src/HOL/List.thy
author paulson
Wed Sep 01 15:03:41 2004 +0200 (2004-09-01)
changeset 15168 33a08cfc3ae5
parent 15140 322485b816ac
child 15176 2fd60846f485
permissions -rw-r--r--
new functions for sets of lists
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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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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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begin
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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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  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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  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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  remove1 :: "'a => '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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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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syntax (HTML output)
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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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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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"remove1 x [] = []"
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"remove1 x (y#xs) = (if x=y then xs else y # remove1 x 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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 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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proof (rule mono_Int [THEN equalityI])
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  show "mono lists" by (simp add: mono_def lists_mono)
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  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
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qed
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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)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@14495
   333
apply (induct xs)
nipkow@14495
   334
 apply fastsimp
nipkow@14495
   335
apply(case_tac zs)
nipkow@14495
   336
 apply simp
nipkow@14495
   337
apply fastsimp
nipkow@14495
   338
done
nipkow@14495
   339
wenzelm@13142
   340
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   341
by simp
wenzelm@13142
   342
wenzelm@13142
   343
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   344
by simp
wenzelm@13114
   345
wenzelm@13142
   346
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   347
by simp
wenzelm@13114
   348
wenzelm@13142
   349
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   350
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   351
wenzelm@13142
   352
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   353
using append_same_eq [of "[]"] by auto
wenzelm@13114
   354
wenzelm@13142
   355
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   356
by (induct xs) auto
wenzelm@13114
   357
wenzelm@13142
   358
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   359
by (induct xs) auto
wenzelm@13114
   360
wenzelm@13142
   361
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   362
by (simp add: hd_append split: list.split)
wenzelm@13114
   363
wenzelm@13142
   364
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   365
by (simp split: list.split)
wenzelm@13114
   366
wenzelm@13142
   367
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   368
by (simp add: tl_append split: list.split)
wenzelm@13114
   369
wenzelm@13114
   370
nipkow@14300
   371
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   372
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   373
by(cases ys) auto
nipkow@14300
   374
nipkow@14300
   375
wenzelm@13142
   376
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   377
wenzelm@13114
   378
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   379
by simp
wenzelm@13114
   380
wenzelm@13142
   381
lemma Cons_eq_appendI:
nipkow@13145
   382
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   383
by (drule sym) simp
wenzelm@13114
   384
wenzelm@13142
   385
lemma append_eq_appendI:
nipkow@13145
   386
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   387
by (drule sym) simp
wenzelm@13114
   388
wenzelm@13114
   389
wenzelm@13142
   390
text {*
nipkow@13145
   391
Simplification procedure for all list equalities.
nipkow@13145
   392
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   393
- both lists end in a singleton list,
nipkow@13145
   394
- or both lists end in the same list.
wenzelm@13142
   395
*}
wenzelm@13142
   396
wenzelm@13142
   397
ML_setup {*
nipkow@3507
   398
local
nipkow@3507
   399
wenzelm@13122
   400
val append_assoc = thm "append_assoc";
wenzelm@13122
   401
val append_Nil = thm "append_Nil";
wenzelm@13122
   402
val append_Cons = thm "append_Cons";
wenzelm@13122
   403
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   404
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   405
wenzelm@13114
   406
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   407
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   408
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   409
  | last t = t;
wenzelm@13114
   410
wenzelm@13114
   411
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   412
  | list1 _ = false;
wenzelm@13114
   413
wenzelm@13114
   414
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   415
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   416
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   417
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   418
wenzelm@13114
   419
val rearr_tac =
wenzelm@13462
   420
  simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114
   421
wenzelm@13114
   422
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   423
  let
wenzelm@13462
   424
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   425
    fun rearr conv =
wenzelm@13462
   426
      let
wenzelm@13462
   427
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   428
        val Type(_,listT::_) = eqT
wenzelm@13462
   429
        val appT = [listT,listT] ---> listT
wenzelm@13462
   430
        val app = Const("List.op @",appT)
wenzelm@13462
   431
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   432
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480
   433
        val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462
   434
      in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   435
wenzelm@13462
   436
  in
wenzelm@13462
   437
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   438
    else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462
   439
    else None
wenzelm@13462
   440
  end;
wenzelm@13462
   441
wenzelm@13114
   442
in
wenzelm@13462
   443
wenzelm@13462
   444
val list_eq_simproc =
wenzelm@13462
   445
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462
   446
wenzelm@13114
   447
end;
wenzelm@13114
   448
wenzelm@13114
   449
Addsimprocs [list_eq_simproc];
wenzelm@13114
   450
*}
wenzelm@13114
   451
wenzelm@13114
   452
wenzelm@13142
   453
subsection {* @{text map} *}
wenzelm@13114
   454
wenzelm@13142
   455
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   456
by (induct xs) simp_all
wenzelm@13114
   457
wenzelm@13142
   458
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   459
by (rule ext, induct_tac xs) auto
wenzelm@13114
   460
wenzelm@13142
   461
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   462
by (induct xs) auto
wenzelm@13114
   463
wenzelm@13142
   464
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   465
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   466
wenzelm@13142
   467
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   468
by (induct xs) auto
wenzelm@13114
   469
nipkow@13737
   470
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   471
by (induct xs) auto
nipkow@13737
   472
wenzelm@13366
   473
lemma map_cong [recdef_cong]:
nipkow@13145
   474
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   475
-- {* a congruence rule for @{text map} *}
nipkow@13737
   476
by simp
wenzelm@13114
   477
wenzelm@13142
   478
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   479
by (cases xs) auto
wenzelm@13114
   480
wenzelm@13142
   481
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   482
by (cases xs) auto
wenzelm@13114
   483
