src/HOL/List.thy
author nipkow
Wed Oct 29 01:17:06 2003 +0100 (2003-10-29)
changeset 14247 cb32eb89bddd
parent 14208 144f45277d5a
child 14300 bf8b8c9425c3
permissions -rw-r--r--
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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, simp, 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, auto)
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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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, simp, force)
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apply (case_tac ys, force, simp)
nipkow@13145
   332
done
wenzelm@13142
   333
wenzelm@13142
   334
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   335
by simp
wenzelm@13142
   336
wenzelm@13142
   337
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   338
by simp
wenzelm@13114
   339
wenzelm@13142
   340
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   341
by simp
wenzelm@13114
   342
wenzelm@13142
   343
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   344
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   345
wenzelm@13142
   346
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   347
using append_same_eq [of "[]"] by auto
wenzelm@13114
   348
wenzelm@13142
   349
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   350
by (induct xs) auto
wenzelm@13114
   351
wenzelm@13142
   352
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   353
by (induct xs) auto
wenzelm@13114
   354
wenzelm@13142
   355
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   356
by (simp add: hd_append split: list.split)
wenzelm@13114
   357
wenzelm@13142
   358
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   359
by (simp split: list.split)
wenzelm@13114
   360
wenzelm@13142
   361
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   362
by (simp add: tl_append split: list.split)
wenzelm@13114
   363
wenzelm@13114
   364
wenzelm@13142
   365
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   366
wenzelm@13114
   367
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   368
by simp
wenzelm@13114
   369
wenzelm@13142
   370
lemma Cons_eq_appendI:
nipkow@13145
   371
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   372
by (drule sym) simp
wenzelm@13114
   373
wenzelm@13142
   374
lemma append_eq_appendI:
nipkow@13145
   375
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   376
by (drule sym) simp
wenzelm@13114
   377
wenzelm@13114
   378
wenzelm@13142
   379
text {*
nipkow@13145
   380
Simplification procedure for all list equalities.
nipkow@13145
   381
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   382
- both lists end in a singleton list,
nipkow@13145
   383
- or both lists end in the same list.
wenzelm@13142
   384
*}
wenzelm@13142
   385
wenzelm@13142
   386
ML_setup {*
nipkow@3507
   387
local
nipkow@3507
   388
wenzelm@13122
   389
val append_assoc = thm "append_assoc";
wenzelm@13122
   390
val append_Nil = thm "append_Nil";
wenzelm@13122
   391
val append_Cons = thm "append_Cons";
wenzelm@13122
   392
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   393
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   394
wenzelm@13114
   395
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   396
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   397
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   398
  | last t = t;
wenzelm@13114
   399
wenzelm@13114
   400
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   401
  | list1 _ = false;
wenzelm@13114
   402
wenzelm@13114
   403
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   404
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   405
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   406
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   407
wenzelm@13114
   408
val rearr_tac =
wenzelm@13462
   409
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   410
wenzelm@13114
   411
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   412
  let
wenzelm@13462
   413
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   414
    fun rearr conv =
wenzelm@13462
   415
      let
wenzelm@13462
   416
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   417
        val Type(_,listT::_) = eqT
wenzelm@13462
   418
        val appT = [listT,listT] ---> listT
wenzelm@13462
   419
        val app = Const("List.op @",appT)
wenzelm@13462
   420
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   421
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   422
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   423
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   424
wenzelm@13462
   425
  in
wenzelm@13462
   426
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   427
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   428
    else None
wenzelm@13462
   429
  end;
wenzelm@13462
   430
wenzelm@13114
   431
in
wenzelm@13462
   432
wenzelm@13462
   433
val list_eq_simproc =
wenzelm@13462
   434
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   435
wenzelm@13114
   436
end;
wenzelm@13114
   437
wenzelm@13114
   438
Addsimprocs [list_eq_simproc];
wenzelm@13114
   439
*}
wenzelm@13114
   440
wenzelm@13114
   441
wenzelm@13142
   442
subsection {* @{text map} *}
wenzelm@13114
   443
wenzelm@13142
   444
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   445
by (induct xs) simp_all
wenzelm@13114
   446
wenzelm@13142
   447
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   448
by (rule ext, induct_tac xs) auto
wenzelm@13114
   449
wenzelm@13142
   450
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   451
by (induct xs) auto
wenzelm@13114
   452
wenzelm@13142
   453
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   454
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   455
wenzelm@13142
   456
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   457
by (induct xs) auto
wenzelm@13114
   458
nipkow@13737
   459
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   460
by (induct xs) auto
nipkow@13737
   461
wenzelm@13366
   462
lemma map_cong [recdef_cong]:
nipkow@13145
   463
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   464
-- {* a congruence rule for @{text map} *}
nipkow@13737
   465
by simp
wenzelm@13114
   466
wenzelm@13142
   467
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   468
by (cases xs) auto
wenzelm@13114
   469
wenzelm@13142
   470
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   471
by (cases xs) auto
wenzelm@13114
   472
nipkow@14025
   473
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   474
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   475
by (cases xs) auto
wenzelm@13114
   476
nipkow@14025
   477
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   478
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   479
by (cases ys) auto
nipkow@14025
   480
nipkow@14111
   481
lemma ex_map_conv:
nipkow@14111
   482
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   483
by(induct ys, auto)
nipkow@14111
   484
wenzelm@13114
   485
lemma map_injective:
nipkow@14025
   486
 "!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys"
nipkow@14025
   487
by (induct ys) auto
wenzelm@13114
   488
