  1967
nipkow@13913
  1968
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1969
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1970
by(induct xs, auto)
nipkow@13913
  1971
nipkow@31077
  1972
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  1973
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  1974
nipkow@31077
  1975
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  1976
by (induct xs) auto
nipkow@31077
  1977
hoelzl@33639
  1978
lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)"
hoelzl@33639
  1979
by (induct xs) auto
hoelzl@33639
  1980
hoelzl@33639
  1981
lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)"
hoelzl@33639
  1982
by (induct xs) auto
hoelzl@33639
  1983
hoelzl@33639
  1984
lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs"
hoelzl@33639
  1985
by (induct xs) auto
hoelzl@33639
  1986
hoelzl@33639
  1987
lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs"
hoelzl@33639
  1988
by (induct xs) auto
hoelzl@33639
  1989
hoelzl@33639
  1990
lemma hd_dropWhile:
hoelzl@33639
  1991
  "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))"
hoelzl@33639
  1992
using assms by (induct xs) auto
hoelzl@33639
  1993
hoelzl@33639
  1994
lemma takeWhile_eq_filter:
hoelzl@33639
  1995
  assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x"
hoelzl@33639
  1996
  shows "takeWhile P xs = filter P xs"
hoelzl@33639
  1997
proof -
hoelzl@33639
  1998
  have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
hoelzl@33639
  1999
    by simp
hoelzl@33639
  2000
  have B: "filter P (dropWhile P xs) = []"
hoelzl@33639
  2001
    unfolding filter_empty_conv using assms by blast
hoelzl@33639
  2002
  have "filter P xs = takeWhile P xs"
hoelzl@33639
  2003
    unfolding A filter_append B
hoelzl@33639
  2004
    by (auto simp add: filter_id_conv dest: set_takeWhileD)
hoelzl@33639
  2005
  thus ?thesis ..
hoelzl@33639
  2006
qed
hoelzl@33639
  2007
hoelzl@33639
  2008
lemma takeWhile_eq_take_P_nth:
hoelzl@33639
  2009
  "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow>
hoelzl@33639
  2010
  takeWhile P xs = take n xs"
hoelzl@33639
  2011
proof (induct xs arbitrary: n)
hoelzl@33639
  2012
  case (Cons x xs)
hoelzl@33639
  2013
  thus ?case
hoelzl@33639
  2014
  proof (cases n)
hoelzl@33639
  2015
    case (Suc n') note this[simp]
hoelzl@33639
  2016
    have "P x" using Cons.prems(1)[of 0] by simp
hoelzl@33639
  2017
    moreover have "takeWhile P xs = take n' xs"
hoelzl@33639
  2018
    proof (rule Cons.hyps)
hoelzl@33639
  2019
      case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp
hoelzl@33639
  2020
    next case goal2 thus ?case using Cons by auto
hoelzl@33639
  2021
    qed
hoelzl@33639
  2022
    ultimately show ?thesis by simp
hoelzl@33639
  2023
   qed simp
hoelzl@33639
  2024
qed simp
hoelzl@33639
  2025
hoelzl@33639
  2026
lemma nth_length_takeWhile:
hoelzl@33639
  2027
  "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))"
hoelzl@33639
  2028
by (induct xs) auto
hoelzl@33639
  2029
hoelzl@33639
  2030
lemma length_takeWhile_less_P_nth:
hoelzl@33639
  2031
  assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs"
hoelzl@33639
  2032
  shows "j \<le> length (takeWhile P xs)"
hoelzl@33639
  2033
proof (rule classical)
hoelzl@33639
  2034
  assume "\<not> ?thesis"
hoelzl@33639
  2035
  hence "length (takeWhile P xs) < length xs" using assms by simp
hoelzl@33639
  2036
  thus ?thesis using all `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto
hoelzl@33639
  2037
qed
nipkow@31077
  2038
nipkow@17501
  2039
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  2040
property. *}
nipkow@17501
  2041
nipkow@17501
  2042
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2043
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  2044
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  2045
nipkow@17501
  2046
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  2047
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  2048
