src/HOL/List.thy
author haftmann
Sat, 20 Oct 2012 09:12:16 +0200
changeset 49948 744934b818c7
parent 49808 418991ce7567
child 49963 326f87427719
permissions -rw-r--r--
moved quite generic material from theory Enum to more appropriate places
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
     1
(*  Title:      HOL/List.thy
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
     2
    Author:     Tobias Nipkow
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     3
*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
     5
header {* The datatype of finite lists *}
13122
wenzelm
parents: 13114
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15113
diff changeset
     7
theory List
44013
5cfc1c36ae97 moved recdef package to HOL/Library/Old_Recdef.thy
krauss
parents: 43594
diff changeset
     8
imports Plain Presburger Code_Numeral Quotient ATP
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15113
diff changeset
     9
begin
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    10
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
    11
datatype 'a list =
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
    12
    Nil    ("[]")
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
    13
  | Cons 'a  "'a list"    (infixr "#" 65)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    14
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    15
syntax
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    16
  -- {* list Enumeration *}
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    17
  "_list" :: "args => 'a list"    ("[(_)]")
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    18
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    19
translations
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    20
  "[x, xs]" == "x#[xs]"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    21
  "[x]" == "x#[]"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    22
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    23
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    24
subsection {* Basic list processing functions *}
15302
a643fcbc3468 Restructured List and added "rotate"
nipkow
parents: 15281
diff changeset
    25
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    26
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    27
  hd :: "'a list \<Rightarrow> 'a" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    28
  "hd (x # xs) = x"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    29
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    30
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    31
  tl :: "'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    32
    "tl [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    33
  | "tl (x # xs) = xs"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    34
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    35
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    36
  last :: "'a list \<Rightarrow> 'a" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    37
  "last (x # xs) = (if xs = [] then x else last xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    38
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    39
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    40
  butlast :: "'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    41
    "butlast []= []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    42
  | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    43
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    44
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    45
  set :: "'a list \<Rightarrow> 'a set" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    46
    "set [] = {}"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    47
  | "set (x # xs) = insert x (set xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    48
46133
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
    49
definition
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
    50
  coset :: "'a list \<Rightarrow> 'a set" where
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
    51
  [simp]: "coset xs = - set xs"
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
    52
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    53
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    54
  map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    55
    "map f [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    56
  | "map f (x # xs) = f x # map f xs"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    57
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    58
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    59
  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    60
    append_Nil:"[] @ ys = ys"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    61
  | append_Cons: "(x#xs) @ ys = x # xs @ ys"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    62
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    63
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    64
  rev :: "'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    65
    "rev [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    66
  | "rev (x # xs) = rev xs @ [x]"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    67
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    68
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    69
  filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    70
    "filter P [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    71
  | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    72
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    73
syntax
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    74
  -- {* Special syntax for filter *}
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    75
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    76
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    77
translations
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    78
  "[x<-xs . P]"== "CONST filter (%x. P) xs"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    79
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    80
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    81
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    82
syntax (HTML output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
    83
  "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    84
47397
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    85
primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    86
where
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    87
  fold_Nil:  "fold f [] = id"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    88
| fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x" -- {* natural argument order *}
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    89
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    90
primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    91
where
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    92
  foldr_Nil:  "foldr f [] = id"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    93
| foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs" -- {* natural argument order *}
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    94
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    95
primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    96
where
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    97
  foldl_Nil:  "foldl f a [] = a"
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
    98
| foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
    99
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   100
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   101
  concat:: "'a list list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   102
    "concat [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   103
  | "concat (x # xs) = x @ concat xs"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   104
39774
30cf9d80939e localized listsum
haftmann
parents: 39728
diff changeset
   105
definition (in monoid_add)
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   106
  listsum :: "'a list \<Rightarrow> 'a" where
39774
30cf9d80939e localized listsum
haftmann
parents: 39728
diff changeset
   107
  "listsum xs = foldr plus xs 0"
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   108
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   109
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   110
  drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   111
    drop_Nil: "drop n [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   112
  | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   113
  -- {*Warning: simpset does not contain this definition, but separate
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   114
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   115
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   116
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   117
  take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   118
    take_Nil:"take n [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   119
  | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   120
  -- {*Warning: simpset does not contain this definition, but separate
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   121
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   122
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   123
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   124
  nth :: "'a list => nat => 'a" (infixl "!" 100) where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   125
  nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   126
  -- {*Warning: simpset does not contain this definition, but separate
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   127
       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   128
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   129
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   130
  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   131
    "list_update [] i v = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   132
  | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   133
41229
