src/HOL/List.thy
author nipkow
Tue Jul 15 08:25:20 2003 +0200 (2003-07-15)
changeset 14111 993471c762b8
parent 14099 55d244f3c86d
child 14187 26dfcd0ac436
permissions -rw-r--r--
Some new thm (ex_map_conv?)
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* The datatype of finite lists *}
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theory List = PreList:
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  o2l :: "'a option => 'a list"
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  list_all:: "('a => bool) => ('a list => bool)"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  mem :: "'a => 'a list => bool"    (infixl 55)
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_../_'(])")
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  remdups :: "'a list => 'a list"
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  null:: "'a list => bool"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  postfix :: "'a list => 'a list => bool"
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syntax (xsymbols)
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  postfix :: "'a list => 'a list => bool"             ("(_/ \<sqsupseteq> _)" [51, 51] 50)
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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  upto:: "nat => nat => nat list"    ("(1[_../_])")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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  "[i..j]" == "[i..(Suc j)(]"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes.Users may
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  refer to the list version as @{text length}. *}
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syntax length :: "'a list => nat"
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translations "length" => "size :: _ list => nat"
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typed_print_translation {*
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  let
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    fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
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          Syntax.const "length" $ t
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      | size_tr' _ _ _ = raise Match;
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  in [("size", size_tr')] end
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*}
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primrec
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"hd(x#xs) = x"
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primrec
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"tl([]) = []"
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"tl(x#xs) = xs"
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primrec
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"null([]) = True"
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"null(x#xs) = False"
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primrec
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"last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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"butlast []= []"
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"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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"x mem [] = False"
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"x mem (y#ys) = (if y=x then True else x mem ys)"
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primrec
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"set [] = {}"
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"set (x#xs) = insert x (set xs)"
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primrec
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 "o2l  None    = []"
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 "o2l (Some x) = [x]"
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primrec
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list_all_Nil:"list_all P [] = True"
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list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
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primrec
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"map f [] = []"
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"map f (x#xs) = f(x)#map f xs"
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primrec
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append_Nil:"[]@ys = ys"
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append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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"rev([]) = []"
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"rev(x#xs) = rev(xs) @ [x]"
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primrec
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"filter P [] = []"
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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foldl_Nil:"foldl f a [] = a"
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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"foldr f [] a = a"
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"foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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"concat([]) = []"
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"concat(x#xs) = x @ concat(xs)"
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primrec
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drop_Nil:"drop n [] = []"
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drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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take_Nil:"take n [] = []"
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take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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"[][i:=v] = []"
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"(x#xs)[i:=v] =
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(case i of 0 => v # xs
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| Suc j => x # xs[j:=v])"
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primrec
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"takeWhile P [] = []"
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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"dropWhile P [] = []"
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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"zip xs [] = []"
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zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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-- {* Warning: simpset does not contain this definition *}
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-- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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upt_0: "[i..0(] = []"
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upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
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primrec
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"distinct [] = True"
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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"remdups [] = []"
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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replicate_0: "replicate 0 x = []"
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replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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defs
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 postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys"
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defs
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 list_all2_def:
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 "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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subsection {* Lexicographic orderings on lists *}
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consts
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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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primrec
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"lexn r 0 = {}"
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"lexn r (Suc n) =
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(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
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constdefs
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lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lex r == \<Union>n. lexn r n"
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lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
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sublist :: "'a list => nat set => 'a list"
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) rules
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subsection {* @{text lists}: the list-forming operator over sets *}
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consts lists :: "'a set => 'a list set"
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inductive "lists A"
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intros
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Nil [intro!]: "[]: lists A"
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Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
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inductive_cases listsE [elim!]: "x#l : lists A"
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lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
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by (unfold lists.defs) (blast intro!: lfp_mono)
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lemma lists_IntI:
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  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
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  by induct blast+
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
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apply (rule mono_Int [THEN equalityI])
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apply (simp add: mono_def lists_mono)
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apply (blast intro!: lists_IntI)
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done
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lemma append_in_lists_conv [iff]:
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
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by (induct xs) auto
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subsection {* @{text length} *}
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text {*
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Needs to come before @{text "@"} because of theorem @{text
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append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs)
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 apply simp
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apply simp
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs)
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apply auto
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done
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subsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct xs)
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 apply (case_tac ys)
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apply simp
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 apply force
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apply (case_tac ys)
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 apply force
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apply simp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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wenzelm@13142
   336
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   337
by simp
wenzelm@13114
   338
wenzelm@13142
   339
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   340
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   341
wenzelm@13142
   342
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   343
using append_same_eq [of "[]"] by auto
wenzelm@13114
   344
wenzelm@13142
   345
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   346
by (induct xs) auto
wenzelm@13114
   347
wenzelm@13142
   348
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   349
by (induct xs) auto
wenzelm@13114
   350
wenzelm@13142
   351
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   352
by (simp add: hd_append split: list.split)
wenzelm@13114
   353
wenzelm@13142
   354
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   355
by (simp split: list.split)
wenzelm@13114
   356
wenzelm@13142
   357
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   358
by (simp add: tl_append split: list.split)
wenzelm@13114
   359
wenzelm@13114
   360
wenzelm@13142
   361
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   362
wenzelm@13114
   363
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   364
by simp
wenzelm@13114
   365
wenzelm@13142
   366
lemma Cons_eq_appendI:
nipkow@13145
   367
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   368
by (drule sym) simp
wenzelm@13114
   369
wenzelm@13142
   370
lemma append_eq_appendI:
nipkow@13145
   371
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   372
by (drule sym) simp
wenzelm@13114
   373
wenzelm@13114
   374
wenzelm@13142
   375
text {*
nipkow@13145
   376
Simplification procedure for all list equalities.
nipkow@13145
   377
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   378
- both lists end in a singleton list,
nipkow@13145
   379
- or both lists end in the same list.