wenzelm@13114
   489
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   490
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   491
wenzelm@13114
   492
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   493
apply (unfold inj_on_def, clarify)
nipkow@13145
   494
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   495
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   496
apply blast
nipkow@13145
   497
done
wenzelm@13114
   498
wenzelm@13114
   499
lemma inj_map: "inj (map f) = inj f"
nipkow@13145
   500
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   501
wenzelm@13114
   502
wenzelm@13142
   503
subsection {* @{text rev} *}
wenzelm@13114
   504
wenzelm@13142
   505
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   506
by (induct xs) auto
wenzelm@13114
   507
wenzelm@13142
   508
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   509
by (induct xs) auto
wenzelm@13114
   510
wenzelm@13142
   511
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   512
by (induct xs) auto
wenzelm@13114
   513
wenzelm@13142
   514
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   515
by (induct xs) auto
wenzelm@13114
   516
wenzelm@13142
   517
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   518
apply (induct xs, force)
paulson@14208
   519
apply (case_tac ys, simp, force)
nipkow@13145
   520
done
wenzelm@13114
   521
wenzelm@13366
   522
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   523
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   524
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   525
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   526
done
wenzelm@13114
   527
nipkow@13145
   528
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   529
wenzelm@13366
   530
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   531
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   532
by (induct xs rule: rev_induct) auto
wenzelm@13114
   533
wenzelm@13366
   534
lemmas rev_cases = rev_exhaust
wenzelm@13366
   535
wenzelm@13114
   536
wenzelm@13142
   537
subsection {* @{text set} *}
wenzelm@13114
   538
wenzelm@13142
   539
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   540
by (induct xs) auto
wenzelm@13114
   541
wenzelm@13142
   542
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   543
by (induct xs) auto
wenzelm@13114
   544
oheimb@14099
   545
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   546
by (case_tac l, auto)
oheimb@14099
   547
wenzelm@13142
   548
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   549
by auto
wenzelm@13114
   550
oheimb@14099
   551
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   552
by auto
oheimb@14099
   553
wenzelm@13142
   554
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   555
by (induct xs) auto
wenzelm@13114
   556
wenzelm@13142
   557
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   558
by (induct xs) auto
wenzelm@13114
   559
wenzelm@13142
   560
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   561
by (induct xs) auto
wenzelm@13114
   562
wenzelm@13142
   563
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   564
by (induct xs) auto
wenzelm@13114
   565
wenzelm@13142
   566
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
paulson@14208
   567
apply (induct j, simp_all)
paulson@14208
   568
apply (erule ssubst, auto)
nipkow@13145
   569
done
wenzelm@13114
   570
wenzelm@13142
   571
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@14208
   572
apply (induct xs, simp, simp)
nipkow@13145
   573
apply (rule iffI)
nipkow@13145
   574
 apply (blast intro: eq_Nil_appendI Cons_eq_appendI)
nipkow@13145
   575
apply (erule exE)+
paulson@14208
   576
apply (case_tac ys, auto)
nipkow@13145
   577
done
wenzelm@13142
   578
wenzelm@13142
   579
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   580
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   581
by (induct xs) auto
wenzelm@13142
   582
wenzelm@13142
   583
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   584
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   585
wenzelm@13142
   586
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   587
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   588
paulson@13508
   589
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   590
apply (erule finite_induct, auto)
paulson@13508
   591
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   592
done
paulson@13508
   593
wenzelm@13114
   594
wenzelm@13142
   595
subsection {* @{text mem} *}
wenzelm@13114
   596
wenzelm@13114
   597
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   598
by (induct xs) auto
wenzelm@13114
   599
wenzelm@13114
   600
wenzelm@13142
   601
subsection {* @{text list_all} *}
wenzelm@13114
   602
wenzelm@13142
   603
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   604
by (induct xs) auto
wenzelm@13114
   605
wenzelm@13142
   606
lemma list_all_append [simp]:
nipkow@13145
   607
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   608
by (induct xs) auto
wenzelm@13114
   609
wenzelm@13114
   610
wenzelm@13142
   611
subsection {* @{text filter} *}
wenzelm@13114
   612
wenzelm@13142
   613
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   614
by (induct xs) auto
wenzelm@13114
   615
wenzelm@13142
   616
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13142
   619
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   620
by (induct xs) auto
wenzelm@13114
   621
wenzelm@13142
   622
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   623
by (induct xs) auto
wenzelm@13114
   624
wenzelm@13142
   625
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   626
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   627
wenzelm@13142
   628
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   629
by auto
wenzelm@13114
   630
wenzelm@13114
   631
wenzelm@13142
   632
subsection {* @{text concat} *}
wenzelm@13114
   633
wenzelm@13142
   634
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   635
by (induct xs) auto
wenzelm@13114
   636
wenzelm@13142
   637
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   638
by (induct xss) auto
wenzelm@13114
   639
wenzelm@13142
   640
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   641
by (induct xss) auto
wenzelm@13114
   642
wenzelm@13142
   643
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   644
by (induct xs) auto
wenzelm@13114
   645
wenzelm@13142
   646
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   647
by (induct xs) auto
wenzelm@13114
   648
wenzelm@13142
   649
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   650
by (induct xs) auto
wenzelm@13114
   651
wenzelm@13142
   652
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   653
by (induct xs) auto
wenzelm@13114
   654
wenzelm@13114
   655
wenzelm@13142
   656
subsection {* @{text nth} *}
wenzelm@13114
   657
wenzelm@13142
   658
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   659
by auto
wenzelm@13114
   660
wenzelm@13142
   661
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   662
by auto
wenzelm@13114
   663
wenzelm@13142
   664
declare nth.simps [simp del]
wenzelm@13114
   665
wenzelm@13114
   666
lemma nth_append:
nipkow@13145
   667
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   668
