src/HOL/List.thy
author nipkow
Fri Oct 15 18:16:03 2004 +0200 (2004-10-15)
changeset 15246 0984a2c2868b
parent 15245 5a21d9a8f14b
child 15251 bb6f072c8d10
permissions -rw-r--r--
added and renamed
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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
nipkow@15245
   610
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   611
by(induct xs) auto
nipkow@15245
   612
wenzelm@13142
   613
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   614
by (induct xs) auto
wenzelm@13114
   615
wenzelm@13142
   616
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13142
   619
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   620
by (induct xs) auto
wenzelm@13114
   621
wenzelm@13142
   622
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}"
paulson@14208
   623
apply (induct j, simp_all)
paulson@14208
   624
apply (erule ssubst, auto)
nipkow@13145
   625
done
wenzelm@13114
   626
wenzelm@13142
   627
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   628
proof (induct xs)
paulson@15113
   629
  case Nil show ?case by simp
paulson@15113
   630
  case (Cons a xs)
paulson@15113
   631
  show ?case
paulson@15113
   632
  proof 
paulson@15113
   633
    assume "x \<in> set (a # xs)"
paulson@15113
   634
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   635
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   636
  next
paulson@15113
   637
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   638
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   639
    show "x \<in> set (a # xs)" 
paulson@15113
   640
      by (cases ys, auto simp add: eq)
paulson@15113
   641
  qed
paulson@15113
   642
qed
wenzelm@13142
   643
wenzelm@13142
   644
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145
   645
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145
   646
by (induct xs) auto
wenzelm@13142
   647
wenzelm@13142
   648
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145
   649
by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142
   650
wenzelm@13142
   651
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145
   652
by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114
   653
paulson@15168
   654
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
paulson@15168
   655
by auto
paulson@15168
   656
paulson@13508
   657
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   658
apply (erule finite_induct, auto)
paulson@13508
   659
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   660
done
paulson@13508
   661
kleing@14388
   662
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   663
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   664
paulson@15168
   665
paulson@15168
   666
subsection{*Sets of Lists*}
paulson@15168
   667
paulson@15168
   668
text{*Resembles a Cartesian product: it denotes the set of lists with head
paulson@15168
   669
  drawn from @{term A} and tail drawn from @{term Xs}.*}
paulson@15168
   670
constdefs
paulson@15168
   671
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
paulson@15168
   672
   "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
paulson@15168
   673
paulson@15168
   674
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
paulson@15168
   675
by (auto simp add: set_Cons_def)
paulson@15168
   676
paulson@15168
   677
text{*Yields the set of lists, all of the same length as the argument and
paulson@15168
   678
with elements drawn from the corresponding element of the argument.*}
paulson@15168
   679
consts  listset :: "'a set list \<Rightarrow> 'a list set"
paulson@15168
   680
primrec
paulson@15168
   681
   "listset []    = {[]}"
paulson@15168
   682
   "listset(A#As) = set_Cons A (listset As)"
paulson@15168
   683
paulson@15168
   684
paulson@15176
   685
subsection{*Lifting a Relation on List Elements to the Lists*}
paulson@15176
   686
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
paulson@15176
   687
paulson@15176
   688
inductive "listrel(r)"
paulson@15176
   689
 intros
paulson@15176
   690
   Nil:  "([],[]) \<in> listrel r"
paulson@15176
   691
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
paulson@15176
   692
paulson@15176
   693
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
paulson@15176
   694
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
paulson@15176
   695
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
paulson@15176
   696
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
paulson@15176
   697
paulson@15176
   698
paulson@15176
   699
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
paulson@15176
   700
apply clarify  
paulson@15176
   701
apply (erule listrel.induct)
paulson@15176
   702
apply (blast intro: listrel.intros)+
paulson@15176
   703
done
paulson@15176
   704
paulson@15176
   705
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
paulson@15176
   706
apply clarify 
paulson@15176
   707
apply (erule listrel.induct, auto) 
paulson@15176
   708
done
paulson@15176
   709
paulson@15176
   710
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
paulson@15176
   711
apply (simp add: refl_def listrel_subset Ball_def)
paulson@15176
   712
apply (rule allI) 
paulson@15176
   713
apply (induct_tac x) 
paulson@15176
   714