wenzelm@13142
   380
*}
wenzelm@13142
   381
wenzelm@13142
   382
ML_setup {*
nipkow@3507
   383
local
nipkow@3507
   384
wenzelm@13122
   385
val append_assoc = thm "append_assoc";
wenzelm@13122
   386
val append_Nil = thm "append_Nil";
wenzelm@13122
   387
val append_Cons = thm "append_Cons";
wenzelm@13122
   388
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   389
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   390
wenzelm@13114
   391
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   392
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   393
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   394
  | last t = t;
wenzelm@13114
   395
wenzelm@13114
   396
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   397
  | list1 _ = false;
wenzelm@13114
   398
wenzelm@13114
   399
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   400
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   401
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   402
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   403
wenzelm@13114
   404
val rearr_tac =
wenzelm@13462
   405
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   406
wenzelm@13114
   407
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   408
  let
wenzelm@13462
   409
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   410
    fun rearr conv =
wenzelm@13462
   411
      let
wenzelm@13462
   412
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   413
        val Type(_,listT::_) = eqT
wenzelm@13462
   414
        val appT = [listT,listT] ---> listT
wenzelm@13462
   415
        val app = Const("List.op @",appT)
wenzelm@13462
   416
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   417
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   418
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   419
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   420
wenzelm@13462
   421
  in
wenzelm@13462
   422
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   423
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   424
    else None
wenzelm@13462
   425
  end;
wenzelm@13462
   426
wenzelm@13114
   427
in
wenzelm@13462
   428
wenzelm@13462
   429
val list_eq_simproc =
wenzelm@13462
   430
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   431
wenzelm@13114
   432
end;
wenzelm@13114
   433
wenzelm@13114
   434
Addsimprocs [list_eq_simproc];
wenzelm@13114
   435
*}
wenzelm@13114
   436
wenzelm@13114
   437
wenzelm@13142
   438
subsection {* @{text map} *}
wenzelm@13114
   439
wenzelm@13142
   440
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   441
by (induct xs) simp_all
wenzelm@13114
   442
wenzelm@13142
   443
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   444
by (rule ext, induct_tac xs) auto
wenzelm@13114
   445
wenzelm@13142
   446
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   447
by (induct xs) auto
wenzelm@13114
   448
wenzelm@13142
   449
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   450
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   451
wenzelm@13142
   452
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   453
by (induct xs) auto
wenzelm@13114
   454
nipkow@13737
   455
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   456
by (induct xs) auto
nipkow@13737
   457
wenzelm@13366
   458
lemma map_cong [recdef_cong]:
nipkow@13145
   459
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   460
-- {* a congruence rule for @{text map} *}
nipkow@13737
   461
by simp
wenzelm@13114
   462
wenzelm@13142
   463
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   464
by (cases xs) auto
wenzelm@13114
   465
wenzelm@13142
   466
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   467
by (cases xs) auto
wenzelm@13114
   468
nipkow@14025
   469
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   470
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   471
by (cases xs) auto
wenzelm@13114
   472
nipkow@14025
   473
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   474
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   475
by (cases ys) auto
nipkow@14025
   476
nipkow@14111
   477
lemma ex_map_conv:
nipkow@14111
   478
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   479
by(induct ys, auto)
nipkow@14111
   480
wenzelm@13114
   481
lemma map_injective:
nipkow@14025
   482
 "!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys"
nipkow@14025
   483
by (induct ys) auto
wenzelm@13114
   484
wenzelm@13114
   485
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   486
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   487
wenzelm@13114
   488
lemma inj_mapD: "inj (map f) ==> inj f"
nipkow@13145
   489
apply (unfold inj_on_def)
nipkow@13145
   490
apply clarify
nipkow@13145
   491
apply (erule_tac x = "[x]" in ballE)
nipkow@13145
   492
 apply (erule_tac x = "[y]" in ballE)
nipkow@13145
   493
apply simp
nipkow@13145
   494
 apply blast
nipkow@13145
   495
apply blast
nipkow@13145
   496
done
wenzelm@13114
   497
wenzelm@13114
   498
lemma inj_map: "inj (map f) = inj f"
nipkow@13145
   499
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   500
wenzelm@13114
   501
wenzelm@13142
   502
subsection {* @{text rev} *}
wenzelm@13114
   503
wenzelm@13142
   504
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   505
by (induct xs) auto
wenzelm@13114
   506
wenzelm@13142
   507
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   508
by (induct xs) auto
wenzelm@13114
   509
wenzelm@13142
   510
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   511
by (induct xs) auto
wenzelm@13114
   512
wenzelm@13142
   513
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   514
by (induct xs) auto
wenzelm@13114
   515
wenzelm@13142
   516
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
nipkow@13145
   517
apply (induct xs)
nipkow@13145
   518
 apply force
nipkow@13145
   519
apply (case_tac ys)
nipkow@13145
   520
 apply simp
nipkow@13145
   521
apply force
nipkow@13145
   522
done
wenzelm@13114
   523
wenzelm@13366
   524
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   525
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   526
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   527
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   528
done
wenzelm@13114
   529
nipkow@13145
   530
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   531
wenzelm@13366
   532
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   533
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   534
by (induct xs rule: rev_induct) auto
wenzelm@13114
   535
wenzelm@13366
   536
lemmas rev_cases = rev_exhaust
wenzelm@13366
   537
wenzelm@13114
   538
wenzelm@13142
   539
subsection {* @{text set} *}
wenzelm@13114
   540
wenzelm@13142
   541
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   542
by (induct xs) auto
wenzelm@13114
   543
wenzelm@13142
   544
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   545
by (induct xs) auto
wenzelm@13114
   546
oheimb@14099
   547
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
oheimb@14099
   548
apply (case_tac l)
oheimb@14099
   549
apply auto
oheimb@14099
   550
done
oheimb@14099
   551
wenzelm@13142
   552
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   553
by auto
wenzelm@13114
   554
oheimb@14099
   555
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   556
by auto
oheimb@14099
   557
wenzelm@13142
   558
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   559
by (induct xs) auto
wenzelm@13114
   560
wenzelm@13142
   561
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   562