apply (induct "xs", simp)
paulson@14208
   669
apply (case_tac n, auto)
nipkow@13145
   670
done
wenzelm@13114
   671
wenzelm@13142
   672
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   673
apply (induct xs, simp)
paulson@14208
   674
apply (case_tac n, auto)
nipkow@13145
   675
done
wenzelm@13114
   676
wenzelm@13142
   677
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@14208
   678
apply (induct_tac xs, simp, simp)
nipkow@13145
   679
apply safe
paulson@14208
   680
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   681
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   682
apply (case_tac i, simp)
nipkow@13145
   683
apply (rename_tac j)
paulson@14208
   684
apply (rule_tac x = j in exI, simp)
nipkow@13145
   685
done
wenzelm@13114
   686
nipkow@13145
   687
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   688
by (auto simp add: set_conv_nth)
wenzelm@13114
   689
wenzelm@13142
   690
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   691
by (auto simp add: set_conv_nth)
wenzelm@13114
   692
wenzelm@13114
   693
lemma all_nth_imp_all_set:
nipkow@13145
   694
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   695
by (auto simp add: set_conv_nth)
wenzelm@13114
   696
wenzelm@13114
   697
lemma all_set_conv_all_nth:
nipkow@13145
   698
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   699
by (auto simp add: set_conv_nth)
wenzelm@13114
   700
wenzelm@13114
   701
wenzelm@13142
   702
subsection {* @{text list_update} *}
wenzelm@13114
   703
wenzelm@13142
   704
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   705
by (induct xs) (auto split: nat.split)
wenzelm@13114
   706
wenzelm@13114
   707
lemma nth_list_update:
nipkow@13145
   708
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   709
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   710
wenzelm@13142
   711
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   712
by (simp add: nth_list_update)
wenzelm@13114
   713
wenzelm@13142
   714
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   715
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   716
wenzelm@13142
   717
lemma list_update_overwrite [simp]:
nipkow@13145
   718
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   719
by (induct xs) (auto split: nat.split)
wenzelm@13114
   720
nipkow@14187
   721
lemma list_update_id[simp]: "!!i. i < length xs \<Longrightarrow> xs[i := xs!i] = xs"
paulson@14208
   722
apply (induct xs, simp)
nipkow@14187
   723
apply(simp split:nat.splits)
nipkow@14187
   724
done
nipkow@14187
   725
wenzelm@13114
   726
lemma list_update_same_conv:
nipkow@13145
   727
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   728
by (induct xs) (auto split: nat.split)
wenzelm@13114
   729
nipkow@14187
   730
lemma list_update_append1:
nipkow@14187
   731
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   732
apply (induct xs, simp)
nipkow@14187
   733
apply(simp split:nat.split)
nipkow@14187
   734
done
nipkow@14187
   735
wenzelm@13114
   736
lemma update_zip:
nipkow@13145
   737
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   738
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   739
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   740
wenzelm@13114
   741
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   742
by (induct xs) (auto split: nat.split)
wenzelm@13114
   743
wenzelm@13114
   744
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   745
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   746
wenzelm@13114
   747
wenzelm@13142
   748
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   749
wenzelm@13142
   750
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   751
by (induct xs) auto
wenzelm@13114
   752
wenzelm@13142
   753
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   754
by (induct xs) auto
wenzelm@13114
   755
wenzelm@13142
   756
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   757
by (induct xs rule: rev_induct) auto
wenzelm@13114
   758
wenzelm@13114
   759
lemma butlast_append:
nipkow@13145
   760
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   761
by (induct xs) auto
wenzelm@13114
   762
wenzelm@13142
   763
lemma append_butlast_last_id [simp]:
nipkow@13145
   764
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   765
by (induct xs) auto
wenzelm@13114
   766
wenzelm@13142
   767
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   768
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   769
wenzelm@13114
   770
lemma in_set_butlast_appendI:
nipkow@13145
   771
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   772
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   773
wenzelm@13142
   774
wenzelm@13142
   775
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   776
wenzelm@13142
   777
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   778
by (induct xs) auto
wenzelm@13114
   779
wenzelm@13142
   780
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   781
by (induct xs) auto
wenzelm@13114
   782
wenzelm@13142
   783
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   784
by simp
wenzelm@13114
   785
wenzelm@13142
   786
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   787
by simp
wenzelm@13114
   788
wenzelm@13142
   789
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   790
nipkow@14187
   791
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   792
by(cases xs, simp_all)
nipkow@14187
   793
nipkow@14187
   794
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   795
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   796
nipkow@14187
   797
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   798
apply (induct xs, simp)
nipkow@14187
   799
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   800
done
nipkow@14187
   801
nipkow@13913
   802
lemma take_Suc_conv_app_nth:
nipkow@13913
   803
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   804
apply (induct xs, simp)
paulson@14208
   805
apply (case_tac i, auto)
nipkow@13913
   806
done
nipkow@13913
   807
wenzelm@13142
   808
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   809
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   810
wenzelm@13142
   811
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   812
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   813
wenzelm@13142
   814
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   815
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   816
wenzelm@13142
   817
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   818
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   819
wenzelm@13142
   820
lemma take_append [simp]:
nipkow@13145
   821
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   822
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   823
wenzelm@13142
   824
lemma drop_append [simp]:
nipkow@13145
   825
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   826
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   827
wenzelm@13142
   828
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
   829
apply (induct m, auto)
paulson@14208
   830
apply (case_tac xs, auto)
paulson@14208
   831
apply (case_tac na, auto)
nipkow@13145
   832
done
wenzelm@13114
   833
wenzelm@13142