apply (auto intro: listrel.intros)
paulson@15176
   715
done
paulson@15176
   716
paulson@15176
   717
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
paulson@15176
   718
apply (auto simp add: sym_def)
paulson@15176
   719
apply (erule listrel.induct) 
paulson@15176
   720
apply (blast intro: listrel.intros)+
paulson@15176
   721
done
paulson@15176
   722
paulson@15176
   723
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
paulson@15176
   724
apply (simp add: trans_def)
paulson@15176
   725
apply (intro allI) 
paulson@15176
   726
apply (rule impI) 
paulson@15176
   727
apply (erule listrel.induct) 
paulson@15176
   728
apply (blast intro: listrel.intros)+
paulson@15176
   729
done
paulson@15176
   730
paulson@15176
   731
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
paulson@15176
   732
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
paulson@15176
   733
paulson@15176
   734
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
paulson@15176
   735
by (blast intro: listrel.intros)
paulson@15176
   736
paulson@15176
   737
lemma listrel_Cons:
paulson@15176
   738
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
paulson@15176
   739
by (auto simp add: set_Cons_def intro: listrel.intros) 
paulson@15176
   740
paulson@15176
   741
wenzelm@13142
   742
subsection {* @{text mem} *}
wenzelm@13114
   743
wenzelm@13114
   744
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145
   745
by (induct xs) auto
wenzelm@13114
   746
wenzelm@13114
   747
wenzelm@13142
   748
subsection {* @{text list_all} *}
wenzelm@13114
   749
wenzelm@13142
   750
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145
   751
by (induct xs) auto
wenzelm@13114
   752
wenzelm@13142
   753
lemma list_all_append [simp]:
nipkow@13145
   754
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145
   755
by (induct xs) auto
wenzelm@13114
   756
wenzelm@13114
   757
wenzelm@13142
   758
subsection {* @{text filter} *}
wenzelm@13114
   759
wenzelm@13142
   760
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   761
by (induct xs) auto
wenzelm@13114
   762
wenzelm@13142
   763
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   764
by (induct xs) auto
wenzelm@13114
   765
wenzelm@13142
   766
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   767
by (induct xs) auto
wenzelm@13114
   768
wenzelm@13142
   769
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   770
by (induct xs) auto
wenzelm@13114
   771
nipkow@15246
   772
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145
   773
by (induct xs) (auto simp add: le_SucI)
wenzelm@13114
   774
wenzelm@13142
   775
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   776
by auto
wenzelm@13114
   777
nipkow@15246
   778
lemma length_filter_less:
nipkow@15246
   779
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   780
proof (induct xs)
nipkow@15246
   781
  case Nil thus ?case by simp
nipkow@15246
   782
next
nipkow@15246
   783
  case (Cons x xs) thus ?case
nipkow@15246
   784
    apply (auto split:split_if_asm)
nipkow@15246
   785
    using length_filter_le[of P xs] apply arith
nipkow@15246
   786
  done
nipkow@15246
   787
qed
wenzelm@13114
   788
wenzelm@13142
   789
subsection {* @{text concat} *}
wenzelm@13114
   790
wenzelm@13142
   791
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   792
by (induct xs) auto
wenzelm@13114
   793
wenzelm@13142
   794
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   795
by (induct xss) auto
wenzelm@13114
   796
wenzelm@13142
   797
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   798
by (induct xss) auto
wenzelm@13114
   799
wenzelm@13142
   800
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   801
by (induct xs) auto
wenzelm@13114
   802
wenzelm@13142
   803
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   804
by (induct xs) auto
wenzelm@13114
   805
wenzelm@13142
   806
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   807
by (induct xs) auto
wenzelm@13114
   808
wenzelm@13142
   809
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   810
by (induct xs) auto
wenzelm@13114
   811
wenzelm@13114
   812
wenzelm@13142
   813
subsection {* @{text nth} *}
wenzelm@13114
   814
wenzelm@13142
   815
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   816
by auto
wenzelm@13114
   817
wenzelm@13142
   818
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   819
by auto
wenzelm@13114
   820
wenzelm@13142
   821
declare nth.simps [simp del]
wenzelm@13114
   822
wenzelm@13114
   823
lemma nth_append:
nipkow@13145
   824
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   825
apply (induct "xs", simp)
paulson@14208
   826
apply (case_tac n, auto)
nipkow@13145
   827
done
wenzelm@13114
   828
nipkow@14402
   829
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   830
by (induct "xs") auto
nipkow@14402
   831
nipkow@14402
   832
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   833
by (induct "xs") auto
nipkow@14402
   834
wenzelm@13142
   835
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   836
apply (induct xs, simp)
paulson@14208
   837
apply (case_tac n, auto)
nipkow@13145
   838
done
wenzelm@13114
   839
wenzelm@13142
   840
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@14208
   841