nipkow@14025
   484
lemma map_eq_Cons_conv[iff]:
nipkow@14025
   485
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   486
by (cases xs) auto
wenzelm@13114
   487
nipkow@14025
   488
lemma Cons_eq_map_conv[iff]:
nipkow@14025
   489
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   490
by (cases ys) auto
nipkow@14025
   491
nipkow@14111
   492
lemma ex_map_conv:
nipkow@14111
   493
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111
   494
by(induct ys, auto)
nipkow@14111
   495
nipkow@15110
   496
lemma map_eq_imp_length_eq:
nipkow@15110
   497
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   498
apply (induct ys)
nipkow@15110
   499
 apply simp
nipkow@15110
   500
apply(simp (no_asm_use))
nipkow@15110
   501
apply clarify
nipkow@15110
   502
apply(simp (no_asm_use))
nipkow@15110
   503
apply fast
nipkow@15110
   504
done
nipkow@15110
   505
nipkow@15110
   506
lemma map_inj_on:
nipkow@15110
   507
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   508
  ==> xs = ys"
nipkow@15110
   509
apply(frule map_eq_imp_length_eq)
nipkow@15110
   510
apply(rotate_tac -1)
nipkow@15110
   511
apply(induct rule:list_induct2)
nipkow@15110
   512
 apply simp
nipkow@15110
   513
apply(simp)
nipkow@15110
   514
apply (blast intro:sym)
nipkow@15110
   515
done
nipkow@15110
   516
nipkow@15110
   517
lemma inj_on_map_eq_map:
nipkow@15110
   518
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   519
by(blast dest:map_inj_on)
nipkow@15110
   520
wenzelm@13114
   521
lemma map_injective:
nipkow@14338
   522
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   523
by (induct ys) (auto dest!:injD)
wenzelm@13114
   524
nipkow@14339
   525
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   526
by(blast dest:map_injective)
nipkow@14339
   527
wenzelm@13114
   528
lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585
   529
by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114
   530
wenzelm@13114
   531
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   532
apply (unfold inj_on_def, clarify)
nipkow@13145
   533
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   534
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   535
apply blast
nipkow@13145
   536
done
wenzelm@13114
   537
nipkow@14339
   538
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   539
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   540
kleing@14343
   541
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   542
by (induct xs, auto)
wenzelm@13114
   543
nipkow@14402
   544
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   545
by (induct xs) auto
nipkow@14402
   546
nipkow@15110
   547
lemma map_fst_zip[simp]:
nipkow@15110
   548
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   549
by (induct rule:list_induct2, simp_all)
nipkow@15110
   550
nipkow@15110
   551
lemma map_snd_zip[simp]:
nipkow@15110
   552
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   553
by (induct rule:list_induct2, simp_all)
nipkow@15110
   554
nipkow@15110
   555
wenzelm@13142
   556
subsection {* @{text rev} *}
wenzelm@13114
   557
wenzelm@13142
   558
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   559
by (induct xs) auto
wenzelm@13114
   560
wenzelm@13142
   561
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   562
by (induct xs) auto
wenzelm@13114
   563
wenzelm@13142
   564
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   565
by (induct xs) auto
wenzelm@13114
   566
wenzelm@13142
   567
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   568
by (induct xs) auto
wenzelm@13114
   569
wenzelm@13142
   570
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208
   571
apply (induct xs, force)
paulson@14208
   572
apply (case_tac ys, simp, force)
nipkow@13145
   573
done
wenzelm@13114
   574
wenzelm@13366
   575
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   576
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145
   577
apply(subst rev_rev_ident[symmetric])
nipkow@13145
   578
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   579
done
wenzelm@13114
   580
nipkow@13145
   581
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   582
wenzelm@13366
   583
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   584
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   585
by (induct xs rule: rev_induct) auto
wenzelm@13114
   586
wenzelm@13366
   587
lemmas rev_cases = rev_exhaust
wenzelm@13366
   588
wenzelm@13114
   589
wenzelm@13142
   590
subsection {* @{text set} *}
wenzelm@13114
   591
wenzelm@13142
   592
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   593
by (induct xs) auto
wenzelm@13114
   594
wenzelm@13142
   595
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   596
by (induct xs) auto
wenzelm@13114
   597
oheimb@14099
   598
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208
   599
by (case_tac l, auto)
oheimb@14099
   600
wenzelm@13142
   601
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   602
by auto
wenzelm@13114
   603
oheimb@14099
   604
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   605
by auto
oheimb@14099
   606
wenzelm@13142
   607
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   608
by (induct xs) auto
wenzelm@13114
   609
wenzelm@13142
   610
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   611
by (induct xs) auto
wenzelm@13114
   612
wenzelm@13142
   613
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   614
by (induct xs) auto
wenzelm@13114
   615
wenzelm@13142
   616
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13142
   619
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
paulson@14208
   620
apply (induct j, simp_all)
paulson@14208
   621
apply (erule ssubst, auto)
nipkow@13145
   622
done
wenzelm@13114
   623
wenzelm@13142
   624
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   625
proof (induct xs)
paulson@15113
   626
  case Nil show ?case by simp
paulson@15113
   627
  case (Cons a xs)
paulson@15113
   628
  show ?case
paulson@15113
   629
  proof 
paulson@15113
   630
    assume "x \<in> set (a # xs)"
paulson@15113
   631
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   632
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   633
  next
paulson@15113
   634
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   635
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   636
    show "x \<in> set (a # xs)" 
paulson@15113
   637
      by (cases ys, auto simp add: eq)
paulson@15113
   638
  qed
paulson@15113
   639
qed
wenzelm@13142
   640
wenzelm@13142
   641
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   642
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   643
by (induct xs) auto
wenzelm@13142
   644
wenzelm@13142
   645
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   646
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   647
wenzelm@13142
   648
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   649
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   650
paulson@15168
   651
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
paulson@15168
   652
by auto
paulson@15168
   653
paulson@13508
   654
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   655
apply (erule finite_induct, auto)
paulson@13508
   656
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   657
done
paulson@13508
   658
kleing@14388
   659
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   660
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   661
paulson@15168
   662
paulson@15168
   663
subsection{*Sets of Lists*}
paulson@15168
   664
paulson@15168
   665
text{*Resembles a Cartesian product: it denotes the set of lists with head
paulson@15168
   666
  drawn from @{term A} and tail drawn from @{term Xs}.*}
paulson@15168
   667
constdefs
paulson@15168
   668
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
paulson@15168
   669
   "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
paulson@15168
   670
paulson@15168
   671
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
paulson@15168
   672
by (auto simp add: set_Cons_def)
paulson@15168
   673
paulson@15168
   674
text{*Yields the set of lists, all of the same length as the argument and
paulson@15168
   675
with elements drawn from the corresponding element of the argument.*}
paulson@15168
   676
consts  listset :: "'a set list \<Rightarrow> 'a list set"
paulson@15168
   677
primrec
paulson@15168
   678
   "listset []    = {[]}"
paulson@15168
   679
   "listset(A#As) = set_Cons A (listset As)"
paulson@15168
   680
paulson@15168