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 41075
diff changeset
   134
nonterminal lupdbinds and lupdbind
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 4643
diff changeset
   135
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   136
syntax
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   137
  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   138
  "" :: "lupdbind => lupdbinds"    ("_")
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   139
  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   140
  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
5077
71043526295f * HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents: 4643
diff changeset
   141
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   142
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
   143
  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   144
  "xs[i:=x]" == "CONST list_update xs i x"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   145
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   146
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   147
  takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   148
    "takeWhile P [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   149
  | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   150
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   151
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   152
  dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   153
    "dropWhile P [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   154
  | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   155
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   156
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   157
  zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   158
    "zip xs [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   159
  | zip_Cons: "zip xs (y # ys) = (case xs of [] => [] | z # zs => (z, y) # zip zs ys)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   160
  -- {*Warning: simpset does not contain this definition, but separate
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   161
       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   162
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   163
primrec
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   164
  product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   165
    "product [] _ = []"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   166
  | "product (x#xs) ys = map (Pair x) ys @ product xs ys"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   167
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   168
hide_const (open) product
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   169
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   170
primrec 
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   171
  upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   172
    upt_0: "[i..<0] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   173
  | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   174
34978
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   175
definition
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   176
  insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   177
  "insert x xs = (if x \<in> set xs then xs else x # xs)"
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   178
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36154
diff changeset
   179
hide_const (open) insert
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36154
diff changeset
   180
hide_fact (open) insert_def
34978
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   181
47122
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   182
primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   183
  "find _ [] = None"
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   184
| "find P (x#xs) = (if P x then Some x else find P xs)"
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   185
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   186
hide_const (open) find
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   187
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   188
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   189
  remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   190
    "remove1 x [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   191
  | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   192
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   193
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   194
  removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   195
    "removeAll x [] = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   196
  | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   197
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
   198
primrec
39915
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   199
  distinct :: "'a list \<Rightarrow> bool" where
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
   200
    "distinct [] \<longleftrightarrow> True"
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
   201
  | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
39915
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   202
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   203
primrec
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   204
  remdups :: "'a list \<Rightarrow> 'a list" where
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   205
    "remdups [] = []"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   206
  | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   207
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   208
primrec
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   209
  replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   210
    replicate_0: "replicate 0 x = []"
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   211
  | replicate_Suc: "replicate (Suc n) x = x # replicate n x"
3342
ec3b55fcb165 New operator "lists" for formalizing sets of lists
paulson
parents: 3320
diff changeset
   212
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   213
text {*
14589
feae7b5fd425 tuned document;
wenzelm
parents: 14565
diff changeset
   214
  Function @{text size} is overloaded for all datatypes. Users may
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   215
  refer to the list version as @{text length}. *}
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   216
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19302
diff changeset
   217
abbreviation
34941
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   218
  length :: "'a list \<Rightarrow> nat" where
156925dd67af dropped some old primrecs and some constdefs
haftmann
parents: 34886
diff changeset
   219
  "length \<equiv> size"
15307
10dd989282fd indentation
paulson
parents: 15305
diff changeset
   220
46440
d4994e2e7364 use 'primrec' to define "rotate1", for uniformity (and to help first-order tools that rely on "Spec_Rules")
blanchet
parents: 46439
diff changeset
   221
primrec rotate1 :: "'a list \<Rightarrow> 'a list" where
d4994e2e7364 use 'primrec' to define "rotate1", for uniformity (and to help first-order tools that rely on "Spec_Rules")
blanchet
parents: 46439
diff changeset
   222
  "rotate1 [] = []" |
d4994e2e7364 use 'primrec' to define "rotate1", for uniformity (and to help first-order tools that rely on "Spec_Rules")
blanchet
parents: 46439
diff changeset
   223
  "rotate1 (x # xs) = xs @ [x]"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   224
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   225
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   226
  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   227
  "rotate n = rotate1 ^^ n"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   228
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   229
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   230
  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37605
diff changeset
   231
  "list_all2 P xs ys =
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   232
    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   233
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   234
definition
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   235
  sublist :: "'a list => nat set => 'a list" where
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21211
diff changeset
   236
  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
17086
0eb0c9259dd7 added quite a few functions for code generation
nipkow
parents: 16998
diff changeset
   237
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   238
primrec
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   239
  sublists :: "'a list \<Rightarrow> 'a list list" where
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   240
  "sublists [] = [[]]"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   241
| "sublists (x#xs) = (let xss = sublists xs in map (Cons x) xss @ xss)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   242
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   243
primrec
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   244
  n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   245
  "n_lists 0 xs = [[]]"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   246
| "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   247
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   248
hide_const (open) n_lists
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   249
40593
1e57b18d27b1 code eqn for slice was missing; redefined splice with fun
nipkow
parents: 40365
diff changeset
   250
fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
1e57b18d27b1 code eqn for slice was missing; redefined splice with fun
nipkow
parents: 40365
diff changeset
   251
"splice [] ys = ys" |
1e57b18d27b1 code eqn for slice was missing; redefined splice with fun
nipkow
parents: 40365
diff changeset
   252
"splice xs [] = xs" |
1e57b18d27b1 code eqn for slice was missing; redefined splice with fun
nipkow
parents: 40365
diff changeset
   253
"splice (x#xs) (y#ys) = x # y # splice xs ys"
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   254
26771
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   255
text{*
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   256