   834
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
   835
apply (induct m, auto)
paulson@14208
   836
apply (case_tac xs, auto)
nipkow@13145
   837
done
wenzelm@13114
   838
wenzelm@13114
   839
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
   840
apply (induct m, auto)
paulson@14208
   841
apply (case_tac xs, auto)
nipkow@13145
   842
done
wenzelm@13114
   843
wenzelm@13142
   844
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
   845
apply (induct n, auto)
paulson@14208
   846
apply (case_tac xs, auto)
nipkow@13145
   847
done
wenzelm@13114
   848
wenzelm@13114
   849
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
   850
apply (induct n, auto)
paulson@14208
   851
apply (case_tac xs, auto)
nipkow@13145
   852
done
wenzelm@13114
   853
wenzelm@13142
   854
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
   855
apply (induct n, auto)
paulson@14208
   856
apply (case_tac xs, auto)
nipkow@13145
   857
done
wenzelm@13114
   858
wenzelm@13114
   859
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
   860
apply (induct xs, auto)
paulson@14208
   861
apply (case_tac i, auto)
nipkow@13145
   862
done
wenzelm@13114
   863
wenzelm@13114
   864
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
   865
apply (induct xs, auto)
paulson@14208
   866
apply (case_tac i, auto)
nipkow@13145
   867
done
wenzelm@13114
   868
wenzelm@13142
   869
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
   870
apply (induct xs, auto)
paulson@14208
   871
apply (case_tac n, blast)
paulson@14208
   872
apply (case_tac i, auto)
nipkow@13145
   873
done
wenzelm@13114
   874
wenzelm@13142
   875
lemma nth_drop [simp]:
nipkow@13145
   876
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
   877
apply (induct n, auto)
paulson@14208
   878
apply (case_tac xs, auto)
nipkow@13145
   879
done
nipkow@3507
   880
nipkow@14025
   881
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
   882
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
   883
nipkow@14025
   884
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
   885
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
   886
nipkow@14187
   887
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
   888
using set_take_subset by fast
nipkow@14187
   889
nipkow@14187
   890
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
   891
using set_drop_subset by fast
nipkow@14187
   892
wenzelm@13114
   893
lemma append_eq_conv_conj:
nipkow@13145
   894
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
   895
apply (induct xs, simp, clarsimp)
paulson@14208
   896
apply (case_tac zs, auto)
nipkow@13145
   897
done
wenzelm@13142
   898
paulson@14050
   899
lemma take_add [rule_format]: 
paulson@14050
   900
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
   901
apply (induct xs, auto) 
paulson@14050
   902
apply (case_tac i, simp_all) 
paulson@14050
   903
done
paulson@14050
   904
wenzelm@13114
   905
wenzelm@13142
   906
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
   907
wenzelm@13142
   908
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
   909
by (induct xs) auto
wenzelm@13114
   910
wenzelm@13142
   911
lemma takeWhile_append1 [simp]:
nipkow@13145
   912
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
   913
by (induct xs) auto
wenzelm@13114
   914
wenzelm@13142
   915
lemma takeWhile_append2 [simp]:
nipkow@13145
   916
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
   917
by (induct xs) auto
wenzelm@13114
   918
wenzelm@13142
   919
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
   920
by (induct xs) auto
wenzelm@13114
   921
wenzelm@13142
   922
lemma dropWhile_append1 [simp]:
nipkow@13145
   923
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
   924
by (induct xs) auto
wenzelm@13114
   925
wenzelm@13142
   926
lemma dropWhile_append2 [simp]:
nipkow@13145
   927
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
   928
by (induct xs) auto
wenzelm@13114
   929
wenzelm@13142
   930
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
   931
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   932
nipkow@13913
   933
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
   934
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   935
by(induct xs, auto)
nipkow@13913
   936
nipkow@13913
   937
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
   938
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
   939
by(induct xs, auto)
nipkow@13913
   940
nipkow@13913
   941
lemma dropWhile_eq_Cons_conv:
nipkow@13913
   942
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
   943
by(induct xs, auto)
nipkow@13913
   944
wenzelm@13114
   945
wenzelm@13142
   946
subsection {* @{text zip} *}
wenzelm@13114
   947
wenzelm@13142
   948
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
   949
by (induct ys) auto
wenzelm@13114
   950
wenzelm@13142
   951
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
   952
by simp
wenzelm@13114
   953
wenzelm@13142
   954
declare zip_Cons [simp del]
wenzelm@13114
   955
wenzelm@13142
   956
lemma length_zip [simp]:
nipkow@13145
   957
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
   958
apply (induct ys, simp)
paulson@14208
   959
apply (case_tac xs, auto)
nipkow@13145
   960
done
wenzelm@13114
   961
wenzelm@13114
   962
lemma zip_append1:
nipkow@13145
   963
"!!xs. zip (xs @ ys) zs =
nipkow@13145
   964
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
   965
apply (induct zs, simp)
paulson@14208
   966
apply (case_tac xs, simp_all)
nipkow@13145
   967
done
wenzelm@13114
   968
wenzelm@13114
   969
lemma zip_append2:
nipkow@13145
   970
"!!ys. zip xs (ys @ zs) =
nipkow@13145
   971
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
   972
apply (induct xs, simp)
paulson@14208
   973
apply (case_tac ys, simp_all)
nipkow@13145
   974
done
wenzelm@13114
   975
wenzelm@13142
   976
lemma zip_append [simp]:
wenzelm@13142
   977
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
   978
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
   979
by (simp add: zip_append1)
wenzelm@13114
   980
wenzelm@13114
   981
lemma zip_rev:
nipkow@14247
   982
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
   983
by (induct rule:list_induct2, simp_all)
wenzelm@13114
   984
wenzelm@13142
   985
lemma nth_zip [simp]:
nipkow@13145
   986
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
   987
apply (induct ys, simp)
nipkow@13145
   988
apply (case_tac xs)
nipkow@13145
   989
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
   990
done
wenzelm@13114
   991
wenzelm@13114
   992
lemma set_zip:
nipkow@13145
   993
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
   994
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
   995
wenzelm@13114
   996
lemma zip_update:
nipkow@13145
   997
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
   998
by (rule sym, simp add: update_zip)
wenzelm@13114
   999
wenzelm@13142
  1000
lemma zip_replicate [simp]:
nipkow@13145
  1001
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1002
apply (induct i, auto)
paulson@14208
  1003
apply (case_tac j, auto)
nipkow@13145
  1004
done
wenzelm@13114
  1005
wenzelm@13142
  1006
wenzelm@13142
  1007
subsection {* @{text list_all2} *}
wenzelm@13114