   681
wenzelm@13142
   682
subsection {* @{text mem} *}
wenzelm@13114
   683
wenzelm@13114
   684
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   685
by (induct xs) auto
wenzelm@13114
   686
wenzelm@13114
   687
wenzelm@13142
   688
subsection {* @{text list_all} *}
wenzelm@13114
   689
wenzelm@13142
   690
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   691
by (induct xs) auto
wenzelm@13114
   692
wenzelm@13142
   693
lemma list_all_append [simp]:
nipkow@13145
   694
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   695
by (induct xs) auto
wenzelm@13114
   696
wenzelm@13114
   697
wenzelm@13142
   698
subsection {* @{text filter} *}
wenzelm@13114
   699
wenzelm@13142
   700
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   701
by (induct xs) auto
wenzelm@13114
   702
wenzelm@13142
   703
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   704
by (induct xs) auto
wenzelm@13114
   705
wenzelm@13142
   706
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   707
by (induct xs) auto
wenzelm@13114
   708
wenzelm@13142
   709
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   710
by (induct xs) auto
wenzelm@13114
   711
wenzelm@13142
   712
lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   713
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   714
wenzelm@13142
   715
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   716
by auto
wenzelm@13114
   717
wenzelm@13114
   718
wenzelm@13142
   719
subsection {* @{text concat} *}
wenzelm@13114
   720
wenzelm@13142
   721
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   722
by (induct xs) auto
wenzelm@13114
   723
wenzelm@13142
   724
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   725
by (induct xss) auto
wenzelm@13114
   726
wenzelm@13142
   727
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   728
by (induct xss) auto
wenzelm@13114
   729
wenzelm@13142
   730
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   731
by (induct xs) auto
wenzelm@13114
   732
wenzelm@13142
   733
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   734
by (induct xs) auto
wenzelm@13114
   735
wenzelm@13142
   736
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   737
by (induct xs) auto
wenzelm@13114
   738
wenzelm@13142
   739
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   740
by (induct xs) auto
wenzelm@13114
   741
wenzelm@13114
   742
wenzelm@13142
   743
subsection {* @{text nth} *}
wenzelm@13114
   744
wenzelm@13142
   745
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   746
by auto
wenzelm@13114
   747
wenzelm@13142
   748
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   749
by auto
wenzelm@13114
   750
wenzelm@13142
   751
declare nth.simps [simp del]
wenzelm@13114
   752
wenzelm@13114
   753
lemma nth_append:
nipkow@13145
   754
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   755
apply (induct "xs", simp)
paulson@14208
   756
apply (case_tac n, auto)
nipkow@13145
   757
done
wenzelm@13114
   758
nipkow@14402
   759
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   760
by (induct "xs") auto
nipkow@14402
   761
nipkow@14402
   762
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   763
by (induct "xs") auto
nipkow@14402
   764
wenzelm@13142
   765
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   766
apply (induct xs, simp)
paulson@14208
   767
apply (case_tac n, auto)
nipkow@13145
   768
done
wenzelm@13114
   769
wenzelm@13142
   770
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@14208
   771
apply (induct_tac xs, simp, simp)
nipkow@13145
   772
apply safe
paulson@14208
   773
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   774
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   775
apply (case_tac i, simp)
nipkow@13145
   776
apply (rename_tac j)
paulson@14208
   777
apply (rule_tac x = j in exI, simp)
nipkow@13145
   778
done
wenzelm@13114
   779
nipkow@13145
   780
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   781
by (auto simp add: set_conv_nth)
wenzelm@13114
   782
wenzelm@13142
   783
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   784
by (auto simp add: set_conv_nth)
wenzelm@13114
   785
wenzelm@13114
   786
lemma all_nth_imp_all_set:
nipkow@13145
   787
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   788
by (auto simp add: set_conv_nth)
wenzelm@13114
   789
wenzelm@13114
   790
lemma all_set_conv_all_nth:
nipkow@13145
   791
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   792
by (auto simp add: set_conv_nth)
wenzelm@13114
   793
wenzelm@13114
   794
wenzelm@13142
   795
subsection {* @{text list_update} *}
wenzelm@13114
   796
wenzelm@13142
   797
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   798
by (induct xs) (auto split: nat.split)
wenzelm@13114
   799
wenzelm@13114
   800
lemma nth_list_update:
nipkow@13145
   801
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   802
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   803
wenzelm@13142
   804
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   805
by (simp add: nth_list_update)
wenzelm@13114
   806
wenzelm@13142
   807
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   808
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   809
wenzelm@13142
   810
lemma list_update_overwrite [simp]:
nipkow@13145
   811
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   812
by (induct xs) (auto split: nat.split)
wenzelm@13114
   813
nipkow@14402
   814
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   815
apply (induct xs, simp)
nipkow@14187
   816
apply(simp split:nat.splits)
nipkow@14187
   817
done
nipkow@14187
   818
wenzelm@13114
   819
lemma list_update_same_conv:
nipkow@13145
   820
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   821
by (induct xs) (auto split: nat.split)
wenzelm@13114
   822
nipkow@14187
   823
lemma list_update_append1:
nipkow@14187
   824
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   825
apply (induct xs, simp)
nipkow@14187
   826
apply(simp split:nat.split)
nipkow@14187
   827
done
nipkow@14187
   828
nipkow@14402
   829
lemma list_update_length [simp]:
nipkow@14402
   830
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   831
by (induct xs, auto)
nipkow@14402
   832
wenzelm@13114
   833
lemma update_zip:
nipkow@13145
   834
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   835
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   836
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   837
wenzelm@13114
   838
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   839
by (induct xs) (auto split: nat.split)
wenzelm@13114
   840
wenzelm@13114
   841
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   842
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   843
wenzelm@13114
   844
wenzelm@13142
   845
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   846
wenzelm@13142
   847
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   848
by (induct xs) auto
wenzelm@13114
   849
wenzelm@13142
   850
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   851
by (induct xs) auto
wenzelm@13114
   852
nipkow@14302
   853
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   854
by(simp add:last.simps)
nipkow@14302
   855
nipkow@14302
   856
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   857
by(simp add:last.simps)
nipkow@14302
   858
nipkow@14302
   859
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   860
by (induct xs) (auto)
nipkow@14302
   861
nipkow@14302
   862
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   863
by(simp add:last_append)
nipkow@14302
   864
nipkow@14302
   865
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   866
by(simp add:last_append)
nipkow@14302
   867
nipkow@14302
   868
wenzelm@13142
   869
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   870
by (induct xs rule: rev_induct) auto
wenzelm@13114
   871
wenzelm@13114
   872
lemma butlast_append:
nipkow@13145
   873
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   874
by (induct xs) auto
wenzelm@13114
   875
wenzelm@13142
   876
lemma append_butlast_last_id [simp]:
nipkow@13145
   877
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   878
by (induct xs) auto
wenzelm@13114
   879
wenzelm@13142
   880
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   881
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   882
wenzelm@13114
   883
lemma in_set_butlast_appendI:
nipkow@13145
   884
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   885
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   886