apply(induct xs)
nipkow@17501
  2049
 apply simp
nipkow@17501
  2050
apply auto
nipkow@17501
  2051
apply(subst dropWhile_append2)
nipkow@17501
  2052
apply auto
nipkow@17501
  2053
done
nipkow@17501
  2054
nipkow@18423
  2055
lemma takeWhile_not_last:
bulwahn@46500
  2056
 "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  2057
apply(induct xs)
nipkow@18423
  2058
 apply simp
nipkow@18423
  2059
apply(case_tac xs)
nipkow@18423
  2060
apply(auto)
nipkow@18423
  2061
done
nipkow@18423
  2062
krauss@44013
  2063
lemma takeWhile_cong [fundef_cong]:
krauss@18336
  2064
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2065
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  2066
by (induct k arbitrary: l) (simp_all)
krauss@18336
  2067
krauss@44013
  2068
lemma dropWhile_cong [fundef_cong]:
krauss@18336
  2069
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  2070
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  2071
by (induct k arbitrary: l, simp_all)
krauss@18336
  2072
wenzelm@13114
  2073
nipkow@15392
  2074
subsubsection {* @{text zip} *}
wenzelm@13114
  2075
wenzelm@13142
  2076
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  2077
by (induct ys) auto
wenzelm@13114
  2078
wenzelm@13142
  2079
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  2080
by simp
wenzelm@13114
  2081
wenzelm@13142
  2082
declare zip_Cons [simp del]
wenzelm@13114
  2083
haftmann@36198
  2084
lemma [code]:
haftmann@36198
  2085
  "zip [] ys = []"
haftmann@36198
  2086
  "zip xs [] = []"
haftmann@36198
  2087
  "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
haftmann@36198
  2088
  by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+
haftmann@36198
  2089
nipkow@15281
  2090
lemma zip_Cons1:
nipkow@15281
  2091
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  2092
by(auto split:list.split)
nipkow@15281
  2093
wenzelm@13142
  2094
lemma length_zip [simp]:
krauss@22493
  2095
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  2096
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2097
haftmann@34978
  2098
lemma zip_obtain_same_length:
haftmann@34978
  2099
  assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys)
haftmann@34978
  2100
    \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)"
haftmann@34978
  2101
  shows "P (zip xs ys)"
haftmann@34978
  2102
proof -
haftmann@34978
  2103
  let ?n = "min (length xs) (length ys)"
haftmann@34978
  2104
  have "P (zip (take ?n xs) (take ?n ys))"
haftmann@34978
  2105
    by (rule assms) simp_all
haftmann@34978
  2106
  moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)"
haftmann@34978
  2107
  proof (induct xs arbitrary: ys)
haftmann@34978
  2108
    case Nil then show ?case by simp
haftmann@34978
  2109
  next
haftmann@34978
  2110
    case (Cons x xs) then show ?case by (cases ys) simp_all
haftmann@34978
  2111
  qed
haftmann@34978
  2112
  ultimately show ?thesis by simp
haftmann@34978
  2113
qed
haftmann@34978
  2114
wenzelm@13114
  2115
lemma zip_append1:
krauss@22493
  2116
"zip (xs @ ys) zs =
nipkow@13145
  2117
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  2118
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  2119
wenzelm@13114
  2120
lemma zip_append2:
krauss@22493
  2121
"zip xs (ys @ zs) =
nipkow@13145
  2122
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  2123
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  2124
wenzelm@13142
  2125
lemma zip_append [simp]:
bulwahn@46500
  2126
 "[| length xs = length us |] ==>
nipkow@13145
  2127
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  2128
by (simp add: zip_append1)
wenzelm@13114
  2129
wenzelm@13114
  2130
lemma zip_rev:
nipkow@14247
  2131
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  2132
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  2133
hoelzl@33639
  2134
lemma zip_map_map:
hoelzl@33639
  2135
  "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)"
hoelzl@33639
  2136