by (induct xs) auto
wenzelm@13114
   563
wenzelm@13142
   564
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   565
by (induct xs) auto
wenzelm@13114
   566
wenzelm@13142
   567
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   568
by (induct xs) auto
wenzelm@13114
   569
wenzelm@13142
   570
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
nipkow@13145
   571
apply (induct j)
nipkow@13145
   572
 apply simp_all
nipkow@13145
   573
apply(erule ssubst)
nipkow@13145
   574
apply auto
nipkow@13145
   575
done
wenzelm@13114
   576
wenzelm@13142
   577
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
nipkow@13145
   578
apply (induct xs)
nipkow@13145
   579
 apply simp
nipkow@13145
   580
apply simp
nipkow@13145
   581
apply (rule iffI)
nipkow@13145
   582
 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
nipkow@13145
   583
apply (erule exE)+
nipkow@13145
   584
apply (case_tac ys)
nipkow@13145
   585
apply auto
nipkow@13145
   586
done
wenzelm@13142
   587
wenzelm@13142
   588
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   589
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   590
by (induct xs) auto
wenzelm@13142
   591
wenzelm@13142
   592
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   593
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   594
wenzelm@13142
   595
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   596
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   597
paulson@13508
   598
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   599
apply (erule finite_induct, auto)
paulson@13508
   600
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   601
done
paulson@13508
   602
wenzelm@13114
   603
wenzelm@13142
   604
subsection {* @{text mem} *}
wenzelm@13114
   605
wenzelm@13114
   606
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   607
by (induct xs) auto
wenzelm@13114
   608
wenzelm@13114
   609
wenzelm@13142
   610
subsection {* @{text list_all} *}
wenzelm@13114
   611
wenzelm@13142
   612
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   613
by (induct xs) auto
wenzelm@13114
   614
wenzelm@13142
   615
lemma list_all_append [simp]:
nipkow@13145
   616
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13114
   619
wenzelm@13142
   620
subsection {* @{text filter} *}
wenzelm@13114
   621
wenzelm@13142
   622
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   623
by (induct xs) auto
wenzelm@13114
   624
wenzelm@13142
   625
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   626
by (induct xs) auto
wenzelm@13114
   627
wenzelm@13142
   628
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   629
by (induct xs) auto
wenzelm@13114
   630
wenzelm@13142
   631
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   632
by (induct xs) auto
wenzelm@13114
   633
wenzelm@13142
   634
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   635
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   636
wenzelm@13142
   637
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   638
by auto
wenzelm@13114
   639
wenzelm@13114
   640
wenzelm@13142
   641
subsection {* @{text concat} *}
wenzelm@13114
   642
wenzelm@13142
   643
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   644
by (induct xs) auto
wenzelm@13114
   645
wenzelm@13142
   646
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   647
by (induct xss) auto
wenzelm@13114
   648
wenzelm@13142
   649
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   650
by (induct xss) auto
wenzelm@13114
   651
wenzelm@13142
   652
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   653
by (induct xs) auto
wenzelm@13114
   654
wenzelm@13142
   655
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   656
by (induct xs) auto
wenzelm@13114
   657
wenzelm@13142
   658
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   659
by (induct xs) auto
wenzelm@13114
   660
wenzelm@13142
   661
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   662
by (induct xs) auto
wenzelm@13114
   663
wenzelm@13114
   664
wenzelm@13142
   665
subsection {* @{text nth} *}
wenzelm@13114
   666
wenzelm@13142
   667
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   668
by auto
wenzelm@13114
   669
wenzelm@13142
   670
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   671
by auto
wenzelm@13114
   672
wenzelm@13142
   673
declare nth.simps [simp del]
wenzelm@13114
   674
wenzelm@13114
   675
lemma nth_append:
nipkow@13145
   676
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@13145
   677
apply(induct "xs")
nipkow@13145
   678
 apply simp
nipkow@13145
   679
apply (case_tac n)
nipkow@13145
   680
 apply auto
nipkow@13145
   681
done
wenzelm@13114
   682
wenzelm@13142
   683
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@13145
   684
apply(induct xs)
nipkow@13145
   685
 apply simp
nipkow@13145
   686
apply (case_tac n)
nipkow@13145
   687
 apply auto
nipkow@13145
   688
done
wenzelm@13114
   689
wenzelm@13142
   690
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
nipkow@13145
   691
apply (induct_tac xs)
nipkow@13145
   692
 apply simp
nipkow@13145
   693
apply simp
nipkow@13145
   694
apply safe
nipkow@13145
   695
apply (rule_tac x = 0 in exI)
nipkow@13145
   696
apply simp
nipkow@13145
   697
 apply (rule_tac x = "Suc i" in exI)
nipkow@13145
   698
 apply simp
nipkow@13145
   699
apply (case_tac i)
nipkow@13145
   700
 apply simp
nipkow@13145
   701
apply (rename_tac j)
nipkow@13145
   702
apply (rule_tac x = j in exI)
nipkow@13145
   703
apply simp
nipkow@13145
   704
done
wenzelm@13114
   705
nipkow@13145
   706
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   707
by (auto simp add: set_conv_nth)
wenzelm@13114
   708
wenzelm@13142
   709
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   710
by (auto simp add: set_conv_nth)
wenzelm@13114
   711
wenzelm@13114
   712
lemma all_nth_imp_all_set:
nipkow@13145
   713
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   714
by (auto simp add: set_conv_nth)
wenzelm@13114
   715
wenzelm@13114
   716
lemma all_set_conv_all_nth:
nipkow@13145
   717
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   718
by (auto simp add: set_conv_nth)
wenzelm@13114
   719
wenzelm@13114
   720
wenzelm@13142
   721
subsection {* @{text list_update} *}
wenzelm@13114
   722
wenzelm@13142
   723
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   724
by (induct xs) (auto split: nat.split)
wenzelm@13114
   725
wenzelm@13114
   726
lemma nth_list_update:
nipkow@13145
   727
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   728
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   729
wenzelm@13142
   730
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   731
by (simp add: nth_list_update)
wenzelm@13114
   732
wenzelm@13142
   733
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   734
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   735
wenzelm@13142
   736
lemma list_update_overwrite [simp]:
nipkow@13145
   737
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   738
by (induct xs) (auto split: nat.split)
wenzelm@13114
   739
wenzelm@13114
   740
lemma list_update_same_conv:
nipkow@13145
   741
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   742
by (induct xs) (auto split: nat.split)
wenzelm@13114
   743
wenzelm@13114
   744
lemma update_zip:
nipkow@13145
   745
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   746
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   747
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   748
wenzelm@13114
   749
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   750