\begin{figure}[htbp]
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   257
\fbox{
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   258
\begin{tabular}{l}
27381
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   259
@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   260
@{lemma "length [a,b,c] = 3" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   261
@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   262
@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   263
@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   264
@{lemma "hd [a,b,c,d] = a" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   265
@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   266
@{lemma "last [a,b,c,d] = d" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   267
@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   268
@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   269
@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
46133
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   270
@{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\
47397
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
   271
@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
   272
@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
27381
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   273
@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   274
@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   275
@{lemma "List.product [a,b] [c,d] = [(a, c), (a, d), (b, c), (b, d)]" by simp}\\
27381
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   276
@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   277
@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   278
@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   279
@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   280
@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   281
@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   282
@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   283
@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   284
@{lemma "distinct [2,0,1::nat]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   285
@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
34978
874150ddd50a canonical insert operation; generalized lemma foldl_apply_inv to foldl_apply
haftmann
parents: 34942
diff changeset
   286
@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
35295
397295fa8387 lemma distinct_insert
haftmann
parents: 35248
diff changeset
   287
@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
47122
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   288
@{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   289
@{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\
27381
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   290
@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
27693
73253a4e3ee2 added removeAll
nipkow
parents: 27381
diff changeset
   291
@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
27381
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   292
@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   293
@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
19ae7064f00f @{lemma}: 'by' keyword;
wenzelm
parents: 27368
diff changeset
   294
@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   295
@{lemma "sublists [a,b] = [[a, b], [a], [b], []]" by simp}\\
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   296
@{lemma "List.n_lists 2 [a,b,c] = [[a, a], [b, a], [c, a], [a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)}\\
46440
d4994e2e7364 use 'primrec' to define "rotate1", for uniformity (and to help first-order tools that rely on "Spec_Rules")
blanchet
parents: 46439
diff changeset
   297
@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\
d4994e2e7364 use 'primrec' to define "rotate1", for uniformity (and to help first-order tools that rely on "Spec_Rules")
blanchet
parents: 46439
diff changeset
   298
@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\
40077
c8a9eaaa2f59 nat_number -> eval_nat_numeral
nipkow
parents: 39963
diff changeset
   299
@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\
c8a9eaaa2f59 nat_number -> eval_nat_numeral
nipkow
parents: 39963
diff changeset
   300
@{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\
47397
d654c73e4b12 no preference wrt. fold(l/r); prefer fold rather than foldr for iterating over lists in generated code
haftmann
parents: 47131
diff changeset
   301
@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
26771
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   302
\end{tabular}}
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   303
\caption{Characteristic examples}
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   304
\label{fig:Characteristic}
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   305
\end{figure}
29927
ae8f42c245b2 Added nitpick attribute, and fixed typo.
blanchet
parents: 29856
diff changeset
   306
Figure~\ref{fig:Characteristic} shows characteristic examples
26771
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   307
that should give an intuitive understanding of the above functions.
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   308
*}
1d67ab20f358 Added documentation
nipkow
parents: 26749
diff changeset
   309
24616
fac3dd4ade83 sorting
nipkow
parents: 24566
diff changeset
   310
text{* The following simple sort functions are intended for proofs,
fac3dd4ade83 sorting
nipkow
parents: 24566
diff changeset
   311
not for efficient implementations. *}
fac3dd4ade83 sorting
nipkow
parents: 24566
diff changeset
   312
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   313
context linorder
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   314
begin
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   315
39915
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   316
inductive sorted :: "'a list \<Rightarrow> bool" where
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   317
  Nil [iff]: "sorted []"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   318
| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   319
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   320
lemma sorted_single [iff]:
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   321
  "sorted [x]"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   322
  by (rule sorted.Cons) auto
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   323
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   324
lemma sorted_many:
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   325
  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   326
  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   327
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   328
lemma sorted_many_eq [simp, code]:
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   329
  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   330
  by (auto intro: sorted_many elim: sorted.cases)
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   331
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   332
lemma [code]:
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   333
  "sorted [] \<longleftrightarrow> True"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   334
  "sorted [x] \<longleftrightarrow> True"
ecf97cf3d248 turned distinct and sorted into inductive predicates: yields nice induction principles for free; more elegant proofs
haftmann
parents: 39774
diff changeset
   335
  by simp_all
24697
b37d3980da3c fixed haftmann bug
nipkow
parents: 24657
diff changeset
   336
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   337
primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
46133
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   338
  "insort_key f x [] = [x]" |
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   339
  "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   340
35195
5163c2d00904 more lemmas about sort(_key)
haftmann
parents: 35115
diff changeset
   341
definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
46133
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   342
  "sort_key f xs = foldr (insort_key f) xs []"
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   343
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   344
definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   345
  "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   346
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   347
abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   348
abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   349
abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)"
35608
db4045d1406e added insort_insert
haftmann
parents: 35603
diff changeset
   350
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   351
end
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   352
24616
fac3dd4ade83 sorting
nipkow
parents: 24566
diff changeset
   353
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
   354
subsubsection {* List comprehension *}
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   355
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   356
text{* Input syntax for Haskell-like list comprehension notation.
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   357
Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   358
the list of all pairs of distinct elements from @{text xs} and @{text ys}.
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   359
The syntax is as in Haskell, except that @{text"|"} becomes a dot
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   360
(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   361
\verb![e| x <- xs, ...]!.
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   362
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   363
The qualifiers after the dot are
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   364
\begin{description}
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   365
\item[generators] @{text"p \<leftarrow> xs"},
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   366
 where @{text p} is a pattern and @{text xs} an expression of list type, or
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   367
\item[guards] @{text"b"}, where @{text b} is a boolean expression.