proof (induct xs arbitrary: ys)
hoelzl@33639
  2137
  case (Cons x xs) note Cons_x_xs = Cons.hyps
hoelzl@33639
  2138
  show ?case
hoelzl@33639
  2139
  proof (cases ys)
hoelzl@33639
  2140
    case (Cons y ys')
hoelzl@33639
  2141
    show ?thesis unfolding Cons using Cons_x_xs by simp
hoelzl@33639
  2142
  qed simp
hoelzl@33639
  2143
qed simp
hoelzl@33639
  2144
hoelzl@33639
  2145
lemma zip_map1:
hoelzl@33639
  2146
  "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)"
hoelzl@33639
  2147
using zip_map_map[of f xs "\<lambda>x. x" ys] by simp
hoelzl@33639
  2148
hoelzl@33639
  2149
lemma zip_map2:
hoelzl@33639
  2150
  "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)"
hoelzl@33639
  2151
using zip_map_map[of "\<lambda>x. x" xs f ys] by simp
hoelzl@33639
  2152
nipkow@23096
  2153
lemma map_zip_map:
hoelzl@33639
  2154
  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
hoelzl@33639
  2155
unfolding zip_map1 by auto
nipkow@23096
  2156
nipkow@23096
  2157
lemma map_zip_map2:
hoelzl@33639
  2158
  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
hoelzl@33639
  2159
unfolding zip_map2 by auto
nipkow@23096
  2160
nipkow@31080
  2161
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2162
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  2163
by(induct xs) auto
nipkow@31080
  2164
wenzelm@13142
  2165
lemma nth_zip [simp]:
nipkow@24526
  2166
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  2167
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  2168
apply (case_tac xs)
nipkow@13145
  2169
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  2170
done
wenzelm@13114
  2171
wenzelm@13114
  2172
lemma set_zip:
nipkow@13145
  2173
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  2174
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  2175
hoelzl@33639
  2176
lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
hoelzl@33639
  2177
by(induct xs) auto
hoelzl@33639
  2178
wenzelm@13114
  2179
lemma zip_update:
nipkow@31080
  2180
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  2181
by(rule sym, simp add: update_zip)
wenzelm@13114
  2182
wenzelm@13142
  2183
lemma zip_replicate [simp]:
nipkow@24526
  2184
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  2185
apply (induct i arbitrary: j, auto)
paulson@14208
  2186
apply (case_tac j, auto)
nipkow@13145
  2187
done
wenzelm@13114
  2188
nipkow@19487
  2189
lemma take_zip:
nipkow@24526
  2190
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  2191
apply (induct n arbitrary: xs ys)
nipkow@19487
  2192
 apply simp
nipkow@19487
  2193
apply (case_tac xs, simp)
nipkow@19487
  2194
apply (case_tac ys, simp_all)
nipkow@19487
  2195
done
nipkow@19487
  2196
nipkow@19487
  2197
lemma drop_zip:
nipkow@24526
  2198
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  2199
apply (induct n arbitrary: xs ys)
nipkow@19487
  2200
 apply simp
nipkow@19487
  2201
apply (case_tac xs, simp)
nipkow@19487
  2202
apply (case_tac ys, simp_all)
nipkow@19487
  2203
done
nipkow@19487
  2204
hoelzl@33639
  2205
lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)"
hoelzl@33639
  2206
proof (induct xs arbitrary: ys)
hoelzl@33639
  2207
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2208
qed simp
hoelzl@33639
  2209
hoelzl@33639
  2210
lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)"
hoelzl@33639
  2211
proof (induct xs arbitrary: ys)
hoelzl@33639
  2212
  case (Cons x xs) thus ?case by (cases ys) auto
hoelzl@33639
  2213
qed simp
hoelzl@33639
  2214
krauss@22493
  2215
lemma set_zip_leftD:
krauss@22493
  2216
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  2217
by (induct xs ys rule:list_induct2') auto
krauss@22493
  2218
krauss@22493
  2219
lemma set_zip_rightD:
krauss@22493
  2220
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  2221
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  2222
nipkow@23983
  2223
lemma in_set_zipE:
nipkow@23983
  2224