wenzelm@13142
   887
wenzelm@13142
   888
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   889
wenzelm@13142
   890
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   891
by (induct xs) auto
wenzelm@13114
   892
wenzelm@13142
   893
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   894
by (induct xs) auto
wenzelm@13114
   895
wenzelm@13142
   896
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   897
by simp
wenzelm@13114
   898
wenzelm@13142
   899
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   900
by simp
wenzelm@13114
   901
wenzelm@13142
   902
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   903
nipkow@15110
   904
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   905
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   906
nipkow@14187
   907
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   908
by(cases xs, simp_all)
nipkow@14187
   909
nipkow@14187
   910
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   911
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   912
nipkow@14187
   913
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   914
apply (induct xs, simp)
nipkow@14187
   915
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   916
done
nipkow@14187
   917
nipkow@13913
   918
lemma take_Suc_conv_app_nth:
nipkow@13913
   919
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   920
apply (induct xs, simp)
paulson@14208
   921
apply (case_tac i, auto)
nipkow@13913
   922
done
nipkow@13913
   923
mehta@14591
   924
lemma drop_Suc_conv_tl:
mehta@14591
   925
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
   926
apply (induct xs, simp)
mehta@14591
   927
apply (case_tac i, auto)
mehta@14591
   928
done
mehta@14591
   929
wenzelm@13142
   930
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
   931
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   932
wenzelm@13142
   933
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
   934
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   935
wenzelm@13142
   936
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
   937
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   938
wenzelm@13142
   939
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
   940
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   941
wenzelm@13142
   942
lemma take_append [simp]:
nipkow@13145
   943
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
   944
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   945
wenzelm@13142
   946
lemma drop_append [simp]:
nipkow@13145
   947
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
   948
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
   949
wenzelm@13142
   950
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
   951
apply (induct m, auto)
paulson@14208
   952
apply (case_tac xs, auto)
paulson@14208
   953
apply (case_tac na, auto)
nipkow@13145
   954
done
wenzelm@13114
   955
wenzelm@13142
   956
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
   957
apply (induct m, auto)
paulson@14208
   958
apply (case_tac xs, auto)
nipkow@13145
   959
done
wenzelm@13114
   960
wenzelm@13114
   961
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
   962
apply (induct m, auto)
paulson@14208
   963
apply (case_tac xs, auto)
nipkow@13145
   964
done
wenzelm@13114
   965
nipkow@14802
   966
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
   967
apply(induct xs)
nipkow@14802
   968
 apply simp
nipkow@14802
   969
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
   970
done
nipkow@14802
   971
wenzelm@13142
   972
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
   973
apply (induct n, auto)
paulson@14208
   974
apply (case_tac xs, auto)
nipkow@13145
   975
done
wenzelm@13114
   976
nipkow@15110
   977
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
   978
apply(induct xs)
nipkow@15110
   979
 apply simp
nipkow@15110
   980
apply(simp add:take_Cons split:nat.split)
nipkow@15110
   981
done
nipkow@15110
   982
nipkow@15110
   983
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
   984
apply(induct xs)
nipkow@15110
   985
apply simp
nipkow@15110
   986
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
   987
done
nipkow@15110
   988
wenzelm@13114
   989
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
   990
apply (induct n, auto)
paulson@14208
   991
apply (case_tac xs, auto)
nipkow@13145
   992
done
wenzelm@13114
   993
wenzelm@13142
   994
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
   995
apply (induct n, auto)
paulson@14208
   996
apply (case_tac xs, auto)
nipkow@13145
   997
done
wenzelm@13114
   998
wenzelm@13114
   999
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1000
apply (induct xs, auto)
paulson@14208
  1001
apply (case_tac i, auto)
nipkow@13145
  1002
done
wenzelm@13114
  1003
wenzelm@13114
  1004
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1005
apply (induct xs, auto)
paulson@14208
  1006
apply (case_tac i, auto)
nipkow@13145
  1007
done
wenzelm@13114
  1008
wenzelm@13142
  1009
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1010
apply (induct xs, auto)
paulson@14208
  1011
apply (case_tac n, blast)
paulson@14208
  1012
apply (case_tac i, auto)
nipkow@13145
  1013
done
wenzelm@13114
  1014
wenzelm@13142
  1015
lemma nth_drop [simp]:
nipkow@13145
  1016
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1017
apply (induct n, auto)
paulson@14208
  1018
apply (case_tac xs, auto)
nipkow@13145
  1019
done
nipkow@3507
  1020
nipkow@14025
  1021
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1022
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1023
nipkow@14025
  1024
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1025
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1026
nipkow@14187
  1027
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1028
using set_take_subset by fast
nipkow@14187
  1029
nipkow@14187
  1030
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1031
using set_drop_subset by fast
nipkow@14187
  1032
wenzelm@13114
  1033
lemma append_eq_conv_conj:
nipkow@13145
  1034
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1035
apply (induct xs, simp, clarsimp)
paulson@14208
  1036
apply (case_tac zs, auto)
nipkow@13145
  1037
done
wenzelm@13142
  1038
paulson@14050
  1039
lemma take_add [rule_format]: 
paulson@14050
  1040
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1041
apply (induct xs, auto) 
paulson@14050
  1042
apply (case_tac i, simp_all) 
paulson@14050
  1043
done
paulson@14050
  1044
nipkow@14300
  1045
lemma append_eq_append_conv_if:
nipkow@14300
  1046
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1047
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1048
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
nipkow@14300
  1049
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
nipkow@14300
  1050
apply(induct xs\<^isub>1)
nipkow@14300
  1051
 apply simp
nipkow@14300
  1052
apply(case_tac ys\<^isub>1)
nipkow@14300
  1053
apply simp_all
nipkow@14300
  1054
done
nipkow@14300
  1055
nipkow@15110
  1056
lemma take_hd_drop:
nipkow@15110
  1057
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1058
apply(induct xs)
nipkow@15110
  1059
apply simp
nipkow@15110
  1060
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1061
done
nipkow@15110
  1062
wenzelm@13114
  1063
wenzelm@13142
  1064
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1065
wenzelm@13142
  1066
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1067
by (induct xs) auto
wenzelm@13114
  1068
wenzelm@13142
  1069
lemma takeWhile_append1 [simp]:
nipkow@13145
  1070
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1071
by (induct xs) auto
wenzelm@13114
  1072
wenzelm@13142
  1073
lemma takeWhile_append2 [simp]:
nipkow@13145
  1074
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1075
by (induct xs) auto
wenzelm@13114
  1076
wenzelm@13142
  1077
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1078
by (induct xs) auto
wenzelm@13114
  1079
wenzelm@13142
  1080
lemma dropWhile_append1 [simp]:
nipkow@13145
  1081
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1082
by (induct xs) auto
wenzelm@13114
  1083
wenzelm@13142
  1084
lemma dropWhile_append2 [simp]:
nipkow@13145
  1085
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1086
by (induct xs) auto
wenzelm@13114
  1087
wenzelm@13142
  1088
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1089
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1090
nipkow@13913
  1091