by (induct xs) (auto split: nat.split)
wenzelm@13114
   751
wenzelm@13114
   752
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   753
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   754
wenzelm@13114
   755
wenzelm@13142
   756
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   757
wenzelm@13142
   758
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   759
by (induct xs) auto
wenzelm@13114
   760
wenzelm@13142
   761
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   762
by (induct xs) auto
wenzelm@13114
   763
wenzelm@13142
   764
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   765
by (induct xs rule: rev_induct) auto
wenzelm@13114
   766
wenzelm@13114
   767
lemma butlast_append:
nipkow@13145
   768
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   769
by (induct xs) auto
wenzelm@13114
   770
wenzelm@13142
   771
lemma append_butlast_last_id [simp]:
nipkow@13145
   772
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   773
by (induct xs) auto
wenzelm@13114
   774
wenzelm@13142
   775
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   776
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   777
wenzelm@13114
   778
lemma in_set_butlast_appendI:
nipkow@13145
   779
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   780
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   781
wenzelm@13142
   782
wenzelm@13142
   783
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   784
wenzelm@13142
   785
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   786
by (induct xs) auto
wenzelm@13114
   787
wenzelm@13142
   788
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   789
by (induct xs) auto
wenzelm@13114
   790
wenzelm@13142
   791
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   792
by simp
wenzelm@13114
   793
wenzelm@13142
   794
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   795
by simp
wenzelm@13114
   796
wenzelm@13142
   797
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   798
nipkow@13913
   799
lemma take_Suc_conv_app_nth:
nipkow@13913
   800
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@13913
   801
apply(induct xs)
nipkow@13913
   802
 apply simp
nipkow@13913
   803
apply(case_tac i)
nipkow@13913
   804
apply auto
nipkow@13913
   805
done
nipkow@13913
   806
wenzelm@13142
   807
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   808
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   809
wenzelm@13142
   810
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   811
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   812
wenzelm@13142
   813
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   814
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   815
wenzelm@13142
   816
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   817
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   818
wenzelm@13142
   819
lemma take_append [simp]:
nipkow@13145
   820
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   821
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   822
wenzelm@13142
   823
lemma drop_append [simp]:
nipkow@13145
   824
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   825
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   826
wenzelm@13142
   827
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
nipkow@13145
   828
apply (induct m)
nipkow@13145
   829
 apply auto
nipkow@13145
   830
apply (case_tac xs)
nipkow@13145
   831
 apply auto
nipkow@13145
   832
apply (case_tac na)
nipkow@13145
   833
 apply auto
nipkow@13145
   834
done
wenzelm@13114
   835
wenzelm@13142
   836
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
nipkow@13145
   837
apply (induct m)
nipkow@13145
   838
 apply auto
nipkow@13145
   839
apply (case_tac xs)
nipkow@13145
   840
 apply auto
nipkow@13145
   841
done
wenzelm@13114
   842
wenzelm@13114
   843
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@13145
   844
apply (induct m)
nipkow@13145
   845
 apply auto
nipkow@13145
   846
apply (case_tac xs)
nipkow@13145
   847
 apply auto
nipkow@13145
   848
done
wenzelm@13114
   849
wenzelm@13142
   850
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
nipkow@13145
   851
apply (induct n)
nipkow@13145
   852
 apply auto
nipkow@13145
   853
apply (case_tac xs)
nipkow@13145
   854
 apply auto
nipkow@13145
   855
done
wenzelm@13114
   856
wenzelm@13114
   857
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
nipkow@13145
   858
apply (induct n)
nipkow@13145
   859
 apply auto
nipkow@13145
   860
apply (case_tac xs)
nipkow@13145
   861
 apply auto
nipkow@13145
   862
done
wenzelm@13114
   863
wenzelm@13142
   864
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
nipkow@13145
   865
apply (induct n)
nipkow@13145
   866
 apply auto
nipkow@13145
   867
apply (case_tac xs)
nipkow@13145
   868
 apply auto
nipkow@13145
   869
done
wenzelm@13114
   870
wenzelm@13114
   871
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@13145
   872
apply (induct xs)
nipkow@13145
   873
 apply auto
nipkow@13145
   874
apply (case_tac i)
nipkow@13145
   875
 apply auto
nipkow@13145
   876
done
wenzelm@13114
   877
wenzelm@13114
   878
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@13145
   879
apply (induct xs)
nipkow@13145
   880
 apply auto
nipkow@13145
   881
apply (case_tac i)
nipkow@13145
   882
 apply auto
nipkow@13145
   883
done
wenzelm@13114
   884
wenzelm@13142
   885
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
nipkow@13145
   886
apply (induct xs)
nipkow@13145
   887
 apply auto
nipkow@13145
   888
apply (case_tac n)
nipkow@13145
   889
 apply(blast )
nipkow@13145
   890
apply (case_tac i)
nipkow@13145
   891
 apply auto
nipkow@13145
   892
done
wenzelm@13114
   893
wenzelm@13142
   894
lemma nth_drop [simp]:
nipkow@13145
   895
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@13145
   896
apply (induct n)
nipkow@13145
   897
 apply auto
nipkow@13145
   898
apply (case_tac xs)
nipkow@13145
   899
 apply auto
nipkow@13145
   900
done
nipkow@3507
   901
nipkow@14025
   902
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
   903
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
   904
nipkow@14025
   905
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
   906
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
   907
wenzelm@13114
   908
lemma append_eq_conv_conj:
nipkow@13145
   909
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@13145
   910
apply(induct xs)
nipkow@13145
   911
 apply simp
nipkow@13145
   912
apply clarsimp
nipkow@13145
   913
apply (case_tac zs)
nipkow@13145
   914
apply auto
nipkow@13145
   915
done
wenzelm@13142
   916
paulson@14050
   917
lemma take_add [rule_format]: 
paulson@14050
   918
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
   919
apply (induct xs, auto) 
paulson@14050
   920
apply (case_tac i, simp_all) 
paulson@14050
   921
done
paulson@14050
   922
wenzelm@13114
   923
wenzelm@13142
   924
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
   925
wenzelm@13142
   926
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
   927
by (induct xs) auto
wenzelm@13114
   928
wenzelm@13142
   929
lemma takeWhile_append1 [simp]:
nipkow@13145
   930
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
   931
by (induct xs) auto
wenzelm@13114
   932
wenzelm@13142
   933
lemma takeWhile_append2 [simp]:
nipkow@13145
   934
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
   935
by (induct xs) auto
wenzelm@13114
   936
wenzelm@13142
   937
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
   938
by (induct xs) auto
wenzelm@13114