apply (induct_tac xs, simp, simp)
nipkow@13145
   842
apply safe
paulson@14208
   843
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   844
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   845
apply (case_tac i, simp)
nipkow@13145
   846
apply (rename_tac j)
paulson@14208
   847
apply (rule_tac x = j in exI, simp)
nipkow@13145
   848
done
wenzelm@13114
   849
nipkow@13145
   850
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   851
by (auto simp add: set_conv_nth)
wenzelm@13114
   852
wenzelm@13142
   853
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   854
by (auto simp add: set_conv_nth)
wenzelm@13114
   855
wenzelm@13114
   856
lemma all_nth_imp_all_set:
nipkow@13145
   857
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   858
by (auto simp add: set_conv_nth)
wenzelm@13114
   859
wenzelm@13114
   860
lemma all_set_conv_all_nth:
nipkow@13145
   861
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   862
by (auto simp add: set_conv_nth)
wenzelm@13114
   863
wenzelm@13114
   864
wenzelm@13142
   865
subsection {* @{text list_update} *}
wenzelm@13114
   866
wenzelm@13142
   867
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   868
by (induct xs) (auto split: nat.split)
wenzelm@13114
   869
wenzelm@13114
   870
lemma nth_list_update:
nipkow@13145
   871
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   872
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   873
wenzelm@13142
   874
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   875
by (simp add: nth_list_update)
wenzelm@13114
   876
wenzelm@13142
   877
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   878
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   879
wenzelm@13142
   880
lemma list_update_overwrite [simp]:
nipkow@13145
   881
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   882
by (induct xs) (auto split: nat.split)
wenzelm@13114
   883
nipkow@14402
   884
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   885
apply (induct xs, simp)
nipkow@14187
   886
apply(simp split:nat.splits)
nipkow@14187
   887
done
nipkow@14187
   888
wenzelm@13114
   889
lemma list_update_same_conv:
nipkow@13145
   890
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   891
by (induct xs) (auto split: nat.split)
wenzelm@13114
   892
nipkow@14187
   893
lemma list_update_append1:
nipkow@14187
   894
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   895
apply (induct xs, simp)
nipkow@14187
   896
apply(simp split:nat.split)
nipkow@14187
   897
done
nipkow@14187
   898
nipkow@14402
   899
lemma list_update_length [simp]:
nipkow@14402
   900
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   901
by (induct xs, auto)
nipkow@14402
   902
wenzelm@13114
   903
lemma update_zip:
nipkow@13145
   904
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   905
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   906
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   907
wenzelm@13114
   908
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   909
by (induct xs) (auto split: nat.split)
wenzelm@13114
   910
wenzelm@13114
   911
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   912
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   913
wenzelm@13114
   914
wenzelm@13142
   915
subsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   916
wenzelm@13142
   917
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   918
by (induct xs) auto
wenzelm@13114
   919
wenzelm@13142
   920
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   921
by (induct xs) auto
wenzelm@13114
   922
nipkow@14302
   923
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   924
by(simp add:last.simps)
nipkow@14302
   925
nipkow@14302
   926
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   927
by(simp add:last.simps)
nipkow@14302
   928
nipkow@14302
   929
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   930
by (induct xs) (auto)
nipkow@14302
   931
nipkow@14302
   932
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   933
by(simp add:last_append)
nipkow@14302
   934
nipkow@14302
   935
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
   936
by(simp add:last_append)
nipkow@14302
   937
nipkow@14302
   938
wenzelm@13142
   939
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
   940
by (induct xs rule: rev_induct) auto
wenzelm@13114
   941
wenzelm@13114
   942
lemma butlast_append:
nipkow@13145
   943
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
   944
by (induct xs) auto
wenzelm@13114
   945
wenzelm@13142
   946
lemma append_butlast_last_id [simp]:
nipkow@13145
   947
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
   948
by (induct xs) auto
wenzelm@13114
   949
wenzelm@13142
   950
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
   951
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
   952
wenzelm@13114
   953
lemma in_set_butlast_appendI:
nipkow@13145
   954
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
   955
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
   956
wenzelm@13142
   957
wenzelm@13142
   958
subsection {* @{text take} and @{text drop} *}
wenzelm@13114
   959
wenzelm@13142
   960
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
   961
by (induct xs) auto