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1092
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1093
by(induct xs, auto)
nipkow@13913
  1094
nipkow@13913
  1095
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1096
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1097
by(induct xs, auto)
nipkow@13913
  1098
nipkow@13913
  1099
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1100
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1101
by(induct xs, auto)
nipkow@13913
  1102
wenzelm@13114
  1103
wenzelm@13142
  1104
subsection {* @{text zip} *}
wenzelm@13114
  1105
wenzelm@13142
  1106
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1107
by (induct ys) auto
wenzelm@13114
  1108
wenzelm@13142
  1109
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1110
by simp
wenzelm@13114
  1111
wenzelm@13142
  1112
declare zip_Cons [simp del]
wenzelm@13114
  1113
wenzelm@13142
  1114
lemma length_zip [simp]:
nipkow@13145
  1115
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1116
apply (induct ys, simp)
paulson@14208
  1117
apply (case_tac xs, auto)
nipkow@13145
  1118
done
wenzelm@13114
  1119
wenzelm@13114
  1120
lemma zip_append1:
nipkow@13145
  1121
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1122
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1123
apply (induct zs, simp)
paulson@14208
  1124
apply (case_tac xs, simp_all)
nipkow@13145
  1125
done
wenzelm@13114
  1126
wenzelm@13114
  1127
lemma zip_append2:
nipkow@13145
  1128
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1129
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1130
apply (induct xs, simp)
paulson@14208
  1131
apply (case_tac ys, simp_all)
nipkow@13145
  1132
done
wenzelm@13114
  1133
wenzelm@13142
  1134
lemma zip_append [simp]:
wenzelm@13142
  1135
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1136
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1137
by (simp add: zip_append1)
wenzelm@13114
  1138
wenzelm@13114
  1139
lemma zip_rev:
nipkow@14247
  1140
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1141
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1142
wenzelm@13142
  1143
lemma nth_zip [simp]:
nipkow@13145
  1144
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1145
apply (induct ys, simp)
nipkow@13145
  1146
apply (case_tac xs)
nipkow@13145
  1147
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1148
done
wenzelm@13114
  1149
wenzelm@13114
  1150
lemma set_zip:
nipkow@13145
  1151
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1152
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1153
wenzelm@13114
  1154
lemma zip_update:
nipkow@13145
  1155
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1156
by (rule sym, simp add: update_zip)
wenzelm@13114
  1157
wenzelm@13142
  1158
lemma zip_replicate [simp]:
nipkow@13145
  1159
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1160
apply (induct i, auto)
paulson@14208
  1161
apply (case_tac j, auto)
nipkow@13145
  1162
done
wenzelm@13114
  1163
wenzelm@13142
  1164
wenzelm@13142
  1165
subsection {* @{text list_all2} *}
wenzelm@13114
  1166
kleing@14316
  1167
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1168
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1169
by (simp add: list_all2_def)
wenzelm@13114
  1170
wenzelm@13142
  1171
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1172
by (simp add: list_all2_def)
wenzelm@13114
  1173
wenzelm@13142
  1174
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1175
by (simp add: list_all2_def)
wenzelm@13114
  1176
wenzelm@13142
  1177
lemma list_all2_Cons [iff]:
nipkow@13145
  1178
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1179
by (auto simp add: list_all2_def)
wenzelm@13114
  1180
wenzelm@13114
  1181
lemma list_all2_Cons1:
nipkow@13145
  1182
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1183
by (cases ys) auto
wenzelm@13114
  1184
wenzelm@13114
  1185
lemma list_all2_Cons2:
nipkow@13145
  1186
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1187
by (cases xs) auto
wenzelm@13114
  1188
wenzelm@13142
  1189
lemma list_all2_rev [iff]:
nipkow@13145
  1190
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1191
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1192
kleing@13863
  1193
lemma list_all2_rev1:
kleing@13863
  1194
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1195
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1196
wenzelm@13114
  1197
lemma list_all2_append1:
nipkow@13145
  1198
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1199
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1200
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1201
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1202
apply (rule iffI)
nipkow@13145
  1203
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1204
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1205
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1206
apply (simp add: ball_Un)
nipkow@13145
  1207
done
wenzelm@13114
  1208
wenzelm@13114
  1209
lemma list_all2_append2:
nipkow@13145
  1210
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1211
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1212
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1213
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1214
apply (rule iffI)
nipkow@13145
  1215
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1216
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1217
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1218
apply (simp add: ball_Un)
nipkow@13145
  1219
done
wenzelm@13114
  1220
kleing@13863
  1221
lemma list_all2_append:
nipkow@14247
  1222
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1223
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1224
by (induct rule:list_induct2, simp_all)
kleing@13863
  1225
kleing@13863
  1226
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1227
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1228
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1229
wenzelm@13114
  1230
lemma list_all2_conv_all_nth:
nipkow@13145
  1231
"list_all2 P xs ys =
nipkow@13145
  1232
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1233
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1234
berghofe@13883
  1235
lemma list_all2_trans:
berghofe@13883
  1236
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1237
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1238
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1239
proof (induct as)
berghofe@13883
  1240
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1241
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1242
  proof (induct bs)
berghofe@13883
  1243
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1244
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1245
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1246
  qed simp
berghofe@13883
  1247
qed simp
berghofe@13883
  1248
kleing@13863
  1249
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1250
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1251
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1252
paulson@14395
  1253
lemma list_all2I:
paulson@14395
  1254
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1255
  by (simp add: list_all2_def)
paulson@14395
  1256
kleing@14328
  1257
lemma list_all2_nthD:
kleing@13863
  1258
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1259
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1260
nipkow@14302
  1261
lemma list_all2_nthD2:
nipkow@14302
  1262
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1263
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1264
kleing@13863
  1265
lemma list_all2_map1: 
kleing@13863
  1266
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1267
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1268
kleing@13863
  1269
lemma list_all2_map2: 
kleing@13863
  1270
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1271
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1272
kleing@14316
  1273
lemma list_all2_refl [intro?]:
kleing@13863
  1274
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1275
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1276
kleing@13863
  1277
lemma list_all2_update_cong:
kleing@13863
  1278
  "\<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
  1279
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1280
kleing@13863
  1281
lemma list_all2_update_cong2:
kleing@13863
  1282
  "\<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
  1283
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1284