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   368
%\item[local bindings] @ {text"let x = e"}.
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   369
\end{description}
23240
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   370
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   371
Just like in Haskell, list comprehension is just a shorthand. To avoid
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   372
misunderstandings, the translation into desugared form is not reversed
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   373
upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   374
optmized to @{term"map (%x. e) xs"}.
23240
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   375
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   376
It is easy to write short list comprehensions which stand for complex
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   377
expressions. During proofs, they may become unreadable (and
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   378
mangled). In such cases it can be advisable to introduce separate
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   379
definitions for the list comprehensions in question.  *}
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   380
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   381
nonterminal lc_qual and lc_quals
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   382
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   383
syntax
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   384
  "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   385
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ <- _")
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   386
  "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   387
  (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   388
  "_lc_end" :: "lc_quals" ("]")
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   389
  "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals"  (", __")
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   390
  "_lc_abs" :: "'a => 'b list => 'b list"
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   391
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   392
(* These are easier than ML code but cannot express the optimized
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   393
   translation of [e. p<-xs]
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   394
translations
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   395
  "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   396
  "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   397
   => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   398
  "[e. P]" => "if P then [e] else []"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   399
  "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   400
   => "if P then (_listcompr e Q Qs) else []"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   401
  "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   402
   => "_Let b (_listcompr e Q Qs)"
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
   403
*)
23240
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   404
23279
e39dd93161d9 tuned list comprehension, changed filter syntax from : to <-
nipkow
parents: 23246
diff changeset
   405
syntax (xsymbols)
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   406
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
23279
e39dd93161d9 tuned list comprehension, changed filter syntax from : to <-
nipkow
parents: 23246
diff changeset
   407
syntax (HTML output)
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   408
  "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual"  ("_ \<leftarrow> _")
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   409
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   410
parse_translation (advanced) {*
46138
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   411
  let
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   412
    val NilC = Syntax.const @{const_syntax Nil};
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   413
    val ConsC = Syntax.const @{const_syntax Cons};
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   414
    val mapC = Syntax.const @{const_syntax map};
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   415
    val concatC = Syntax.const @{const_syntax concat};
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   416
    val IfC = Syntax.const @{const_syntax If};
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   417
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   418
    fun single x = ConsC $ x $ NilC;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   419
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   420
    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   421
      let
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   422
        (* FIXME proper name context!? *)
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   423
        val x =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   424
          Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT);
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   425
        val e = if opti then single e else e;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   426
        val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   427
        val case2 =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   428
          Syntax.const @{syntax_const "_case1"} $
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   429
            Syntax.const @{const_syntax dummy_pattern} $ NilC;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   430
        val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   431
      in Syntax_Trans.abs_tr [x, Datatype_Case.case_tr false ctxt [x, cs]] end;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   432
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   433
    fun abs_tr ctxt p e opti =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   434
      (case Term_Position.strip_positions p of
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   435
        Free (s, T) =>
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   436
          let
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   437
            val thy = Proof_Context.theory_of ctxt;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   438
            val s' = Proof_Context.intern_const ctxt s;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   439
          in
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   440
            if Sign.declared_const thy s'
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   441
            then (pat_tr ctxt p e opti, false)
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   442
            else (Syntax_Trans.abs_tr [p, e], true)
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   443
          end
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   444
      | _ => (pat_tr ctxt p e opti, false));
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   445
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   446
    fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   447
          let
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   448
            val res =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   449
              (case qs of
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   450
                Const (@{syntax_const "_lc_end"}, _) => single e
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   451
              | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]);
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   452
          in IfC $ b $ res $ NilC end
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   453
      | lc_tr ctxt
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   454
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   455
              Const(@{syntax_const "_lc_end"}, _)] =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   456
          (case abs_tr ctxt p e true of
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   457
            (f, true) => mapC $ f $ es
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   458
          | (f, false) => concatC $ (mapC $ f $ es))
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   459
      | lc_tr ctxt
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   460
            [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   461
              Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   462
          let val e' = lc_tr ctxt [e, q, qs];
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   463
          in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   464
85f8d8a8c711 improved list comprehension syntax: more careful treatment of position constraints, which enables PIDE markup;
wenzelm
parents: 46133
diff changeset
   465
  in [(@{syntax_const "_listcompr"}, lc_tr)] end
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   466
*}
23279
e39dd93161d9 tuned list comprehension, changed filter syntax from : to <-
nipkow
parents: 23246
diff changeset
   467
42167
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   468
ML {*
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   469
  let
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   470
    val read = Syntax.read_term @{context};
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   471
    fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1);
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   472
  in
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   473
    check "[(x,y,z). b]" "if b then [(x, y, z)] else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   474
    check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   475
    check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   476
    check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   477
    check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   478
    check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   479
    check "[(x,y). Cons True x \<leftarrow> xs]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   480
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   481
    check "[(x,y,z). Cons x [] \<leftarrow> xs]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   482
      "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   483
    check "[(x,y,z). x<a, x>b, x=d]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   484
      "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   485
    check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   486
      "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   487
    check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   488
      "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   489
    check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   490
      "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   491
    check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   492
      "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   493
    check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   494
      "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   495
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   496