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  2225
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  2226
haftmann@29829
  2227
lemma zip_map_fst_snd:
haftmann@29829
  2228
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  2229
  by (induct zs) simp_all
haftmann@29829
  2230
haftmann@29829
  2231
lemma zip_eq_conv:
haftmann@29829
  2232
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  2233
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  2234
wenzelm@35115
  2235
nipkow@15392
  2236
subsubsection {* @{text list_all2} *}
wenzelm@13114
  2237
kleing@14316
  2238
lemma list_all2_lengthD [intro?]: 
kleing@14316
  2239
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  2240
by (simp add: list_all2_def)
haftmann@19607
  2241
haftmann@19787
  2242
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  2243
by (simp add: list_all2_def)
haftmann@19607
  2244
haftmann@19787
  2245
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  2246
by (simp add: list_all2_def)
haftmann@19607
  2247
haftmann@19607
  2248
lemma list_all2_Cons [iff, code]:
haftmann@19607
  2249
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  2250
by (auto simp add: list_all2_def)
wenzelm@13114
  2251
wenzelm@13114
  2252
lemma list_all2_Cons1:
nipkow@13145
  2253
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  2254
by (cases ys) auto
wenzelm@13114
  2255
wenzelm@13114
  2256
lemma list_all2_Cons2:
nipkow@13145
  2257
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  2258
by (cases xs) auto
wenzelm@13114
  2259
huffman@45794
  2260
lemma list_all2_induct
huffman@45794
  2261
  [consumes 1, case_names Nil Cons, induct set: list_all2]:
huffman@45794
  2262
  assumes P: "list_all2 P xs ys"
huffman@45794
  2263
  assumes Nil: "R [] []"
huffman@47640
  2264
  assumes Cons: "\<And>x xs y ys.
huffman@47640
  2265
    \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)"
huffman@45794
  2266
  shows "R xs ys"
huffman@45794
  2267
using P
huffman@45794
  2268
by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons)
huffman@45794
  2269
wenzelm@13142
  2270
lemma list_all2_rev [iff]:
nipkow@13145
  2271
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  2272
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  2273
kleing@13863
  2274
lemma list_all2_rev1:
kleing@13863
  2275
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  2276
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  2277
wenzelm@13114
  2278
lemma list_all2_append1:
nipkow@13145
  2279
"list_all2 P (xs @ ys) zs =
nipkow@13145
  2280
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  2281
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  2282
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  2283
apply (rule iffI)
nipkow@13145
  2284
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  2285
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  2286
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2287
apply (simp add: ball_Un)
nipkow@13145
  2288
done
wenzelm@13114
  2289
wenzelm@13114
  2290
lemma list_all2_append2:
nipkow@13145
  2291
"list_all2 P xs (ys @ zs) =
nipkow@13145
  2292
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  2293
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  2294
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  2295
apply (rule iffI)
nipkow@13145
  2296
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  2297
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  2298
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  2299
apply (simp add: ball_Un)
nipkow@13145
  2300
done
wenzelm@13114
  2301
kleing@13863
  2302
lemma list_all2_append:
nipkow@14247
  2303
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  2304
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  2305
by (induct rule:list_induct2, simp_all)
kleing@13863
  2306
kleing@13863
  2307
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  2308
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  2309