nipkow@14302
  1285
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1286
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1287
  apply (induct xs)
nipkow@14302
  1288
   apply simp
nipkow@14302
  1289
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1290
  apply (case_tac n)
nipkow@14302
  1291
  apply auto
nipkow@14302
  1292
  done
nipkow@14302
  1293
nipkow@14302
  1294
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1295
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1296
  apply (induct as, simp)
kleing@13863
  1297
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1298
  apply (case_tac n, simp, simp)
kleing@13863
  1299
  done
kleing@13863
  1300
kleing@14327
  1301
lemma list_all2_mono [intro?]:
kleing@13863
  1302
  "\<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
  1303
  apply (induct x, simp)
paulson@14208
  1304
  apply (case_tac y, auto)
kleing@13863
  1305
  done
kleing@13863
  1306
wenzelm@13142
  1307
nipkow@14402
  1308
subsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1309
wenzelm@13142
  1310
lemma foldl_append [simp]:
nipkow@13145
  1311
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1312
by (induct xs) auto
wenzelm@13142
  1313
nipkow@14402
  1314
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1315
by (induct xs) auto
nipkow@14402
  1316
nipkow@14402
  1317
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1318
by (induct xs) auto
nipkow@14402
  1319
nipkow@14402
  1320
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1321
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1322
wenzelm@13142
  1323
text {*
nipkow@13145
  1324
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1325
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1326
*}
wenzelm@13142
  1327
wenzelm@13142
  1328
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1329
by (induct ns) auto
wenzelm@13142
  1330
wenzelm@13142
  1331
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1332
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1333
wenzelm@13142
  1334
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1335
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1336
by (induct ns) auto
wenzelm@13114
  1337
wenzelm@13114
  1338
wenzelm@13142
  1339
subsection {* @{text upto} *}
wenzelm@13114
  1340
wenzelm@13142
  1341
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1342
-- {* Does not terminate! *}
nipkow@13145
  1343
by (induct j) auto
wenzelm@13142
  1344
wenzelm@13142
  1345
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1346
by (subst upt_rec) simp
wenzelm@13114
  1347
wenzelm@13142
  1348
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1349
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1350
by simp
wenzelm@13114
  1351
wenzelm@13142
  1352
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1353
apply(rule trans)
nipkow@13145
  1354
apply(subst upt_rec)
paulson@14208
  1355
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1356
done
wenzelm@13114
  1357
wenzelm@13142
  1358
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1359
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1360
by (induct k) auto
wenzelm@13114
  1361
wenzelm@13142
  1362
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1363
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1364
wenzelm@13142
  1365
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1366
apply (induct j)
nipkow@13145
  1367
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1368
done
wenzelm@13114
  1369
wenzelm@13142
  1370
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208
  1371
apply (induct m, simp)
nipkow@13145
  1372
apply (subst upt_rec)
nipkow@13145
  1373
apply (rule sym)
nipkow@13145
  1374
apply (subst upt_rec)
nipkow@13145
  1375
apply (simp del: upt.simps)
nipkow@13145
  1376
done
nipkow@3507
  1377
wenzelm@13114
  1378
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1379
by (induct n) auto
wenzelm@13114
  1380
wenzelm@13114
  1381
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1382
apply (induct n m rule: diff_induct)
nipkow@13145
  1383
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1384
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1385
done
wenzelm@13114
  1386
berghofe@13883
  1387
lemma nth_take_lemma:
berghofe@13883
  1388
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1389
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1390
apply (atomize, induct k)
paulson@14208
  1391
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1392
txt {* Both lists must be non-empty *}
paulson@14208
  1393
apply (case_tac xs, simp)
paulson@14208
  1394
apply (case_tac ys, clarify)
nipkow@13145
  1395
 apply (simp (no_asm_use))
nipkow@13145
  1396
apply clarify
nipkow@13145
  1397
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1398
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1399
apply blast
nipkow@13145
  1400
done
wenzelm@13114
  1401
wenzelm@13114
  1402
lemma nth_equalityI:
wenzelm@13114
  1403
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1404
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1405
apply (simp_all add: take_all)
nipkow@13145
  1406
done
wenzelm@13142
  1407
kleing@13863
  1408
(* needs nth_equalityI *)
kleing@13863
  1409
lemma list_all2_antisym:
kleing@13863
  1410
  "\<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
  1411
  \<Longrightarrow> xs = ys"
kleing@13863
  1412
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1413
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1414
  done
kleing@13863
  1415
wenzelm@13142
  1416
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1417
-- {* The famous take-lemma. *}
nipkow@13145
  1418
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1419
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1420
done
wenzelm@13142
  1421
wenzelm@13142
  1422
wenzelm@13142
  1423
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1424
wenzelm@13142
  1425
lemma distinct_append [simp]:
nipkow@13145
  1426
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1427
by (induct xs) auto
wenzelm@13142
  1428
wenzelm@13142
  1429
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1430
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1431
wenzelm@13142
  1432
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1433
by (induct xs) auto
wenzelm@13142
  1434
paulson@15072
  1435
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15072
  1436
  by (induct_tac x, auto) 
paulson@15072
  1437
paulson@15072
  1438
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15072
  1439
  by (induct_tac x, auto)
paulson@15072
  1440
wenzelm@13142
  1441
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1442
by (induct xs) auto
wenzelm@13114
  1443
wenzelm@13142
  1444
text {*
nipkow@13145
  1445
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1446
it is useful. *}
wenzelm@13142
  1447
lemma distinct_conv_nth:
nipkow@13145
  1448
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@14208
  1449
apply (induct_tac xs, simp, simp)
paulson@14208
  1450
apply (rule iffI, clarsimp)
nipkow@13145
  1451
 apply (case_tac i)
paulson@14208
  1452
apply (case_tac j, simp)
nipkow@13145
  1453
apply (simp add: set_conv_nth)
nipkow@13145
  1454
 apply (case_tac j)
paulson@14208
  1455
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1456
apply (rule conjI)
nipkow@13145
  1457
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1458
 apply (erule_tac x = 0 in allE)
paulson@14208
  1459
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1460
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1461
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1462
done
wenzelm@13114
  1463
nipkow@15110
  1464
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1465
  by (induct xs) auto
kleing@14388
  1466
nipkow@15110
  1467
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1468
proof (induct xs)
kleing@14388
  1469
  case Nil thus ?case by simp
kleing@14388
  1470
next
kleing@14388
  1471
  case (Cons x xs)
kleing@14388
  1472
  show ?case
kleing@14388
  1473
  proof (cases "x \<in> set xs")
kleing@14388
  1474
    case False with Cons show ?thesis by simp
kleing@14388
  1475
  next
kleing@14388
  1476
    case True with Cons.prems
kleing@14388
  1477
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1478
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1479
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1480
    ultimately have False by simp
kleing@14388
  1481
    thus ?thesis ..