      "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)";
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   497
    check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   498
      "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)"
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   499
  end;
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   500
*}
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   501
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
   502
(*
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   503
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   504
*)
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   505
42167
7d8cb105373c actually check list comprehension examples;
wenzelm
parents: 42144
diff changeset
   506
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48828
diff changeset
   507
ML_file "Tools/list_to_set_comprehension.ML"
41463
edbf0a86fb1c adding simproc to rewrite list comprehensions to set comprehensions; adopting proofs
bulwahn
parents: 41372
diff changeset
   508
edbf0a86fb1c adding simproc to rewrite list comprehensions to set comprehensions; adopting proofs
bulwahn
parents: 41372
diff changeset
   509
simproc_setup list_to_set_comprehension ("set xs") = {* K List_to_Set_Comprehension.simproc *}
edbf0a86fb1c adding simproc to rewrite list comprehensions to set comprehensions; adopting proofs
bulwahn
parents: 41372
diff changeset
   510
46133
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   511
code_datatype set coset
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   512
d9fe85d3d2cd incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents: 46125
diff changeset
   513
hide_const (open) coset
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
   514
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   515
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   516
subsubsection {* @{const Nil} and @{const Cons} *}
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   517
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   518
lemma not_Cons_self [simp]:
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   519
  "xs \<noteq> x # xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   520
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   521
41697
19890332febc explicit is better than implicit;
wenzelm
parents: 41505
diff changeset
   522
lemma not_Cons_self2 [simp]:
19890332febc explicit is better than implicit;
wenzelm
parents: 41505
diff changeset
   523
  "x # xs \<noteq> xs"
19890332febc explicit is better than implicit;
wenzelm
parents: 41505
diff changeset
   524
by (rule not_Cons_self [symmetric])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   525
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   526
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   527
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   528
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   529
lemma length_induct:
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   530
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   531
by (rule measure_induct [of length]) iprover
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   532
37289
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   533
lemma list_nonempty_induct [consumes 1, case_names single cons]:
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   534
  assumes "xs \<noteq> []"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   535
  assumes single: "\<And>x. P [x]"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   536
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   537
  shows "P xs"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   538
using `xs \<noteq> []` proof (induct xs)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   539
  case Nil then show ?case by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   540
next
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   541
  case (Cons x xs) show ?case proof (cases xs)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   542
    case Nil with single show ?thesis by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   543
  next
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   544
    case Cons then have "xs \<noteq> []" by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   545
    moreover with Cons.hyps have "P xs" .
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   546
    ultimately show ?thesis by (rule cons)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   547
  qed
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   548
qed
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   549
45714
ad4242285560 cardinality of sets of lists
hoelzl
parents: 45607
diff changeset
   550
lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X"
ad4242285560 cardinality of sets of lists
hoelzl
parents: 45607
diff changeset
   551
  by (auto intro!: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   552
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   553
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   554
subsubsection {* @{const length} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   555
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   556
text {*
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   557
  Needs to come before @{text "@"} because of theorem @{text
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   558
  append_eq_append_conv}.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   559
*}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   560
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   561
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   562
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   563
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   564
lemma length_map [simp]: "length (map f xs) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   565
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   566
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   567
lemma length_rev [simp]: "length (rev xs) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   568
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   569
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   570
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   571
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   572
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   573
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   574
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   575
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   576
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   577
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   578
23479
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   579
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   580
by auto
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   581
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   582
lemma length_Suc_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   583
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   584
by (induct xs) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   585
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   586
lemma Suc_length_conv:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   587
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   588
apply (induct xs, simp, simp)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   589
apply blast
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   590
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   591
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   592
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   593
  by (induct xs) auto
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   594
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   595
lemma list_induct2 [consumes 1, case_names Nil Cons]:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   596
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   597
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   598
   \<Longrightarrow> P xs ys"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   599
proof (induct xs arbitrary: ys)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   600
  case Nil then show ?case by simp
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   601
next
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   602
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   603
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   604
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   605
lemma list_induct3 [consumes 2, case_names Nil Cons]:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   606
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   607
   (\<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))
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   608
   \<Longrightarrow> P xs ys zs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   609
proof (induct xs arbitrary: ys zs)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   610
  case Nil then show ?case by simp
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   611
next
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   612
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   613
    (cases zs, simp_all)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   614
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   615
36154
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   616
lemma list_induct4 [consumes 3, case_names Nil Cons]:
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   617
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   618
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   619
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   620
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   621
proof (induct xs arbitrary: ys zs ws)
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   622
  case Nil then show ?case by simp
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   623
next
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   624
  case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   625
qed
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   626
22493
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   627
lemma list_induct2': 
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   628
  "\<lbrakk> P [] [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   629
  \<And>x xs. P (x#xs) [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   630
  \<And>y ys. P [] (y#ys);
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   631
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   632
 \<Longrightarrow> P xs ys"
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   633
by (induct xs arbitrary: ys) (case_tac x, auto)+
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   634
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   635
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   636
by (rule Eq_FalseI) auto
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   637
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   638
simproc_setup list_neq ("(xs::'a list) = ys") = {*
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   639
(*
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   640
Reduces xs=ys to False if xs and ys cannot be of the same length.