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  2310
wenzelm@13114
  2311
lemma list_all2_conv_all_nth:
nipkow@13145
  2312
"list_all2 P xs ys =
nipkow@13145
  2313
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  2314
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  2315
berghofe@13883
  2316
lemma list_all2_trans:
berghofe@13883
  2317
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  2318
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  2319
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  2320
proof (induct as)
berghofe@13883
  2321
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  2322
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  2323
  proof (induct bs)
berghofe@13883
  2324
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  2325
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  2326
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  2327
  qed simp
berghofe@13883
  2328
qed simp
berghofe@13883
  2329
kleing@13863
  2330
lemma list_all2_all_nthI [intro?]:
kleing@13863
  2331
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2332
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2333
paulson@14395
  2334
lemma list_all2I:
paulson@14395
  2335
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2336
by (simp add: list_all2_def)
paulson@14395
  2337
kleing@14328
  2338
lemma list_all2_nthD:
kleing@13863
  2339
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2340
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2341
nipkow@14302
  2342
lemma list_all2_nthD2:
nipkow@14302
  2343
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2344
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  2345
kleing@13863
  2346
lemma list_all2_map1: 
kleing@13863
  2347
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  2348
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2349
kleing@13863
  2350
lemma list_all2_map2: 
kleing@13863
  2351
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  2352
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  2353
kleing@14316
  2354
lemma list_all2_refl [intro?]:
kleing@13863
  2355
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  2356
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2357
kleing@13863
  2358
lemma list_all2_update_cong:
bulwahn@46669
  2359
  "\<lbrakk> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
bulwahn@46669
  2360
by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  2361
nipkow@14302
  2362
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  2363
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  2364
apply (induct xs arbitrary: n ys)
nipkow@24526
  2365
 apply simp
nipkow@24526
  2366
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2367
apply (case_tac n)
nipkow@24526
  2368
apply auto
nipkow@24526
  2369
done
nipkow@14302
  2370
nipkow@14302
  2371
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  2372
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  2373
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  2374
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2375
apply (case_tac n, simp, simp)
nipkow@24526
  2376
done
kleing@13863
  2377
kleing@14327
  2378
lemma list_all2_mono [intro?]:
nipkow@24526
  2379
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  2380
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  2381
apply (case_tac ys, auto)
nipkow@24526
  2382
done
kleing@13863
  2383
haftmann@22551
  2384
lemma list_all2_eq:
haftmann@22551
  2385
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2386
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2387
nipkow@40230
  2388
lemma list_eq_iff_zip_eq:
nipkow@40230
  2389
  "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)"
nipkow@40230
  2390
by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong)
nipkow@40230
  2391
wenzelm@13142
  2392
haftmann@47397
  2393
subsubsection {* @{const fold} with natural argument order *}