  1008
wenzelm@13114
  1009
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1010
by (simp add: list_all2_def)
wenzelm@13114
  1011
wenzelm@13142
  1012
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1013
by (simp add: list_all2_def)
wenzelm@13114
  1014
wenzelm@13142
  1015
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1016
by (simp add: list_all2_def)
wenzelm@13114
  1017
wenzelm@13142
  1018
lemma list_all2_Cons [iff]:
nipkow@13145
  1019
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1020
by (auto simp add: list_all2_def)
wenzelm@13114
  1021
wenzelm@13114
  1022
lemma list_all2_Cons1:
nipkow@13145
  1023
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1024
by (cases ys) auto
wenzelm@13114
  1025
wenzelm@13114
  1026
lemma list_all2_Cons2:
nipkow@13145
  1027
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1028
by (cases xs) auto
wenzelm@13114
  1029
wenzelm@13142
  1030
lemma list_all2_rev [iff]:
nipkow@13145
  1031
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1032
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1033
kleing@13863
  1034
lemma list_all2_rev1:
kleing@13863
  1035
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1036
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1037
wenzelm@13114
  1038
lemma list_all2_append1:
nipkow@13145
  1039
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1040
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1041
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1042
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1043
apply (rule iffI)
nipkow@13145
  1044
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1045
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1046
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1047
apply (simp add: ball_Un)
nipkow@13145
  1048
done
wenzelm@13114
  1049
wenzelm@13114
  1050
lemma list_all2_append2:
nipkow@13145
  1051
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1052
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1053
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1054
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1055
apply (rule iffI)
nipkow@13145
  1056
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1057
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1058
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1059
apply (simp add: ball_Un)
nipkow@13145
  1060
done
wenzelm@13114
  1061
kleing@13863
  1062
lemma list_all2_append:
nipkow@14247
  1063
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1064
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1065
by (induct rule:list_induct2, simp_all)
kleing@13863
  1066
kleing@13863
  1067
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1068
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1069
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1070
wenzelm@13114
  1071
lemma list_all2_conv_all_nth:
nipkow@13145
  1072
"list_all2 P xs ys =
nipkow@13145
  1073
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1074
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1075
berghofe@13883
  1076
lemma list_all2_trans:
berghofe@13883
  1077
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1078
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1079
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1080
proof (induct as)
berghofe@13883
  1081
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1082
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1083
  proof (induct bs)
berghofe@13883
  1084
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1085
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1086
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1087
  qed simp
berghofe@13883
  1088
qed simp
berghofe@13883
  1089
kleing@13863
  1090
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1091
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1092
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1093
kleing@13863
  1094
lemma list_all2_nthD [dest?]:
kleing@13863
  1095
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1096
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1097
kleing@13863
  1098
lemma list_all2_map1: 
kleing@13863
  1099
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1100
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1101
kleing@13863
  1102
lemma list_all2_map2: 
kleing@13863
  1103
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1104
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1105
kleing@13863
  1106
lemma list_all2_refl:
kleing@13863
  1107
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1108
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1109
kleing@13863
  1110
lemma list_all2_update_cong:
kleing@13863
  1111
  "\<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
  1112
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1113
kleing@13863
  1114
lemma list_all2_update_cong2:
kleing@13863
  1115
  "\<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
  1116
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1117
kleing@13863
  1118
lemma list_all2_dropI [intro?]:
kleing@13863
  1119
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1120
  apply (induct as, simp)
kleing@13863
  1121
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1122
  apply (case_tac n, simp, simp)
kleing@13863
  1123
  done
kleing@13863
  1124
kleing@13863
  1125
lemma list_all2_mono [intro?]:
kleing@13863
  1126
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208
  1127
  apply (induct x, simp)
paulson@14208
  1128
  apply (case_tac y, auto)
kleing@13863
  1129
  done
kleing@13863
  1130
wenzelm@13142
  1131
wenzelm@13142
  1132
subsection {* @{text foldl} *}
wenzelm@13142
  1133
wenzelm@13142
  1134
lemma foldl_append [simp]:
nipkow@13145
  1135
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1136
by (induct xs) auto
wenzelm@13142
  1137
wenzelm@13142
  1138
text {*
nipkow@13145
  1139
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1140
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1141
*}
wenzelm@13142
  1142
wenzelm@13142
  1143
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1144
by (induct ns) auto
wenzelm@13142
  1145
wenzelm@13142
  1146
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1147
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1148
wenzelm@13142
  1149
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1150
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1151
by (induct ns) auto
wenzelm@13114
  1152
wenzelm@13114
  1153
oheimb@14099
  1154
subsection {* folding a relation over a list *}
oheimb@14099
  1155
oheimb@14099
  1156
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*)
oheimb@14099
  1157
inductive "fold_rel R" intros
oheimb@14099
  1158
  Nil:  "(a, [],a) : fold_rel R"
oheimb@14099
  1159
  Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R"
oheimb@14099
  1160
inductive_cases fold_rel_elim_case [elim!]:
paulson@14208
  1161
   "(a, [] , b) : fold_rel R"
oheimb@14099
  1162
   "(a, x#xs, b) : fold_rel R"
oheimb@14099
  1163
oheimb@14099
  1164
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" 
oheimb@14099
  1165
by (simp add: fold_rel.Nil)
oheimb@14099
  1166
declare fold_rel.Cons [intro!]