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   641
This is the case if the atomic sublists of one are a submultiset
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   642
of those of the other list and there are fewer Cons's in one than the other.
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   643
*)
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   644
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   645
let
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   646
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   647
fun len (Const(@{const_name Nil},_)) acc = acc
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   648
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   649
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   650
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   651
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   652
  | len t (ts,n) = (t::ts,n);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   653
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   654
fun list_neq _ ss ct =
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   655
  let
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   656
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   657
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   658
    fun prove_neq() =
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   659
      let
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   660
        val Type(_,listT::_) = eqT;
22994
02440636214f abstract size function in hologic.ML
haftmann
parents: 22940
diff changeset
   661
        val size = HOLogic.size_const listT;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   662
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   663
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   664
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
22633
haftmann
parents: 22551
diff changeset
   665
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann
parents: 22551
diff changeset
   666
      in SOME (thm RS @{thm neq_if_length_neq}) end
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   667
  in
23214
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   668
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   669
       n < m andalso submultiset (op aconv) (rs,ls)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   670
    then prove_neq() else NONE
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   671
  end;
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   672
in list_neq end;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   673
*}
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   674
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   675
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   676
subsubsection {* @{text "@"} -- append *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   677
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   678
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   679
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   680
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   681
lemma append_Nil2 [simp]: "xs @ [] = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   682
by (induct xs) auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   683
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   684
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   685
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   686
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   687
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   688
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   689
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   690
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   691
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   692
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   693
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   694
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   695
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35827
diff changeset
   696
lemma append_eq_append_conv [simp, no_atp]:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   697
 "length xs = length ys \<or> length us = length vs
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
   698
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   699
apply (induct xs arbitrary: ys)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   700
 apply (case_tac ys, simp, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   701
apply (case_tac ys, force, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   702
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   703
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   704
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   705
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   706
apply (induct xs arbitrary: ys zs ts)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
   707
 apply fastforce
14495
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   708
apply(case_tac zs)
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   709
 apply simp
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
   710
apply fastforce
14495
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   711
done
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   712
34910
b23bd3ee4813 same_append_eq / append_same_eq are now used for simplifying induction rules.
berghofe
parents: 34064
diff changeset
   713
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   714
by simp
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   715
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   716
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   717
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   718
34910
b23bd3ee4813 same_append_eq / append_same_eq are now used for simplifying induction rules.
berghofe
parents: 34064
diff changeset
   719
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   720
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   721
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   722
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   723
using append_same_eq [of _ _ "[]"] by auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   724
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   725
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   726
using append_same_eq [of "[]"] by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   727
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35827
diff changeset
   728
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   729
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   730
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   731
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   732
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   733
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   734
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   735
by (simp add: hd_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   736
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   737
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   738
by (simp split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   739
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   740
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   741
by (simp add: tl_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   742
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   743
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   744
lemma Cons_eq_append_conv: "x#xs = ys@zs =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   745
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   746
by(cases ys) auto
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   747
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   748
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   749
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   750
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   751
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   752
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   753
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   754
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   755
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   756
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   757
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   758
lemma Cons_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   759
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   760
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   761
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   762
lemma append_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   763
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   764
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   765
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   766
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   767
text {*
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   768
Simplification procedure for all list equalities.
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   769
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   770
- both lists end in a singleton list,
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   771
- or both lists end in the same list.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   772
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   773
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   774
simproc_setup list_eq ("(xs::'a list) = ys")  = {*
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   775
  let
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   776
    fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) =
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   777
          (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs)
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   778
      | last (Const(@{const_name append},_) $ _ $ ys) = last ys
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   779
      | last t = t;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   780
    
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   781
    fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   782
      | list1 _ = false;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   783
    
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   784
    fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   785
          (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs)
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   786
      | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   787
      | butlast xs = Const(@{const_name Nil}, fastype_of xs);
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   788
    
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   789
    val rearr_ss =
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   790
      HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}];
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   791
    
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   792
    fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   793
      let
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   794
        val lastl = last lhs and lastr = last rhs;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   795
        fun rearr conv =
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   796
          let
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   797
            val lhs1 = butlast lhs and rhs1 = butlast rhs;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   798
            val Type(_,listT::_) = eqT
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   799
            val appT = [listT,listT] ---> listT
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   800
            val app = Const(@{const_name append},appT)
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   801
            val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   802
            val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   803
            val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   804
              (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   805
          in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   806
      in
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   807
        if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   808
        else if lastl aconv lastr then rearr @{thm append_same_eq}
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   809
        else NONE
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   810
      end;
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 43580
diff changeset
   811
  in fn _ => fn ss => fn ct => list_eq ss (term_of ct) end;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   812
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   813
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   814
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   815
subsubsection {* @{const map} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   816
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   817
lemma hd_map:
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   818
  "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)"
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   819
  by (cases xs) simp_all
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   820
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   821
lemma map_tl:
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   822
  "map f (tl xs) = tl (map f xs)"
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   823
  by (cases xs) simp_all
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 40195
diff changeset
   824
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   825
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   826
by (induct xs) simp_all
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   827
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   828
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   829
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   830
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   831
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   832
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   833
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   834
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   835
by (induct xs) auto
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   836
35208
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   837
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   838
apply(rule ext)
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   839
apply(simp)
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   840
done
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   841
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   842
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   843
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   844
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   845
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   846
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   847
44013
5cfc1c36ae97 moved recdef package to HOL/Library/Old_Recdef.thy
krauss
parents: 43594
diff changeset
   848
lemma map_cong [fundef_cong]:
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
   849
  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
   850
  by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   851