kleing@14388
  1482
  qed
kleing@14388
  1483
qed
kleing@14388
  1484
nipkow@15110
  1485
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1486
apply(induct xs)
nipkow@15110
  1487
 apply simp
nipkow@15110
  1488
apply fastsimp
nipkow@15110
  1489
done
nipkow@15110
  1490
nipkow@15110
  1491
lemma inj_on_set_conv:
nipkow@15110
  1492
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1493
apply(induct xs)
nipkow@15110
  1494
 apply simp
nipkow@15110
  1495
apply fastsimp
nipkow@15110
  1496
done
nipkow@15110
  1497
nipkow@15110
  1498
nipkow@15110
  1499
subsection {* @{text remove1} *}
nipkow@15110
  1500
nipkow@15110
  1501
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1502
apply(induct xs)
nipkow@15110
  1503
 apply simp
nipkow@15110
  1504
apply simp
nipkow@15110
  1505
apply blast
nipkow@15110
  1506
done
nipkow@15110
  1507
nipkow@15110
  1508
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1509
apply(induct xs)
nipkow@15110
  1510
 apply simp
nipkow@15110
  1511
apply simp
nipkow@15110
  1512
apply blast
nipkow@15110
  1513
done
nipkow@15110
  1514
nipkow@15110
  1515
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1516
apply(insert set_remove1_subset)
nipkow@15110
  1517
apply fast
nipkow@15110
  1518
done
nipkow@15110
  1519
nipkow@15110
  1520
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1521
by (induct xs) simp_all
nipkow@15110
  1522
wenzelm@13114
  1523
wenzelm@13142
  1524
subsection {* @{text replicate} *}
wenzelm@13114
  1525
wenzelm@13142
  1526
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1527
by (induct n) auto
nipkow@13124
  1528
wenzelm@13142
  1529
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1530
by (induct n) auto
wenzelm@13114
  1531
wenzelm@13114
  1532
lemma replicate_app_Cons_same:
nipkow@13145
  1533
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1534
by (induct n) auto
wenzelm@13114
  1535
wenzelm@13142
  1536
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1537
apply (induct n, simp)
nipkow@13145
  1538
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1539
done
wenzelm@13114
  1540
wenzelm@13142
  1541
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1542
by (induct n) auto
wenzelm@13114
  1543
wenzelm@13142
  1544
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1545
by (induct n) auto
wenzelm@13114
  1546
wenzelm@13142
  1547
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1548
by (induct n) auto
wenzelm@13114
  1549
wenzelm@13142
  1550
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1551
by (atomize (full), induct n) auto
wenzelm@13114
  1552
wenzelm@13142
  1553
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1554
apply (induct n, simp)
nipkow@13145
  1555
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1556
done
wenzelm@13114
  1557
wenzelm@13142
  1558
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1559
by (induct n) auto
wenzelm@13114
  1560
wenzelm@13142
  1561
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1562
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1563
wenzelm@13142
  1564
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1565
by auto
wenzelm@13114
  1566
wenzelm@13142
  1567
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1568
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1569
wenzelm@13114
  1570
oheimb@14099
  1571
subsection {* Lexicographic orderings on lists *}
nipkow@3507
  1572
wenzelm@13142
  1573
lemma wf_lexn: "wf r ==> wf (lexn r n)"
paulson@14208
  1574
apply (induct_tac n, simp, simp)
nipkow@13145
  1575
apply(rule wf_subset)
nipkow@13145
  1576
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1577
apply(rule wf_prod_fun_image)
paulson@14208
  1578
 prefer 2 apply (rule inj_onI, auto)
nipkow@13145
  1579
done
wenzelm@13114
  1580
wenzelm@13114
  1581
lemma lexn_length:
paulson@15168
  1582
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1583
by (induct n) auto
wenzelm@13114
  1584
wenzelm@13142
  1585
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1586
apply (unfold lex_def)
nipkow@13145
  1587
apply (rule wf_UN)
paulson@14208
  1588
apply (blast intro: wf_lexn, clarify)
nipkow@13145
  1589
apply (rename_tac m n)
nipkow@13145
  1590
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1591
 prefer 2 apply blast
nipkow@13145
  1592
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1593
done
wenzelm@13114
  1594
wenzelm@13114
  1595
lemma lexn_conv:
nipkow@13145
  1596
"lexn r n =
nipkow@13145
  1597
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1598
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
paulson@14208
  1599
apply (induct_tac n, simp, blast)
paulson@14208
  1600
apply (simp add: image_Collect lex_prod_def, safe, blast)
paulson@14208
  1601
 apply (rule_tac x = "ab # xys" in exI, simp)
paulson@14208
  1602
apply (case_tac xys, simp_all, blast)
nipkow@13145
  1603
done
wenzelm@13114
  1604
wenzelm@13114
  1605
lemma lex_conv:
nipkow@13145
  1606
"lex r =
nipkow@13145
  1607
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1608
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1609
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1610
wenzelm@13142
  1611
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1612
by (unfold lexico_def) blast
wenzelm@13114
  1613
wenzelm@13114
  1614
lemma lexico_conv:
nipkow@13145
  1615
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1616
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1617
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1618
wenzelm@13142
  1619
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1620
by (simp add: lex_conv)
wenzelm@13114
  1621
wenzelm@13142
  1622
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1623
by (simp add:lex_conv)
wenzelm@13114
  1624
wenzelm@13142
  1625
lemma Cons_in_lex [iff]:
nipkow@13145
  1626
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1627
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1628
apply (simp add: lex_conv)
nipkow@13145
  1629
apply (rule iffI)
paulson@14208
  1630
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
paulson@14208
  1631
apply (case_tac xys, simp, simp)
nipkow@13145
  1632
apply blast
nipkow@13145
  1633
done
wenzelm@13114
  1634
wenzelm@13114
  1635
wenzelm@13142
  1636
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1637
wenzelm@13142
  1638
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1639
by (auto simp add: sublist_def)
wenzelm@13114
  1640
wenzelm@13142
  1641
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1642
by (auto simp add: sublist_def)
wenzelm@13114
  1643
wenzelm@13114
  1644
lemma sublist_shift_lemma:
paulson@15168
  1645
     "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
paulson@15168
  1646
      map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1647
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1648
wenzelm@13114
  1649
lemma sublist_append:
paulson@15168
  1650
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1651
apply (unfold sublist_def)
paulson@14208
  1652
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1653
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1654
apply (simp add: add_commute)
nipkow@13145
  1655
done
wenzelm@13114
  1656
wenzelm@13114
  1657
lemma sublist_Cons:
nipkow@13145
  1658
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1659
apply (induct l rule: rev_induct)
nipkow@13145
  1660
 apply (simp add: sublist_def)
nipkow@13145
  1661
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1662
done
wenzelm@13114
  1663
wenzelm@13142
  1664