oheimb@14099
  1167
oheimb@14099
  1168
wenzelm@13142
  1169
subsection {* @{text upto} *}
wenzelm@13114
  1170
wenzelm@13142
  1171
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1172
-- {* Does not terminate! *}
nipkow@13145
  1173
by (induct j) auto
wenzelm@13142
  1174
wenzelm@13142
  1175
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1176
by (subst upt_rec) simp
wenzelm@13114
  1177
wenzelm@13142
  1178
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1179
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1180
by simp
wenzelm@13114
  1181
wenzelm@13142
  1182
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1183
apply(rule trans)
nipkow@13145
  1184
apply(subst upt_rec)
paulson@14208
  1185
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1186
done
wenzelm@13114
  1187
wenzelm@13142
  1188
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1189
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1190
by (induct k) auto
wenzelm@13114
  1191
wenzelm@13142
  1192
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1193
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1194
wenzelm@13142
  1195
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1196
apply (induct j)
nipkow@13145
  1197
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1198
done
wenzelm@13114
  1199
wenzelm@13142
  1200
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208
  1201
apply (induct m, simp)
nipkow@13145
  1202
apply (subst upt_rec)
nipkow@13145
  1203
apply (rule sym)
nipkow@13145
  1204
apply (subst upt_rec)
nipkow@13145
  1205
apply (simp del: upt.simps)
nipkow@13145
  1206
done
nipkow@3507
  1207
wenzelm@13114
  1208
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1209
by (induct n) auto
wenzelm@13114
  1210
wenzelm@13114
  1211
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1212
apply (induct n m rule: diff_induct)
nipkow@13145
  1213
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1214
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1215
done
wenzelm@13114
  1216
berghofe@13883
  1217
lemma nth_take_lemma:
berghofe@13883
  1218
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1219
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1220
apply (atomize, induct k)
paulson@14208
  1221
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1222
txt {* Both lists must be non-empty *}
paulson@14208
  1223
apply (case_tac xs, simp)
paulson@14208
  1224
apply (case_tac ys, clarify)
nipkow@13145
  1225
 apply (simp (no_asm_use))
nipkow@13145
  1226
apply clarify
nipkow@13145
  1227
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1228
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1229
apply blast
nipkow@13145
  1230
done
wenzelm@13114
  1231
wenzelm@13114
  1232
lemma nth_equalityI:
wenzelm@13114
  1233
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1234
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1235
apply (simp_all add: take_all)
nipkow@13145
  1236
done
wenzelm@13142
  1237
kleing@13863
  1238
(* needs nth_equalityI *)
kleing@13863
  1239
lemma list_all2_antisym:
kleing@13863
  1240
  "\<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
  1241
  \<Longrightarrow> xs = ys"
kleing@13863
  1242
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1243
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1244
  done
kleing@13863
  1245
wenzelm@13142
  1246
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1247
-- {* The famous take-lemma. *}
nipkow@13145
  1248
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1249
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1250
done
wenzelm@13142
  1251
wenzelm@13142
  1252
wenzelm@13142
  1253
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1254
wenzelm@13142
  1255
lemma distinct_append [simp]:
nipkow@13145
  1256
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1257
by (induct xs) auto
wenzelm@13142
  1258
wenzelm@13142
  1259
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1260
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1261
wenzelm@13142
  1262
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1263
by (induct xs) auto
wenzelm@13142
  1264
wenzelm@13142
  1265
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1266
by (induct xs) auto
wenzelm@13114
  1267
wenzelm@13142
  1268
text {*
nipkow@13145
  1269
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1270
it is useful. *}
wenzelm@13142
  1271
lemma distinct_conv_nth:
nipkow@13145
  1272
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@14208
  1273
apply (induct_tac xs, simp, simp)
paulson@14208
  1274
apply (rule iffI, clarsimp)
nipkow@13145
  1275
 apply (case_tac i)
paulson@14208
  1276
apply (case_tac j, simp)
nipkow@13145
  1277
apply (simp add: set_conv_nth)
nipkow@13145
  1278
 apply (case_tac j)
paulson@14208
  1279
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1280
apply (rule conjI)
nipkow@13145
  1281
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1282
 apply (erule_tac x = 0 in allE)
paulson@14208
  1283
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1284
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1285
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1286
done
wenzelm@13114
  1287
wenzelm@13114
  1288
wenzelm@13142
  1289
subsection {* @{text replicate} *}
wenzelm@13114
  1290
wenzelm@13142
  1291
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1292
by (induct n) auto
nipkow@13124
  1293
wenzelm@13142
  1294
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1295
by (induct n) auto
wenzelm@13114
  1296
wenzelm@13114
  1297
lemma replicate_app_Cons_same:
nipkow@13145
  1298
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1299
by (induct n) auto
wenzelm@13114
  1300
wenzelm@13142
  1301
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1302
apply (induct n, simp)
nipkow@13145
  1303
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1304
done