haftmann@46133
  2394
haftmann@46133
  2395
lemma fold_remove1_split:
haftmann@46133
  2396
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
haftmann@46133
  2397
    and x: "x \<in> set xs"
haftmann@46133
  2398
  shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
haftmann@46133
  2399
  using assms by (induct xs) (auto simp add: o_assoc [symmetric])
haftmann@46133
  2400
haftmann@46133
  2401
lemma fold_cong [fundef_cong]:
haftmann@46133
  2402
  "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)
haftmann@46133
  2403
    \<Longrightarrow> fold f xs a = fold g ys b"
haftmann@46133
  2404
  by (induct ys arbitrary: a b xs) simp_all
haftmann@46133
  2405
haftmann@46133
  2406
lemma fold_id:
haftmann@46133
  2407
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id"
haftmann@46133
  2408
  shows "fold f xs = id"
haftmann@46133
  2409
  using assms by (induct xs) simp_all
haftmann@46133
  2410
haftmann@46133
  2411
lemma fold_commute:
haftmann@46133
  2412
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
haftmann@46133
  2413
  shows "h \<circ> fold g xs = fold f xs \<circ> h"
haftmann@46133
  2414
  using assms by (induct xs) (simp_all add: fun_eq_iff)
haftmann@46133
  2415
haftmann@46133
  2416
lemma fold_commute_apply:
haftmann@46133
  2417
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h"
haftmann@46133
  2418
  shows "h (fold g xs s) = fold f xs (h s)"
haftmann@46133
  2419
proof -
haftmann@46133
  2420
  from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute)
haftmann@46133
  2421
  then show ?thesis by (simp add: fun_eq_iff)
haftmann@37605
  2422
qed
haftmann@37605
  2423
haftmann@46133
  2424
lemma fold_invariant: 
haftmann@46133
  2425
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s"
haftmann@46133
  2426
    and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)"
haftmann@46133
  2427
  shows "P (fold f xs s)"
haftmann@34978
  2428
  using assms by (induct xs arbitrary: s) simp_all
haftmann@34978
  2429
haftmann@46133
  2430
lemma fold_append [simp]:
haftmann@46133
  2431
  "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
haftmann@46133
  2432
  by (induct xs) simp_all
haftmann@46133
  2433
haftmann@46133
  2434
lemma fold_map [code_unfold]:
haftmann@46133
  2435
  "fold g (map f xs) = fold (g o f) xs"
haftmann@46133
  2436
  by (induct xs) simp_all
haftmann@46133
  2437
haftmann@46133
  2438
lemma fold_rev:
haftmann@46133
  2439
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
haftmann@46133
  2440
  shows "fold f (rev xs) = fold f xs"
haftmann@46133
  2441
using assms by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff)
haftmann@46133
  2442
haftmann@46133
  2443
lemma fold_Cons_rev:
haftmann@46133
  2444
  "fold Cons xs = append (rev xs)"
haftmann@46133
  2445
  by (induct xs) simp_all
haftmann@46133
  2446
haftmann@46133
  2447
lemma rev_conv_fold [code]:
haftmann@46133
  2448
  "rev xs = fold Cons xs []"
haftmann@46133
  2449
  by (simp add: fold_Cons_rev)
haftmann@46133
  2450
haftmann@46133
  2451
lemma fold_append_concat_rev:
haftmann@46133
  2452
  "fold append xss = append (concat (rev xss))"
haftmann@46133
  2453
  by (induct xss) simp_all
haftmann@46133
  2454
haftmann@46133
  2455
text {* @{const Finite_Set.fold} and @{const fold} *}
haftmann@46133
  2456
haftmann@46133
  2457
lemma (in comp_fun_commute) fold_set_fold_remdups:
haftmann@46133
  2458
  "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
haftmann@35195
  2459
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm insert_absorb)
haftmann@35195
  2460
haftmann@46133
  2461
lemma (in comp_fun_idem) fold_set_fold:
haftmann@46133
  2462
  "Finite_Set.fold f y (set xs) = fold f xs y"
haftmann@31455
  2463
  by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_comm)
haftmann@31455
  2464
haftmann@46133
  2465
lemma (in ab_semigroup_idem_mult) fold1_set_fold:
haftmann@32681
  2466
  assumes "xs \<noteq> []"
haftmann@46133
  2467