wenzelm@13114
   962
wenzelm@13142
   963
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
   964
by (induct xs) auto
wenzelm@13114
   965
wenzelm@13142
   966
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
   967
by simp
wenzelm@13114
   968
wenzelm@13142
   969
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
   970
by simp
wenzelm@13114
   971
wenzelm@13142
   972
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
   973
nipkow@15110
   974
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
   975
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
   976
nipkow@14187
   977
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
   978
by(cases xs, simp_all)
nipkow@14187
   979
nipkow@14187
   980
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
   981
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
   982
nipkow@14187
   983
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
   984
apply (induct xs, simp)
nipkow@14187
   985
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
   986
done
nipkow@14187
   987
nipkow@13913
   988
lemma take_Suc_conv_app_nth:
nipkow@13913
   989
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
   990
apply (induct xs, simp)
paulson@14208
   991
apply (case_tac i, auto)
nipkow@13913
   992
done
nipkow@13913
   993
mehta@14591
   994
lemma drop_Suc_conv_tl:
mehta@14591
   995
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
   996
apply (induct xs, simp)
mehta@14591
   997
apply (case_tac i, auto)
mehta@14591
   998
done
mehta@14591
   999
wenzelm@13142
  1000
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1001
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1002
wenzelm@13142
  1003
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1004
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1005
wenzelm@13142
  1006
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1007
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1008
wenzelm@13142
  1009
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1010
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1011
wenzelm@13142
  1012
lemma take_append [simp]:
nipkow@13145
  1013
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1014
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1015
wenzelm@13142
  1016
lemma drop_append [simp]:
nipkow@13145
  1017
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1018
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1019
wenzelm@13142
  1020
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1021
apply (induct m, auto)
paulson@14208
  1022
apply (case_tac xs, auto)
nipkow@15236
  1023
apply (case_tac n, auto)
nipkow@13145
  1024
done
wenzelm@13114
  1025
wenzelm@13142
  1026
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1027
apply (induct m, auto)
paulson@14208
  1028
apply (case_tac xs, auto)
nipkow@13145
  1029
done
wenzelm@13114
  1030
wenzelm@13114
  1031
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1032
apply (induct m, auto)
paulson@14208
  1033
apply (case_tac xs, auto)
nipkow@13145
  1034
done
wenzelm@13114
  1035
nipkow@14802
  1036
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1037
apply(induct xs)
nipkow@14802
  1038
 apply simp
nipkow@14802
  1039
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1040
done
nipkow@14802
  1041
wenzelm@13142
  1042
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1043
apply (induct n, auto)
paulson@14208
  1044
apply (case_tac xs, auto)
nipkow@13145
  1045
done
wenzelm@13114
  1046
nipkow@15110
  1047
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1048
apply(induct xs)
nipkow@15110
  1049
 apply simp
nipkow@15110
  1050
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1051
done
nipkow@15110
  1052
nipkow@15110
  1053
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1054
apply(induct xs)
nipkow@15110
  1055
apply simp
nipkow@15110
  1056
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1057
done
nipkow@15110
  1058
wenzelm@13114
  1059
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1060
apply (induct n, auto)
paulson@14208
  1061
apply (case_tac xs, auto)
nipkow@13145
  1062
done
wenzelm@13114
  1063
wenzelm@13142
  1064
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1065
apply (induct n, auto)
paulson@14208
  1066
apply (case_tac xs, auto)
nipkow@13145
  1067
done
wenzelm@13114
  1068
wenzelm@13114
  1069
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1070
apply (induct xs, auto)
paulson@14208
  1071
apply (case_tac i, auto)
nipkow@13145
  1072
done
wenzelm@13114
  1073
wenzelm@13114
  1074
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1075
apply (induct xs, auto)
paulson@14208
  1076
apply (case_tac i, auto)
nipkow@13145
  1077
done
wenzelm@13114
  1078
wenzelm@13142
  1079
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1080
apply (induct xs, auto)
paulson@14208
  1081
apply (case_tac n, blast)
paulson@14208
  1082
apply (case_tac i, auto)
nipkow@13145
  1083
done
wenzelm@13114
  1084
wenzelm@13142
  1085
lemma nth_drop [simp]:
nipkow@13145
  1086
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1087
apply (induct n, auto)
paulson@14208
  1088
apply (case_tac xs, auto)
nipkow@13145
  1089
done
nipkow@3507
  1090
nipkow@14025