wenzelm@13114
  1305
wenzelm@13142
  1306
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1307
by (induct n) auto
wenzelm@13114
  1308
wenzelm@13142
  1309
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1310
by (induct n) auto
wenzelm@13114
  1311
wenzelm@13142
  1312
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1313
by (induct n) auto
wenzelm@13114
  1314
wenzelm@13142
  1315
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1316
by (atomize (full), induct n) auto
wenzelm@13114
  1317
wenzelm@13142
  1318
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1319
apply (induct n, simp)
nipkow@13145
  1320
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1321
done
wenzelm@13114
  1322
wenzelm@13142
  1323
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1324
by (induct n) auto
wenzelm@13114
  1325
wenzelm@13142
  1326
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1327
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1328
wenzelm@13142
  1329
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1330
by auto
wenzelm@13114
  1331
wenzelm@13142
  1332
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1333
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1334
wenzelm@13114
  1335
oheimb@14099
  1336
subsection {* @{text postfix} *}
oheimb@14099
  1337
oheimb@14099
  1338
lemma postfix_refl [simp, intro!]: "xs \<sqsupseteq> xs" by (auto simp add: postfix_def)
oheimb@14099
  1339
lemma postfix_trans: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsupseteq> zs" 
oheimb@14099
  1340
         by (auto simp add: postfix_def)
oheimb@14099
  1341
lemma postfix_antisym: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> xs\<rbrakk> \<Longrightarrow> xs = ys" 
oheimb@14099
  1342
         by (auto simp add: postfix_def)
oheimb@14099
  1343
oheimb@14099
  1344
lemma postfix_emptyI [simp, intro!]: "xs \<sqsupseteq> []" by (auto simp add: postfix_def)
oheimb@14099
  1345
lemma postfix_emptyD [dest!]: "[] \<sqsupseteq> xs \<Longrightarrow> xs = []"by(auto simp add:postfix_def)
oheimb@14099
  1346
lemma postfix_ConsI: "xs \<sqsupseteq> ys \<Longrightarrow> x#xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1347
lemma postfix_ConsD: "xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1348
lemma postfix_appendI: "xs \<sqsupseteq> ys \<Longrightarrow> zs@xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1349
lemma postfix_appendD: "xs \<sqsupseteq> zs@ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def)
oheimb@14099
  1350
oheimb@14099
  1351
lemma postfix_is_subset_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs"
oheimb@14099
  1352
by (induct zs, auto)
oheimb@14099
  1353
lemma postfix_is_subset: "xs \<sqsupseteq> ys \<Longrightarrow> set ys \<subseteq> set xs"
oheimb@14099
  1354
by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma)
oheimb@14099
  1355
oheimb@14099
  1356
lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \<longrightarrow> xs \<sqsupseteq> ys"
oheimb@14099
  1357
by (induct zs, auto intro!: postfix_appendI postfix_ConsI)
oheimb@14099
  1358
lemma postfix_ConsD2: "x#xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys"
oheimb@14099
  1359
by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma)
oheimb@14099
  1360
oheimb@14099
  1361
subsection {* Lexicographic orderings on lists *}
nipkow@3507
  1362
wenzelm@13142
  1363
lemma wf_lexn: "wf r ==> wf (lexn r n)"
paulson@14208
  1364
apply (induct_tac n, simp, simp)
nipkow@13145
  1365
apply(rule wf_subset)
nipkow@13145
  1366
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1367
apply(rule wf_prod_fun_image)
paulson@14208
  1368
 prefer 2 apply (rule inj_onI, auto)
nipkow@13145
  1369
done
wenzelm@13114
  1370
wenzelm@13114
  1371
lemma lexn_length:
nipkow@13145
  1372
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1373
by (induct n) auto
wenzelm@13114
  1374
wenzelm@13142
  1375
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1376
apply (unfold lex_def)
nipkow@13145
  1377
apply (rule wf_UN)
paulson@14208
  1378
apply (blast intro: wf_lexn, clarify)
nipkow@13145
  1379
apply (rename_tac m n)
nipkow@13145
  1380
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1381
 prefer 2 apply blast
nipkow@13145
  1382
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1383
done
wenzelm@13114
  1384
wenzelm@13114
  1385
lemma lexn_conv:
nipkow@13145
  1386
"lexn r n =
nipkow@13145
  1387
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1388
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
paulson@14208
  1389
apply (induct_tac n, simp, blast)
paulson@14208
  1390
apply (simp add: image_Collect lex_prod_def, safe, blast)
paulson@14208
  1391
 apply (rule_tac x = "ab # xys" in exI, simp)
paulson@14208
  1392
apply (case_tac xys, simp_all, blast)
nipkow@13145
  1393
done
wenzelm@13114
  1394
wenzelm@13114
  1395
lemma lex_conv:
nipkow@13145
  1396
"lex r =
nipkow@13145
  1397
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1398
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1399
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1400
wenzelm@13142
  1401
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1402
by (unfold lexico_def) blast
wenzelm@13114
  1403
wenzelm@13114
  1404
lemma lexico_conv:
nipkow@13145
  1405
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1406
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1407
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1408
wenzelm@13142
  1409
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1410
by (simp add: lex_conv)
wenzelm@13114
  1411
wenzelm@13142
  1412
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1413
by (simp add:lex_conv)
wenzelm@13114
  1414
wenzelm@13142
  1415
lemma Cons_in_lex [iff]:
nipkow@13145
  1416
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1417
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1418
apply (simp add: lex_conv)
nipkow@13145