   939
wenzelm@13142
   940
lemma dropWhile_append1 [simp]:
nipkow@13145
   941
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
   942
by (induct xs) auto
wenzelm@13114
   943
wenzelm@13142
   944
lemma dropWhile_append2 [simp]:
nipkow@13145
   945
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
   946
by (induct xs) auto
wenzelm@13114
   947
wenzelm@13142
   948
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
   949
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   950
nipkow@13913
   951
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
   952
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   953
by(induct xs, auto)
nipkow@13913
   954
nipkow@13913
   955
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
   956
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   957
by(induct xs, auto)
nipkow@13913
   958
nipkow@13913
   959
lemma dropWhile_eq_Cons_conv:
nipkow@13913
   960
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
   961
by(induct xs, auto)
nipkow@13913
   962
wenzelm@13114
   963
wenzelm@13142
   964
subsection {* @{text zip} *}
wenzelm@13114
   965
wenzelm@13142
   966
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
   967
by (induct ys) auto
wenzelm@13114
   968
wenzelm@13142
   969
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
   970
by simp
wenzelm@13114
   971
wenzelm@13142
   972
declare zip_Cons [simp del]
wenzelm@13114
   973
wenzelm@13142
   974
lemma length_zip [simp]:
nipkow@13145
   975
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
nipkow@13145
   976
apply(induct ys)
nipkow@13145
   977
 apply simp
nipkow@13145
   978
apply (case_tac xs)
nipkow@13145
   979
 apply auto
nipkow@13145
   980
done
wenzelm@13114
   981
wenzelm@13114
   982
lemma zip_append1:
nipkow@13145
   983
"!!xs. zip (xs @ ys) zs =
nipkow@13145
   984
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
nipkow@13145
   985
apply (induct zs)
nipkow@13145
   986
 apply simp
nipkow@13145
   987
apply (case_tac xs)
nipkow@13145
   988
 apply simp_all
nipkow@13145
   989
done
wenzelm@13114
   990
wenzelm@13114
   991
lemma zip_append2:
nipkow@13145
   992
"!!ys. zip xs (ys @ zs) =
nipkow@13145
   993
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
nipkow@13145
   994
apply (induct xs)
nipkow@13145
   995
 apply simp
nipkow@13145
   996
apply (case_tac ys)
nipkow@13145
   997
 apply simp_all
nipkow@13145
   998
done
wenzelm@13114
   999
wenzelm@13142
  1000
lemma zip_append [simp]:
wenzelm@13142
  1001
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1002
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1003
by (simp add: zip_append1)
wenzelm@13114
  1004
wenzelm@13114
  1005
lemma zip_rev:
nipkow@13145
  1006
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@13145
  1007
apply(induct ys)
nipkow@13145
  1008
 apply simp
nipkow@13145
  1009
apply (case_tac xs)
nipkow@13145
  1010
 apply simp_all
nipkow@13145
  1011
done
wenzelm@13114
  1012
wenzelm@13142
  1013
lemma nth_zip [simp]:
nipkow@13145
  1014
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@13145
  1015
apply (induct ys)
nipkow@13145
  1016
 apply simp
nipkow@13145
  1017
apply (case_tac xs)
nipkow@13145
  1018
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1019
done
wenzelm@13114
  1020
wenzelm@13114
  1021
lemma set_zip:
nipkow@13145
  1022
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1023
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1024
wenzelm@13114
  1025
lemma zip_update:
nipkow@13145
  1026
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1027
by (rule sym, simp add: update_zip)
wenzelm@13114
  1028
wenzelm@13142
  1029
lemma zip_replicate [simp]:
nipkow@13145
  1030
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@13145
  1031
apply (induct i)
nipkow@13145
  1032
 apply auto
nipkow@13145
  1033
apply (case_tac j)
nipkow@13145
  1034
 apply auto
nipkow@13145
  1035
done
wenzelm@13114
  1036
wenzelm@13142
  1037
wenzelm@13142
  1038
subsection {* @{text list_all2} *}
wenzelm@13114
  1039
wenzelm@13114
  1040
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1041
by (simp add: list_all2_def)
wenzelm@13114
  1042
wenzelm@13142
  1043
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1044
by (simp add: list_all2_def)
wenzelm@13114
  1045
wenzelm@13142
  1046
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1047
by (simp add: list_all2_def)
wenzelm@13114
  1048
wenzelm@13142
  1049
lemma list_all2_Cons [iff]:
nipkow@13145
  1050
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1051
by (auto simp add: list_all2_def)
wenzelm@13114
  1052
wenzelm@13114
  1053
lemma list_all2_Cons1:
nipkow@13145
  1054
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1055
by (cases ys) auto
wenzelm@13114
  1056
wenzelm@13114
  1057
lemma list_all2_Cons2:
nipkow@13145
  1058
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1059
by (cases xs) auto
wenzelm@13114
  1060
wenzelm@13142
  1061
lemma list_all2_rev [iff]:
nipkow@13145
  1062
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1063
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1064
kleing@13863
  1065
lemma list_all2_rev1:
kleing@13863
  1066
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1067
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1068
wenzelm@13114
  1069
lemma list_all2_append1:
nipkow@13145
  1070
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1071
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1072
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1073
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1074
apply (rule iffI)
nipkow@13145
  1075
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1076
 apply (rule_tac x = "drop (length xs) zs" in exI)
nipkow@13145
  1077
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
  1078
apply clarify
nipkow@13145
  1079
apply (simp add: ball_Un)
nipkow@13145
  1080
done
wenzelm@13114
  1081
wenzelm@13114
  1082
lemma list_all2_append2:
nipkow@13145
  1083
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1084
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1085
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1086
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1087
apply (rule iffI)
nipkow@13145
  1088
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1089
 apply (rule_tac x = "drop (length ys) xs" in exI)
nipkow@13145
  1090
 apply (force split: nat_diff_split simp add: min_def)
nipkow@13145
  1091
apply clarify
nipkow@13145
  1092
apply (simp add: ball_Un)
nipkow@13145
  1093
done
wenzelm@13114
  1094
kleing@13863
  1095
lemma list_all2_append:
kleing@13863
  1096
  "\<And>b. length a = length b \<Longrightarrow>
kleing@13863
  1097
  list_all2 P (a@c) (b@d) = (list_all2 P a b \<and> list_all2 P c d)"
kleing@13863
  1098
  apply (induct a)
kleing@13863
  1099
   apply simp
kleing@13863
  1100
  apply (case_tac b)
kleing@13863
  1101
  apply auto
kleing@13863
  1102
  done
kleing@13863
  1103
kleing@13863
  1104
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1105
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1106
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1107
wenzelm@13114
  1108
lemma list_all2_conv_all_nth:
nipkow@13145
  1109
"list_all2 P xs ys =
nipkow@13145
  1110
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1111
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1112
berghofe@13883
  1113
lemma list_all2_trans:
berghofe@13883
  1114
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1115
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1116