  shows "Finite_Set.fold1 times (set xs) = fold times (tl xs) (hd xs)"
haftmann@32681
  2468
proof -
haftmann@42871
  2469
  interpret comp_fun_idem times by (fact comp_fun_idem)
haftmann@32681
  2470
  from assms obtain y ys where xs: "xs = y # ys"
haftmann@32681
  2471
    by (cases xs) auto
haftmann@32681
  2472
  show ?thesis
haftmann@32681
  2473
  proof (cases "set ys = {}")
haftmann@32681
  2474
    case True with xs show ?thesis by simp
haftmann@32681
  2475
  next
haftmann@32681
  2476
    case False
haftmann@46034
  2477
    then have "fold1 times (insert y (set ys)) = Finite_Set.fold times y (set ys)"
haftmann@32681
  2478
      by (simp only: finite_set fold1_eq_fold_idem)
haftmann@46133
  2479
    with xs show ?thesis by (simp add: fold_set_fold mult_commute)
haftmann@32681
  2480
  qed
haftmann@32681
  2481
qed
haftmann@32681
  2482
haftmann@47397
  2483
lemma union_set_fold [code]:
haftmann@46147
  2484
  "set xs \<union> A = fold Set.insert xs A"
haftmann@46147
  2485
proof -
haftmann@46147
  2486
  interpret comp_fun_idem Set.insert
haftmann@46147
  2487
    by (fact comp_fun_idem_insert)
haftmann@46147
  2488
  show ?thesis by (simp add: union_fold_insert fold_set_fold)
haftmann@46147
  2489
qed
haftmann@46147
  2490
haftmann@47397
  2491
lemma union_coset_filter [code]:
haftmann@47397
  2492
  "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)"
haftmann@47397
  2493
  by auto
haftmann@47397
  2494
haftmann@47397
  2495
lemma minus_set_fold [code]:
haftmann@46147
  2496
  "A - set xs = fold Set.remove xs A"
haftmann@46147
  2497
proof -
haftmann@46147
  2498
  interpret comp_fun_idem Set.remove
haftmann@46147
  2499
    by (fact comp_fun_idem_remove)
haftmann@46147
  2500
  show ?thesis
haftmann@46147
  2501
    by (simp add: minus_fold_remove [of _ A] fold_set_fold)
haftmann@46147
  2502
qed
haftmann@46147
  2503
haftmann@47397
  2504
lemma minus_coset_filter [code]:
haftmann@47397
  2505
  "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
haftmann@47397
  2506
  by auto
haftmann@47397
  2507
haftmann@47397
  2508
lemma inter_set_filter [code]:
haftmann@47397
  2509
  "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)"
haftmann@47397
  2510
  by auto
haftmann@47397
  2511
haftmann@47397
  2512
lemma inter_coset_fold [code]:
haftmann@47397
  2513
  "A \<inter> List.coset xs = fold Set.remove xs A"
haftmann@47397
  2514
  by (simp add: Diff_eq [symmetric] minus_set_fold)
haftmann@47397
  2515
haftmann@46133
  2516
lemma (in lattice) Inf_fin_set_fold:
haftmann@46133
  2517
  "Inf_fin (set (x # xs)) = fold inf xs x"
haftmann@46133
  2518
proof -
haftmann@46133
  2519
  interpret ab_semigroup_idem_mult "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2520
    by (fact ab_semigroup_idem_mult_inf)
haftmann@46133
  2521
  show ?thesis
haftmann@46133
  2522
    by (simp add: Inf_fin_def fold1_set_fold del: set.simps)
haftmann@46133
  2523
qed
haftmann@46133
  2524
haftmann@47397
  2525
declare Inf_fin_set_fold [code]
haftmann@47397
  2526
haftmann@46133
  2527
lemma (in lattice) Sup_fin_set_fold:
haftmann@46133
  2528
  "Sup_fin (set (x # xs)) = fold sup xs x"
haftmann@46133
  2529
proof -
haftmann@46133
  2530
  interpret ab_semigroup_idem_mult "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2531
    by (fact ab_semigroup_idem_mult_sup)
haftmann@46133
  2532
  show ?thesis
haftmann@46133
  2533
    by (simp add: Sup_fin_def fold1_set_fold del: set.simps)
haftmann@46133
  2534
qed
haftmann@46133
  2535
haftmann@47397
  2536
declare Sup_fin_set_fold [code]
haftmann@47397
  2537
haftmann@46133
  2538
lemma (in linorder) Min_fin_set_fold:
haftmann@46133
  2539
  "Min (set (x # xs)) = fold min xs x"
haftmann@46133
  2540
proof -
haftmann@46133
  2541
  interpret ab_semigroup_idem_mult "min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2542
    by (fact ab_semigroup_idem_mult_min)
haftmann@46133
  2543
  show ?thesis
haftmann@46133
  2544