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   852
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   853
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   854
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   855
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   856
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   857
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   858
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   859
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   860
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   861
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   862
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   863
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   864
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   865
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   866
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   867
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   868
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   869
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   870
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   871
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   872
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   873
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   874
lemma map_eq_imp_length_eq:
35510
64d2d54cbf03 Slightly generalised a theorem
paulson
parents: 35296
diff changeset
   875
  assumes "map f xs = map g ys"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   876
  shows "length xs = length ys"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   877
using assms proof (induct ys arbitrary: xs)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   878
  case Nil then show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   879
next
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   880
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
35510
64d2d54cbf03 Slightly generalised a theorem
paulson
parents: 35296
diff changeset
   881
  from Cons xs have "map f zs = map g ys" by simp
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   882
  moreover with Cons have "length zs = length ys" by blast
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   883
  with xs show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   884
qed
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   885
  
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   886
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   887
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   888
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   889
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   890
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   891
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   892
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   893
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   894
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   895
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   896
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   897
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   898
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   899
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   900
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   901
lemma map_injective:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   902
 "map f xs = map f ys ==> inj f ==> xs = ys"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   903
by (induct ys arbitrary: xs) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   904
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   905
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   906
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   907
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   908
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   909
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   910
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   911
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   912
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   913
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   914
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   915
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   916
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   917
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   918
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   919
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   920
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   921
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   922
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   923
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   924
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   925
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   926
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   927
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   928
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   929
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   930
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   931
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   932
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   933
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   934
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   935
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   936
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   937
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   938
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   939
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   940
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41463
diff changeset
   941
enriched_type map: map
47122
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   942
by (simp_all add: id_def)
790fb5eb5969 Functions and lemmas by Christian Sternagel
nipkow
parents: 47108
diff changeset
   943
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   944
declare map.id [simp]
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   945
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   946
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   947
subsubsection {* @{const rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   948
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   949
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   950
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   951
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   952
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   953
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   954
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   955
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   956
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   957
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   958
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   959
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   960
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   961
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   962
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   963
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   964
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   965
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   966
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   967
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   968
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   969
46439
2388be11cb50 removed fact that confuses SPASS -- better rely on "rev_rev_ident", which is stronger and more ATP friendly
blanchet
parents: 46424
diff changeset
   970
lemma rev_is_rev_conv [iff, no_atp]: "(rev xs = rev ys) = (xs = ys)"
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   971
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   972
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   973
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   974
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   975
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   976
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   977
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   978
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   979
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
15489
d136af442665 Replaced application of subst by simplesubst in proof of rev_induct
berghofe
parents: 15439
diff changeset
   980
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   981
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   982
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   983
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   984
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   985
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   986
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   987
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   988
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   989
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   990
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   991
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   992
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   993
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
   994
subsubsection {* @{const set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   995
46698
f1dfcf8be88d converting "set [...]" to "{...}" in evaluation results
nipkow
parents: 46669
diff changeset
   996
declare set.simps [code_post]  --"pretty output"
f1dfcf8be88d converting "set [...]" to "{...}" in evaluation results
nipkow
parents: 46669
diff changeset
   997
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   998
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   999
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1000
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1001
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1002
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1003
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
  1004
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
  1005
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1006
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1007
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1008
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1009
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1010
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1011
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
  1012
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1013
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1014
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1015
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
  1016
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
  1017
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
  1018
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1019
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1020
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1021
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1022
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1023
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1024
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1025
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1026
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1027
32417
e87d9c78910c tuned code generation for lists
nipkow
parents: 32415
diff changeset
  1028
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
41463
edbf0a86fb1c adding simproc to rewrite list comprehensions to set comprehensions; adopting proofs
bulwahn
parents: 41372
diff changeset
  1029
by (induct j) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1030
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1031
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1032
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1033
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1034
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1035
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1036
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1037
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1038
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1039
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1040
  by (auto elim: split_list)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1041
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1042
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1043
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1044
  case Nil thus ?case by simp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1045
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1046
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1047
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1048
  proof cases
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
  1049
    assume "x = a" thus ?case using Cons by fastforce
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1050
  next
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
  1051
    assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1052
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1053
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1054
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1055
lemma in_set_conv_decomp_first:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1056
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1057
  by (auto dest!: split_list_first)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1058
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
  1059
lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
  1060
proof (induct xs rule: rev_induct)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1061
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1062
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1063
  case (snoc a xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1064
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1065
  proof cases
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
  1066
    assume "x = a" thus ?case using snoc by (metis List.set.simps(1) emptyE)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1067
  next
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
  1068
    assume "x \<noteq> a" thus ?case using snoc by fastforce
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1069
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1070
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1071
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1072
lemma in_set_conv_decomp_last:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1073
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1074
  by (auto dest!: split_list_last)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1075
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1076
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1077
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1078
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1079
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1080
  case Cons thus ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1081
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1082
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1083
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1084
lemma split_list_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1085