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1117
proof (induct as)
berghofe@13883
  1118
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1119
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1120
  proof (induct bs)
berghofe@13883
  1121
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1122
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1123
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1124
  qed simp
berghofe@13883
  1125
qed simp
berghofe@13883
  1126
kleing@13863
  1127
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1128
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1129
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1130
kleing@13863
  1131
lemma list_all2_nthD [dest?]:
kleing@13863
  1132
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1133
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1134
kleing@13863
  1135
lemma list_all2_map1: 
kleing@13863
  1136
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1137
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1138
kleing@13863
  1139
lemma list_all2_map2: 
kleing@13863
  1140
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1141
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1142
kleing@13863
  1143
lemma list_all2_refl:
kleing@13863
  1144
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1145
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1146
kleing@13863
  1147
lemma list_all2_update_cong:
kleing@13863
  1148
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1149
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1150
kleing@13863
  1151
lemma list_all2_update_cong2:
kleing@13863
  1152
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1153
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1154
kleing@13863
  1155
lemma list_all2_dropI [intro?]:
kleing@13863
  1156
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
kleing@13863
  1157
  apply (induct as)
kleing@13863
  1158
   apply simp
kleing@13863
  1159
  apply (clarsimp simp add: list_all2_Cons1)
kleing@13863
  1160
  apply (case_tac n)
kleing@13863
  1161
   apply simp
kleing@13863
  1162
  apply simp
kleing@13863
  1163
  done
kleing@13863
  1164
kleing@13863
  1165
lemma list_all2_mono [intro?]:
kleing@13863
  1166
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
kleing@13863
  1167
  apply (induct x)
kleing@13863
  1168
   apply simp
kleing@13863
  1169
  apply (case_tac y)
kleing@13863
  1170
  apply auto
kleing@13863
  1171
  done
kleing@13863
  1172
wenzelm@13142
  1173
wenzelm@13142
  1174
subsection {* @{text foldl} *}
wenzelm@13142
  1175
wenzelm@13142
  1176
lemma foldl_append [simp]:
nipkow@13145
  1177
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1178
by (induct xs) auto
wenzelm@13142
  1179
wenzelm@13142
  1180
text {*
nipkow@13145
  1181
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1182
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1183
*}
wenzelm@13142
  1184
wenzelm@13142
  1185
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1186
by (induct ns) auto
wenzelm@13142
  1187
wenzelm@13142
  1188
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1189
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1190
wenzelm@13142
  1191
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1192
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1193
by (induct ns) auto
wenzelm@13114
  1194
wenzelm@13114
  1195
oheimb@14099
  1196
subsection {* folding a relation over a list *}
oheimb@14099
  1197
oheimb@14099
  1198
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*)
oheimb@14099
  1199
inductive "fold_rel R" intros
oheimb@14099
  1200
  Nil:  "(a, [],a) : fold_rel R"
oheimb@14099
  1201
  Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R"
oheimb@14099
  1202
inductive_cases fold_rel_elim_case [elim!]:
oheimb@14099
  1203
   "(a, []  , b) : fold_rel R"
oheimb@14099
  1204
   "(a, x#xs, b) : fold_rel R"
oheimb@14099
  1205
oheimb@14099
  1206
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" 
oheimb@14099
  1207
by (simp add: fold_rel.Nil)
oheimb@14099
  1208
declare fold_rel.Cons [intro!]
oheimb@14099
  1209
oheimb@14099
  1210
wenzelm@13142
  1211
subsection {* @{text upto} *}
wenzelm@13114
  1212
wenzelm@13142
  1213
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1214
-- {* Does not terminate! *}
nipkow@13145
  1215
by (induct j) auto
wenzelm@13142
  1216
wenzelm@13142
  1217
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1218
by (subst upt_rec) simp
wenzelm@13114
  1219
wenzelm@13142
  1220
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1221
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1222
by simp
wenzelm@13114
  1223
wenzelm@13142
  1224
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1225
apply(rule trans)
nipkow@13145
  1226
apply(subst upt_rec)
nipkow@13145
  1227
 prefer 2 apply(rule refl)
nipkow@13145
  1228
apply simp
nipkow@13145
  1229
done
wenzelm@13114
  1230
wenzelm@13142
  1231
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1232
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1233
by (induct k) auto
wenzelm@13114
  1234
wenzelm@13142
  1235
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1236
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1237
wenzelm@13142
  1238
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1239
apply (induct j)
nipkow@13145
  1240
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1241
done
wenzelm@13114
  1242
wenzelm@13142
  1243
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
nipkow@13145
  1244
apply (induct m)
nipkow@13145
  1245
 apply simp
nipkow@13145
  1246
apply (subst upt_rec)
nipkow@13145
  1247
apply (rule sym)
nipkow@13145
  1248
apply (subst upt_rec)
nipkow@13145
  1249
apply (simp del: upt.simps)
nipkow@13145
  1250
done
nipkow@3507
  1251
wenzelm@13114
  1252
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1253
by (induct n) auto
wenzelm@13114
  1254
wenzelm@13114
  1255
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1256
apply (induct n m rule: diff_induct)
nipkow@13145
  1257
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1258
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1259
done
wenzelm@13114
  1260
berghofe@13883
  1261
lemma nth_take_lemma:
berghofe@13883
  1262
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1263
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1264
apply (atomize, induct k)
nipkow@13145
  1265
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib)
nipkow@13145
  1266
apply clarify
nipkow@13145
  1267
txt {* Both lists must be non-empty *}
nipkow@13145
  1268
apply (case_tac xs)
nipkow@13145
  1269
 apply simp
nipkow@13145
  1270
apply (case_tac ys)
nipkow@13145
  1271
 apply clarify
nipkow@13145
  1272
 apply (simp (no_asm_use))
nipkow@13145
  1273
apply clarify
nipkow@13145
  1274
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1275
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1276
apply blast
nipkow@13145
  1277
done
wenzelm@13114
  1278
wenzelm@13114
  1279
lemma nth_equalityI:
wenzelm@13114
  1280
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1281
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1282
apply (simp_all add: take_all)
nipkow@13145
  1283
done
wenzelm@13142
  1284
kleing@13863
  1285
(* needs nth_equalityI *)
kleing@13863
  1286
lemma list_all2_antisym:
kleing@13863
  1287
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863
  1288
  \<Longrightarrow> xs = ys"
kleing@13863
  1289
  apply (simp add: list_all2_conv_all_nth) 
kleing@13863
  1290
  apply (rule nth_equalityI)
kleing@13863
  1291