  1091
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1092
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1093
nipkow@14025
  1094
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1095
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1096
nipkow@14187
  1097
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1098
using set_take_subset by fast
nipkow@14187
  1099
nipkow@14187
  1100
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1101
using set_drop_subset by fast
nipkow@14187
  1102
wenzelm@13114
  1103
lemma append_eq_conv_conj:
nipkow@13145
  1104
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1105
apply (induct xs, simp, clarsimp)
paulson@14208
  1106
apply (case_tac zs, auto)
nipkow@13145
  1107
done
wenzelm@13142
  1108
paulson@14050
  1109
lemma take_add [rule_format]: 
paulson@14050
  1110
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1111
apply (induct xs, auto) 
paulson@14050
  1112
apply (case_tac i, simp_all) 
paulson@14050
  1113
done
paulson@14050
  1114
nipkow@14300
  1115
lemma append_eq_append_conv_if:
nipkow@14300
  1116
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1117
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1118
   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
  1119
   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
  1120
apply(induct xs\<^isub>1)
nipkow@14300
  1121
 apply simp
nipkow@14300
  1122
apply(case_tac ys\<^isub>1)
nipkow@14300
  1123
apply simp_all
nipkow@14300
  1124
done
nipkow@14300
  1125
nipkow@15110
  1126
lemma take_hd_drop:
nipkow@15110
  1127
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1128
apply(induct xs)
nipkow@15110
  1129
apply simp
nipkow@15110
  1130
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1131
done
nipkow@15110
  1132
wenzelm@13114
  1133
wenzelm@13142
  1134
subsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1135
wenzelm@13142
  1136
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1137
by (induct xs) auto
wenzelm@13114
  1138
wenzelm@13142
  1139
lemma takeWhile_append1 [simp]:
nipkow@13145
  1140
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1141
by (induct xs) auto
wenzelm@13114
  1142
wenzelm@13142
  1143
lemma takeWhile_append2 [simp]:
nipkow@13145
  1144
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1145
by (induct xs) auto
wenzelm@13114
  1146
wenzelm@13142
  1147
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1148
by (induct xs) auto
wenzelm@13114
  1149
wenzelm@13142
  1150
lemma dropWhile_append1 [simp]:
nipkow@13145
  1151
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1152
by (induct xs) auto
wenzelm@13114
  1153
wenzelm@13142
  1154
lemma dropWhile_append2 [simp]:
nipkow@13145
  1155
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1156
by (induct xs) auto
wenzelm@13114
  1157
wenzelm@13142
  1158
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1159
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1160
nipkow@13913
  1161
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1162
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1163
by(induct xs, auto)
nipkow@13913
  1164
nipkow@13913
  1165
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1166
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1167
by(induct xs, auto)
nipkow@13913
  1168
nipkow@13913
  1169
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1170
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1171
by(induct xs, auto)
nipkow@13913
  1172
wenzelm@13114
  1173
wenzelm@13142
  1174
subsection {* @{text zip} *}
wenzelm@13114
  1175
wenzelm@13142
  1176
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1177
by (induct ys) auto
wenzelm@13114
  1178
wenzelm@13142
  1179
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1180
by simp
wenzelm@13114
  1181
wenzelm@13142
  1182
declare zip_Cons [simp del]
wenzelm@13114
  1183
wenzelm@13142
  1184
lemma length_zip [simp]:
nipkow@13145
  1185
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1186
apply (induct ys, simp)
paulson@14208
  1187
apply (case_tac xs, auto)
nipkow@13145
  1188
done
wenzelm@13114
  1189
wenzelm@13114
  1190
lemma zip_append1:
nipkow@13145
  1191
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1192
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1193
apply (induct zs, simp)
paulson@14208
  1194
apply (case_tac xs, simp_all)
nipkow@13145
  1195
done
wenzelm@13114
  1196
wenzelm@13114
  1197
lemma zip_append2:
nipkow@13145
  1198
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1199
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1200
apply (induct xs, simp)
paulson@14208
  1201
apply (case_tac ys, simp_all)
nipkow@13145
  1202
done
wenzelm@13114
  1203
wenzelm@13142
  1204
lemma zip_append [simp]:
wenzelm@13142
  1205
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1206
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1207
by (simp add: zip_append1)
wenzelm@13114
  1208
wenzelm@13114
  1209
lemma zip_rev:
nipkow@14247
  1210
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1211
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1212
wenzelm@13142
  1213
lemma nth_zip [simp]:
nipkow@13145
  1214
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1215