  1419
apply (rule iffI)
paulson@14208
  1420
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
paulson@14208
  1421
apply (case_tac xys, simp, simp)
nipkow@13145
  1422
apply blast
nipkow@13145
  1423
done
wenzelm@13114
  1424
wenzelm@13114
  1425
wenzelm@13142
  1426
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1427
wenzelm@13142
  1428
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1429
by (auto simp add: sublist_def)
wenzelm@13114
  1430
wenzelm@13142
  1431
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1432
by (auto simp add: sublist_def)
wenzelm@13114
  1433
wenzelm@13114
  1434
lemma sublist_shift_lemma:
nipkow@13145
  1435
"map fst [p:zip xs [i..i + length xs(] . snd p : A] =
nipkow@13145
  1436
map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1437
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1438
wenzelm@13114
  1439
lemma sublist_append:
nipkow@13145
  1440
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1441
apply (unfold sublist_def)
paulson@14208
  1442
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1443
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1444
apply (simp add: add_commute)
nipkow@13145
  1445
done
wenzelm@13114
  1446
wenzelm@13114
  1447
lemma sublist_Cons:
nipkow@13145
  1448
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1449
apply (induct l rule: rev_induct)
nipkow@13145
  1450
 apply (simp add: sublist_def)
nipkow@13145
  1451
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1452
done
wenzelm@13114
  1453
wenzelm@13142
  1454
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1455
by (simp add: sublist_Cons)
wenzelm@13114
  1456
wenzelm@13142
  1457
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l"
paulson@14208
  1458
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1459
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1460
done
wenzelm@13114
  1461
wenzelm@13114
  1462
wenzelm@13142
  1463
lemma take_Cons':
nipkow@13145
  1464
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1465
by (cases n) simp_all
wenzelm@13114
  1466
wenzelm@13142
  1467
lemma drop_Cons':
nipkow@13145
  1468
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1469
by (cases n) simp_all
wenzelm@13114
  1470
wenzelm@13142
  1471
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1472
by (cases n) simp_all
wenzelm@13142
  1473
nipkow@13145
  1474
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1475
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1476
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1477
wenzelm@13462
  1478
wenzelm@13366
  1479
subsection {* Characters and strings *}
wenzelm@13366
  1480
wenzelm@13366
  1481
datatype nibble =
wenzelm@13366
  1482
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1483
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1484
wenzelm@13366
  1485
datatype char = Char nibble nibble
wenzelm@13366
  1486
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1487
wenzelm@13366
  1488
types string = "char list"
wenzelm@13366
  1489
wenzelm@13366
  1490
syntax
wenzelm@13366
  1491
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1492
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1493
wenzelm@13366
  1494
parse_ast_translation {*
wenzelm@13366
  1495
  let
wenzelm@13366
  1496
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1497
wenzelm@13366
  1498
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1499
    fun mk_char c =
wenzelm@13366
  1500
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1501
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1502
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1503
wenzelm@13366
  1504
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1505
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1506
wenzelm@13366
  1507
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1508
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1509
          [c] => mk_char c
wenzelm@13366
  1510
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1511
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1512
wenzelm@13366
  1513
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1514
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1515
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  1516
        | cs => mk_string cs)
wenzelm@13366
  1517
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  1518
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  1519
*}
wenzelm@13366
  1520
wenzelm@13366
  1521
print_ast_translation {*
wenzelm@13366
  1522
  let
wenzelm@13366
  1523
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  1524
        (case explode c of
wenzelm@13366
  1525
          ["N", "i", "b", "b", "l", "e", h] =>
wenzelm@13366
  1526
            if "0" <= h andalso h <= "9" then ord h - ord "0"
wenzelm@13366
  1527
            else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
wenzelm@13366
  1528
            else raise Match
wenzelm@13366
  1529
        | _ => raise Match)
wenzelm@13366
  1530
      | dest_nib _ = raise Match;
wenzelm@13366
  1531
wenzelm@13366
  1532
    fun dest_chr c1 c2 =
wenzelm@13366
  1533
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  1534
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  1535
wenzelm@13366
  1536
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  1537
      | dest_char _ = raise Match;
wenzelm@13366
  1538
wenzelm@13366
  1539
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  1540
wenzelm@13366
  1541
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  1542
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  1543
wenzelm@13366
  1544
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  1545
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  1546
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  1547
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  1548
*}
wenzelm@13366
  1549
wenzelm@13122
  1550
end