   apply blast
kleing@13863
  1292
  apply simp
kleing@13863
  1293
  done
kleing@13863
  1294
wenzelm@13142
  1295
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1296
-- {* The famous take-lemma. *}
nipkow@13145
  1297
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1298
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1299
done
wenzelm@13142
  1300
wenzelm@13142
  1301
wenzelm@13142
  1302
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1303
wenzelm@13142
  1304
lemma distinct_append [simp]:
nipkow@13145
  1305
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1306
by (induct xs) auto
wenzelm@13142
  1307
wenzelm@13142
  1308
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1309
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1310
wenzelm@13142
  1311
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1312
by (induct xs) auto
wenzelm@13142
  1313
wenzelm@13142
  1314
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1315
by (induct xs) auto
wenzelm@13114
  1316
wenzelm@13142
  1317
text {*
nipkow@13145
  1318
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1319
it is useful. *}
wenzelm@13142
  1320
lemma distinct_conv_nth:
nipkow@13145
  1321
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
nipkow@13145
  1322
apply (induct_tac xs)
nipkow@13145
  1323
 apply simp
nipkow@13145
  1324
apply simp
nipkow@13145
  1325
apply (rule iffI)
nipkow@13145
  1326
 apply clarsimp
nipkow@13145
  1327
 apply (case_tac i)
nipkow@13145
  1328
apply (case_tac j)
nipkow@13145
  1329
 apply simp
nipkow@13145
  1330
apply (simp add: set_conv_nth)
nipkow@13145
  1331
 apply (case_tac j)
nipkow@13145
  1332
apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1333
 apply simp
nipkow@13145
  1334
apply (rule conjI)
nipkow@13145
  1335
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1336
 apply (erule_tac x = 0 in allE)
nipkow@13145
  1337
 apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1338
 apply simp
nipkow@13145
  1339
apply clarsimp
nipkow@13145
  1340
apply (erule_tac x = "Suc i" in allE)
nipkow@13145
  1341
apply (erule_tac x = "Suc j" in allE)
nipkow@13145
  1342
apply simp
nipkow@13145
  1343
done
wenzelm@13114
  1344
wenzelm@13114
  1345
wenzelm@13142
  1346
subsection {* @{text replicate} *}
wenzelm@13114
  1347
wenzelm@13142
  1348
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1349
by (induct n) auto
nipkow@13124
  1350
wenzelm@13142
  1351
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1352
by (induct n) auto
wenzelm@13114
  1353
wenzelm@13114
  1354
lemma replicate_app_Cons_same:
nipkow@13145
  1355
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1356
by (induct n) auto
wenzelm@13114
  1357
wenzelm@13142
  1358
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
nipkow@13145
  1359
apply(induct n)
nipkow@13145
  1360
 apply simp
nipkow@13145
  1361
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1362
done
wenzelm@13114
  1363
wenzelm@13142
  1364
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1365
by (induct n) auto
wenzelm@13114
  1366
wenzelm@13142
  1367
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1368
by (induct n) auto
wenzelm@13114
  1369
wenzelm@13142
  1370
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1371
by (induct n) auto
wenzelm@13114
  1372
wenzelm@13142
  1373
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1374
by (atomize (full), induct n) auto
wenzelm@13114
  1375
wenzelm@13142
  1376
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
nipkow@13145
  1377
apply(induct n)
nipkow@13145
  1378
 apply simp
nipkow@13145
  1379
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1380
done
wenzelm@13114
  1381
wenzelm@13142
  1382
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1383
by (induct n) auto
wenzelm@13114
  1384
wenzelm@13142
  1385
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1386
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1387
wenzelm@13142
  1388
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1389
by auto
wenzelm@13114
  1390
wenzelm@13142
  1391
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1392
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1393
wenzelm@13114
  1394
oheimb@14099
  1395
subsection {* @{text postfix} *}
oheimb@14099
  1396
oheimb@14099
  1397
lemma postfix_refl [simp, intro!]: "xs \<sqsupseteq> xs" by (auto simp add: postfix_def)
oheimb@14099
  1398
lemma postfix_trans: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsupseteq> zs" 
oheimb@14099
  1399
         by (auto simp add: postfix_def)
oheimb@14099
  1400
lemma postfix_antisym: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> xs\<rbrakk> \<Longrightarrow> xs = ys" 
oheimb@14099
  1401
         by (auto simp add: postfix_def)
oheimb@14099
  1402
oheimb@14099
  1403
lemma postfix_emptyI [simp, intro!]: "xs \<sqsupseteq> []" by (auto simp add: postfix_def)
oheimb@14099
  1404
lemma postfix_emptyD [dest!]: "[] \<sqsupseteq> xs \<Longrightarrow> xs = []"by(auto simp add:postfix_def)
oheimb@14099
  1405
lemma postfix_ConsI: "xs \<sqsupseteq> ys \<Longrightarrow> x#xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1406
lemma postfix_ConsD: "xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1407
lemma postfix_appendI: "xs \<sqsupseteq> ys \<Longrightarrow> zs@xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1408
lemma postfix_appendD: "xs \<sqsupseteq> zs@ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1409
oheimb@14099
  1410
lemma postfix_is_subset_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs"
oheimb@14099
  1411
by (induct zs, auto)
oheimb@14099
  1412
lemma postfix_is_subset: "xs \<sqsupseteq> ys \<Longrightarrow> set ys \<subseteq> set xs"
oheimb@14099
  1413
by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma)
oheimb@14099
  1414
oheimb@14099
  1415
lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \<longrightarrow> xs \<sqsupseteq> ys"
oheimb@14099
  1416
by (induct zs, auto intro!: postfix_appendI postfix_ConsI)
oheimb@14099
  1417
lemma postfix_ConsD2: "x#xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys"
oheimb@14099
  1418
by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma)
oheimb@14099
  1419
oheimb@14099
  1420
subsection {* Lexicographic orderings on lists *}
nipkow@3507
  1421
wenzelm@13142
  1422
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@13145
  1423
apply (induct_tac n)
nipkow@13145
  1424
 apply simp
nipkow@13145
  1425
apply simp
nipkow@13145
  1426
apply(rule wf_subset)
nipkow@13145
  1427
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1428
apply(rule wf_prod_fun_image)
paulson@13585
  1429
 prefer 2 apply (rule inj_onI)
nipkow@13145
  1430
apply auto
nipkow@13145
  1431
done
wenzelm@13114
  1432
wenzelm@13114
  1433
lemma lexn_length:
nipkow@13145
  1434
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1435
by (induct n) auto
wenzelm@13114
  1436
wenzelm@13142
  1437
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1438
apply (unfold lex_def)
nipkow@13145
  1439
apply (rule wf_UN)
nipkow@13145
  1440
apply (blast intro: wf_lexn)
nipkow@13145
  1441
apply clarify
nipkow@13145
  1442
apply (rename_tac m n)
nipkow@13145
  1443
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1444
 prefer 2 apply blast
nipkow@13145
  1445
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1446
done
wenzelm@13114
  1447
wenzelm@13114
  1448
lemma lexn_conv:
nipkow@13145
  1449
"lexn r n =
nipkow@13145
  1450
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1451
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1452
apply (induct_tac n)