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1086
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1087
using split_list_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1088
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1089
lemma split_list_first_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1090
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1091
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1092
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1093
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1094
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1095
  case (Cons x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1096
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1097
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1098
    assume "P x"
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 40077
diff changeset
  1099
    thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1100
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1101
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1102
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1103
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1104
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1105
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1106
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1107
lemma split_list_first_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1108
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1109
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1110
using split_list_first_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1111
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1112
lemma split_list_first_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1113
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1114
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1115
by (rule, erule split_list_first_prop) auto
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1116
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1117
lemma split_list_last_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1118
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1119
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1120
proof(induct xs rule:rev_induct)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1121
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1122
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1123
  case (snoc x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1124
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1125
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1126
    assume "P x" thus ?thesis by (metis emptyE set_empty)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1127
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1128
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1129
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
  1130
    thus ?thesis using `\<not> P x` snoc(1) by fastforce
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1131
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1132
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1133
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1134
lemma split_list_last_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1135
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1136
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1137
using split_list_last_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1138
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1139
lemma split_list_last_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1140
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1141
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1142
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1143
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1144
lemma finite_list: "finite A ==> EX xs. set xs = A"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1145
  by (erule finite_induct)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1146
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
  1147
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1148
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1149
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1150
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1151
lemma set_minus_filter_out:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1152
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1153
  by (induct xs) auto
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
  1154
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
  1155
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49808
diff changeset
  1156
subsubsection {* @{const filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1157
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1158
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1159
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1160
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1161
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1162
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1163
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1164
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1165
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1166
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1167
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1168
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1169
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1170
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1171
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1172
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1173
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1174
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1175
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1176
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1177
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1178
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1179
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1180
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1181
by (induct xs) simp_all
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1182
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1183
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1184
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1185
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1186
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1187
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1188
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1189
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1190
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1191
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1192
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1193
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1194
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1195
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1196
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1197
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1198
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1199
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1200
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1201
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1202
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1203
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1204
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1205
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1206
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1207
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1208
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1209
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1210
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1211
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1212
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1213
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1214
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1215
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1216
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1217
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1218
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1219
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1220
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1221
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1222
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1223
    hence eq: "?S' = insert 0 (Suc ` ?S)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1224
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1225
    have "length (filter p (x # xs)) = Suc(card ?S)"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1226
      using Cons `p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1227
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
44921
58eef4843641 tuned proofs
huffman
parents: 44916
diff changeset
  1228
      by (simp add: card_image)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1229
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1230
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1231
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1232
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1233
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1234
    hence eq: "?S' = Suc ` ?S"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1235
      by(auto simp add: image_def split:nat.split elim:lessE)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1236
    have "length (filter p (x # xs)) = card ?S"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1237
      using Cons `\<not> p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1238
    also have "\<dots> = card(Suc ` ?S)" using fin
44921
58eef4843641 tuned proofs
huffman
parents: 44916
diff changeset
  1239
      by (simp add: card_image)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1240
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1241
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1242
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1243
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1244
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1245
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1246
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1247
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1248
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
19585
70a1ce3b23ae removed 'concl is' patterns;
wenzelm
parents: 19487
diff changeset
  1249
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1250
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1251
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1252
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1253
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1254
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1255
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1256
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1257
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1258
    proof cases
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1259
      assume "x = y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1260
      with Py Cons.prems have "?Q []" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1261
      then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1262
    next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1263
      assume "x \<noteq> y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1264
      with Py Cons.prems show ?thesis by simp
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1265
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1266
  next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1267
    assume "\<not> P y"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44635
diff changeset
  1268
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1269
    then have "?Q (y#us)" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1270
    then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1271
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1272
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1273
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1274
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1275
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1276
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1277
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1278
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1279
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1280
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1281
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1282
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1283
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1284
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1285
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1286
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1287
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1288
44013
5cfc1c36ae97 moved recdef package to HOL/Library/Old_Recdef.thy
krauss
parents: 43594
diff changeset
  1289
lemma filter_cong[fundef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1290
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1291
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1292
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1293
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1294
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1295
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1296
subsubsection {* List partitioning *}
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1297
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1298
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1299
  "partition P [] = ([], [])"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1300
  | "partition P (x # xs) = 
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1301
      (let (yes, no) = partition P xs
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1302
      in if P x then (x # yes, no) else (yes, x # no))"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1303
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1304
lemma partition_filter1:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1305
    "fst (partition P xs) = filter P xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1306
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1307
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1308
lemma partition_filter2:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1309
    "snd (partition P xs) = filter (Not o P) xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1310
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1311
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1312
lemma partition_P:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1313
  assumes "partition P xs = (yes, no)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1314
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1315
proof -
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1316
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1317
    by simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1318
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1319
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1320
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1321
lemma partition_set:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
ha