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1665
by (simp add: sublist_Cons)
wenzelm@13114
  1666
nipkow@15045
  1667
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1668
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1669
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1670
done
wenzelm@13114
  1671
wenzelm@13114
  1672
wenzelm@13142
  1673
lemma take_Cons':
paulson@15168
  1674
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1675
by (cases n) simp_all
wenzelm@13114
  1676
wenzelm@13142
  1677
lemma drop_Cons':
paulson@15168
  1678
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1679
by (cases n) simp_all
wenzelm@13114
  1680
wenzelm@13142
  1681
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1682
by (cases n) simp_all
wenzelm@13142
  1683
nipkow@13145
  1684
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1685
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1686
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1687
wenzelm@13462
  1688
kleing@14388
  1689
lemma "card (set xs) = size xs \<Longrightarrow> distinct xs"
kleing@14388
  1690
proof (induct xs)
kleing@14388
  1691
  case Nil thus ?case by simp
kleing@14388
  1692
next
kleing@14388
  1693
  case (Cons x xs)
kleing@14388
  1694
  show ?case
kleing@14388
  1695
  proof (cases "x \<in> set xs")
kleing@14388
  1696
    case False with Cons show ?thesis by simp
kleing@14388
  1697
  next
kleing@14388
  1698
    case True with Cons.prems
kleing@14388
  1699
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1700
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1701
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1702
    ultimately have False by simp
kleing@14388
  1703
    thus ?thesis ..
kleing@14388
  1704
  qed
kleing@14388
  1705
qed
kleing@14388
  1706
wenzelm@13366
  1707
subsection {* Characters and strings *}
wenzelm@13366
  1708
wenzelm@13366
  1709
datatype nibble =
wenzelm@13366
  1710
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1711
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1712
wenzelm@13366
  1713
datatype char = Char nibble nibble
wenzelm@13366
  1714
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1715
wenzelm@13366
  1716
types string = "char list"
wenzelm@13366
  1717
wenzelm@13366
  1718
syntax
wenzelm@13366
  1719
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1720
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1721
wenzelm@13366
  1722
parse_ast_translation {*
wenzelm@13366
  1723
  let
wenzelm@13366
  1724
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1725
wenzelm@13366
  1726
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1727
    fun mk_char c =
wenzelm@13366
  1728
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1729
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1730
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1731
wenzelm@13366
  1732
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1733
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1734
wenzelm@13366
  1735
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1736
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1737
          [c] => mk_char c
wenzelm@13366
  1738
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1739
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1740
wenzelm@13366
  1741
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1742
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1743
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  1744
        | cs => mk_string cs)
wenzelm@13366
  1745
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  1746
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  1747
*}
wenzelm@13366
  1748
berghofe@15064
  1749
ML {*
berghofe@15064
  1750
fun int_of_nibble h =
berghofe@15064
  1751
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  1752
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  1753
  else raise Match;
berghofe@15064
  1754
berghofe@15064
  1755
fun nibble_of_int i =
berghofe@15064
  1756
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  1757
*}
berghofe@15064
  1758
wenzelm@13366
  1759
print_ast_translation {*
wenzelm@13366
  1760
  let
wenzelm@13366
  1761
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  1762
        (case explode c of
berghofe@15064
  1763
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  1764
        | _ => raise Match)
wenzelm@13366
  1765
      | dest_nib _ = raise Match;
wenzelm@13366
  1766
wenzelm@13366
  1767
    fun dest_chr c1 c2 =
wenzelm@13366
  1768
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  1769
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  1770
wenzelm@13366
  1771
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  1772
      | dest_char _ = raise Match;
wenzelm@13366
  1773
wenzelm@13366
  1774
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  1775
wenzelm@13366
  1776
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  1777
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  1778
wenzelm@13366
  1779
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  1780
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  1781
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  1782
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  1783
*}
wenzelm@13366
  1784
berghofe@15064
  1785
subsection {* Code generator setup *}
berghofe@15064
  1786
berghofe@15064
  1787
ML {*
berghofe@15064
  1788
local
berghofe@15064
  1789
berghofe@15064
  1790
fun list_codegen thy gr dep b t =
berghofe@15064
  1791
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064
  1792
    (gr, HOLogic.dest_list t)
berghofe@15064
  1793
  in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064
  1794
berghofe@15064
  1795
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  1796
  | dest_nibble _ = raise Match;
berghofe@15064
  1797
berghofe@15064
  1798
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  1799
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064
  1800
     in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064
  1801
       else None
berghofe@15064
  1802
     end handle LIST _ => None | Match => None)
berghofe@15064
  1803
  | char_codegen thy gr dep b _ = None;
berghofe@15064
  1804
berghofe@15064
  1805
in
berghofe@15064
  1806
berghofe@15064
  1807
val list_codegen_setup =
berghofe@15064
  1808
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  1809
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  1810
berghofe@15064
  1811
end;
berghofe@15064
  1812
berghofe@15064
  1813
val term_of_list = HOLogic.mk_list;
berghofe@15064
  1814
berghofe@15064
  1815
fun gen_list' aG i j = frequency
berghofe@15064
  1816
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  1817
and gen_list aG i = gen_list' aG i i;
berghofe@15064
  1818
berghofe@15064
  1819
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  1820
berghofe@15064
  1821
fun term_of_char c =
berghofe@15064
  1822
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  1823
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  1824
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064
  1825
berghofe@15064
  1826
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  1827
*}
berghofe@15064
  1828
berghofe@15064
  1829
types_code
berghofe@15064
  1830
  "list" ("_ list")
berghofe@15064
  1831
  "char" ("string")
berghofe@15064
  1832
berghofe@15064
  1833
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  1834
berghofe@15064
  1835
setup list_codegen_setup
berghofe@15064
  1836
wenzelm@13122
  1837
end