nipkow@13145
  1453
 apply simp
nipkow@13145
  1454
 apply blast
nipkow@13145
  1455
apply (simp add: image_Collect lex_prod_def)
berghofe@13601
  1456
apply safe
nipkow@13145
  1457
apply blast
berghofe@13601
  1458
 apply (rule_tac x = "ab # xys" in exI)
nipkow@13145
  1459
 apply simp
nipkow@13145
  1460
apply (case_tac xys)
nipkow@13145
  1461
 apply simp_all
nipkow@13145
  1462
apply blast
nipkow@13145
  1463
done
wenzelm@13114
  1464
wenzelm@13114
  1465
lemma lex_conv:
nipkow@13145
  1466
"lex r =
nipkow@13145
  1467
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1468
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1469
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1470
wenzelm@13142
  1471
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1472
by (unfold lexico_def) blast
wenzelm@13114
  1473
wenzelm@13114
  1474
lemma lexico_conv:
nipkow@13145
  1475
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1476
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1477
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1478
wenzelm@13142
  1479
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1480
by (simp add: lex_conv)
wenzelm@13114
  1481
wenzelm@13142
  1482
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1483
by (simp add:lex_conv)
wenzelm@13114
  1484
wenzelm@13142
  1485
lemma Cons_in_lex [iff]:
nipkow@13145
  1486
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1487
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1488
apply (simp add: lex_conv)
nipkow@13145
  1489
apply (rule iffI)
nipkow@13145
  1490
 prefer 2 apply (blast intro: Cons_eq_appendI)
nipkow@13145
  1491
apply clarify
nipkow@13145
  1492
apply (case_tac xys)
nipkow@13145
  1493
 apply simp
nipkow@13145
  1494
apply simp
nipkow@13145
  1495
apply blast
nipkow@13145
  1496
done
wenzelm@13114
  1497
wenzelm@13114
  1498
wenzelm@13142
  1499
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1500
wenzelm@13142
  1501
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1502
by (auto simp add: sublist_def)
wenzelm@13114
  1503
wenzelm@13142
  1504
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1505
by (auto simp add: sublist_def)
wenzelm@13114
  1506
wenzelm@13114
  1507
lemma sublist_shift_lemma:
nipkow@13145
  1508
"map fst [p:zip xs [i..i + length xs(] . snd p : A] =
nipkow@13145
  1509
map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1510
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1511
wenzelm@13114
  1512
lemma sublist_append:
nipkow@13145
  1513
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1514
apply (unfold sublist_def)
nipkow@13145
  1515
apply (induct l' rule: rev_induct)
nipkow@13145
  1516
 apply simp
nipkow@13145
  1517
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1518
apply (simp add: add_commute)
nipkow@13145
  1519
done
wenzelm@13114
  1520
wenzelm@13114
  1521
lemma sublist_Cons:
nipkow@13145
  1522
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1523
apply (induct l rule: rev_induct)
nipkow@13145
  1524
 apply (simp add: sublist_def)
nipkow@13145
  1525
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1526
done
wenzelm@13114
  1527
wenzelm@13142
  1528
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1529
by (simp add: sublist_Cons)
wenzelm@13114
  1530
wenzelm@13142
  1531
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
nipkow@13145
  1532
apply (induct l rule: rev_induct)
nipkow@13145
  1533
 apply simp
nipkow@13145
  1534
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1535
done
wenzelm@13114
  1536
wenzelm@13114
  1537
wenzelm@13142
  1538
lemma take_Cons':
nipkow@13145
  1539
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1540
by (cases n) simp_all
wenzelm@13114
  1541
wenzelm@13142
  1542
lemma drop_Cons':
nipkow@13145
  1543
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1544
by (cases n) simp_all
wenzelm@13114
  1545
wenzelm@13142
  1546
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1547
by (cases n) simp_all
wenzelm@13142
  1548
nipkow@13145
  1549
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1550
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1551
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1552
wenzelm@13462
  1553
wenzelm@13366
  1554
subsection {* Characters and strings *}
wenzelm@13366
  1555
wenzelm@13366
  1556
datatype nibble =
wenzelm@13366
  1557
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1558
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1559
wenzelm@13366
  1560
datatype char = Char nibble nibble
wenzelm@13366
  1561
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1562
wenzelm@13366
  1563
types string = "char list"
wenzelm@13366
  1564
wenzelm@13366
  1565
syntax
wenzelm@13366
  1566
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1567
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1568
wenzelm@13366
  1569
parse_ast_translation {*
wenzelm@13366
  1570
  let
wenzelm@13366
  1571
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1572
wenzelm@13366
  1573
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1574
    fun mk_char c =
wenzelm@13366
  1575
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1576
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1577
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1578
wenzelm@13366
  1579
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1580
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1581
wenzelm@13366
  1582
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1583
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1584
          [c] => mk_char c
wenzelm@13366
  1585
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1586
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1587
wenzelm@13366
  1588
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1589
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1590
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  1591
        | cs => mk_string cs)
wenzelm@13366
  1592
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  1593
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  1594
*}
wenzelm@13366
  1595
wenzelm@13366
  1596
print_ast_translation {*
wenzelm@13366
  1597
  let
wenzelm@13366
  1598
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  1599
        (case explode c of
wenzelm@13366
  1600
          ["N", "i", "b", "b", "l", "e", h] =>
wenzelm@13366
  1601
            if "0" <= h andalso h <= "9" then ord h - ord "0"
wenzelm@13366
  1602
            else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
wenzelm@13366
  1603
            else raise Match
wenzelm@13366
  1604
        | _ => raise Match)
wenzelm@13366
  1605
      | dest_nib _ = raise Match;
wenzelm@13366
  1606
wenzelm@13366
  1607
    fun dest_chr c1 c2 =
wenzelm@13366
  1608
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  1609
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  1610
wenzelm@13366
  1611
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  1612
      | dest_char _ = raise Match;
wenzelm@13366
  1613
wenzelm@13366
  1614
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  1615
wenzelm@13366
  1616
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  1617
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  1618
wenzelm@13366
  1619
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  1620
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  1621
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  1622
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  1623
*}
wenzelm@13366
  1624
wenzelm@13122
  1625
end