    by (simp add: Min_def fold1_set_fold del: set.simps)
haftmann@46133
  2545
qed
haftmann@46133
  2546
haftmann@47397
  2547
declare Min_fin_set_fold [code]
haftmann@47397
  2548
haftmann@46133
  2549
lemma (in linorder) Max_fin_set_fold:
haftmann@46133
  2550
  "Max (set (x # xs)) = fold max xs x"
haftmann@46133
  2551
proof -
haftmann@46133
  2552
  interpret ab_semigroup_idem_mult "max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2553
    by (fact ab_semigroup_idem_mult_max)
haftmann@46133
  2554
  show ?thesis
haftmann@46133
  2555
    by (simp add: Max_def fold1_set_fold del: set.simps)
haftmann@46133
  2556
qed
haftmann@46133
  2557
haftmann@47397
  2558
declare Max_fin_set_fold [code]
haftmann@47397
  2559
haftmann@46133
  2560
lemma (in complete_lattice) Inf_set_fold:
haftmann@46133
  2561
  "Inf (set xs) = fold inf xs top"
haftmann@46133
  2562
proof -
haftmann@46133
  2563
  interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2564
    by (fact comp_fun_idem_inf)
haftmann@46133
  2565
  show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute)
haftmann@46133
  2566
qed
haftmann@46133
  2567
haftmann@47397
  2568
declare Inf_set_fold [where 'a = "'a set", code]
haftmann@47397
  2569
haftmann@46133
  2570
lemma (in complete_lattice) Sup_set_fold:
haftmann@46133
  2571
  "Sup (set xs) = fold sup xs bot"
haftmann@46133
  2572
proof -
haftmann@46133
  2573
  interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@46133
  2574
    by (fact comp_fun_idem_sup)
haftmann@46133
  2575
  show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute)
haftmann@46133
  2576
qed
haftmann@46133
  2577
haftmann@47397
  2578
declare Sup_set_fold [where 'a = "'a set", code]
haftmann@47397
  2579
haftmann@46133
  2580
lemma (in complete_lattice) INF_set_fold:
haftmann@46133
  2581
  "INFI (set xs) f = fold (inf \<circ> f) xs top"
haftmann@46133
  2582
  unfolding INF_def set_map [symmetric] Inf_set_fold fold_map ..
haftmann@46133
  2583
haftmann@47397
  2584
declare INF_set_fold [code]
haftmann@47397
  2585
haftmann@46133
  2586
lemma (in complete_lattice) SUP_set_fold:
haftmann@46133
  2587
  "SUPR (set xs) f = fold (sup \<circ> f) xs bot"
haftmann@46133
  2588
  unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map ..
haftmann@46133
  2589
haftmann@47397
  2590
declare SUP_set_fold [code]
haftmann@46133
  2591
haftmann@46133
  2592
subsubsection {* Fold variants: @{const foldr} and @{const foldl} *}
haftmann@46133
  2593
haftmann@46133
  2594
text {* Correspondence *}
haftmann@46133
  2595
haftmann@47397
  2596
lemma foldr_conv_fold [code_abbrev]:
haftmann@47397
  2597
  "foldr f xs = fold f (rev xs)"
haftmann@47397
  2598
  by (induct xs) simp_all
haftmann@47397
  2599
haftmann@47397
  2600
lemma foldl_conv_fold:
haftmann@47397
  2601
  "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
haftmann@47397
  2602
  by (induct xs arbitrary: s) simp_all
haftmann@47397
  2603
haftmann@47397
  2604
lemma foldr_conv_foldl: -- {* The ``Third Duality Theorem'' in Bird \& Wadler: *}
haftmann@46133
  2605
  "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
haftmann@47397
  2606
  by (simp add: foldr_conv_fold foldl_conv_fold)
haftmann@47397
  2607
haftmann@47397
  2608
lemma foldl_conv_foldr:
haftmann@46133
  2609
  "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
haftmann@47397
  2610
  by (simp add: foldr_conv_fold foldl_conv_fold)
haftmann@46133
  2611
haftmann@46133
  2612
lemma foldr_fold:
haftmann@46133
  2613
  assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y"
haftmann@46133
  2614
  shows "foldr f xs = fold f xs"
haftmann@47397
  2615
  using assms unfolding foldr_conv_fold by (rule fold_rev)
haftmann@46133
  2616
haftmann@46133
  2617
lemma foldr_cong [fundef_cong]:
haftmann@46133
  2618
  "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a) \<Longrightarrow> foldr f l a = foldr g k b"
haftmann@47397
  2619
  by (auto simp add: foldr_conv_fold intro!: fold_cong)