apply (induct ys, simp)
nipkow@13145
  1216
apply (case_tac xs)
nipkow@13145
  1217
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1218
done
wenzelm@13114
  1219
wenzelm@13114
  1220
lemma set_zip:
nipkow@13145
  1221
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1222
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1223
wenzelm@13114
  1224
lemma zip_update:
nipkow@13145
  1225
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1226
by (rule sym, simp add: update_zip)
wenzelm@13114
  1227
wenzelm@13142
  1228
lemma zip_replicate [simp]:
nipkow@13145
  1229
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1230
apply (induct i, auto)
paulson@14208
  1231
apply (case_tac j, auto)
nipkow@13145
  1232
done
wenzelm@13114
  1233
wenzelm@13142
  1234
wenzelm@13142
  1235
subsection {* @{text list_all2} *}
wenzelm@13114
  1236
kleing@14316
  1237
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1238
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145
  1239
by (simp add: list_all2_def)
wenzelm@13114
  1240
wenzelm@13142
  1241
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145
  1242
by (simp add: list_all2_def)
wenzelm@13114
  1243
wenzelm@13142
  1244
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145
  1245
by (simp add: list_all2_def)
wenzelm@13114
  1246
wenzelm@13142
  1247
lemma list_all2_Cons [iff]:
nipkow@13145
  1248
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145
  1249
by (auto simp add: list_all2_def)
wenzelm@13114
  1250
wenzelm@13114
  1251
lemma list_all2_Cons1:
nipkow@13145
  1252
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1253
by (cases ys) auto
wenzelm@13114
  1254
wenzelm@13114
  1255
lemma list_all2_Cons2:
nipkow@13145
  1256
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1257
by (cases xs) auto
wenzelm@13114
  1258
wenzelm@13142
  1259
lemma list_all2_rev [iff]:
nipkow@13145
  1260
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1261
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1262
kleing@13863
  1263
lemma list_all2_rev1:
kleing@13863
  1264
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1265
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1266
wenzelm@13114
  1267
lemma list_all2_append1:
nipkow@13145
  1268
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1269
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1270
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1271
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1272
apply (rule iffI)
nipkow@13145
  1273
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1274
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1275
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1276
apply (simp add: ball_Un)
nipkow@13145
  1277
done
wenzelm@13114
  1278
wenzelm@13114
  1279
lemma list_all2_append2:
nipkow@13145
  1280
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1281
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1282
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1283
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1284
apply (rule iffI)
nipkow@13145
  1285
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1286
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1287
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1288
apply (simp add: ball_Un)
nipkow@13145
  1289
done
wenzelm@13114
  1290
kleing@13863
  1291
lemma list_all2_append:
nipkow@14247
  1292
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1293
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1294
by (induct rule:list_induct2, simp_all)
kleing@13863
  1295
kleing@13863
  1296
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1297
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1298
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1299
wenzelm@13114
  1300
lemma list_all2_conv_all_nth:
nipkow@13145
  1301
"list_all2 P xs ys =
nipkow@13145
  1302
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1303
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1304
berghofe@13883
  1305
lemma list_all2_trans:
berghofe@13883
  1306
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1307
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1308
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1309
proof (induct as)
berghofe@13883
  1310
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1311
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1312
  proof (induct bs)
berghofe@13883
  1313
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1314
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1315
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1316
  qed simp
berghofe@13883
  1317
qed simp
berghofe@13883
  1318
kleing@13863
  1319
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1320
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1321
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1322
paulson@14395
  1323
lemma list_all2I:
paulson@14395
  1324
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1325
  by (simp add: list_all2_def)
paulson@14395
  1326
kleing@14328
  1327
lemma list_all2_nthD:
kleing@13863
  1328
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1329
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1330
nipkow@14302
  1331
lemma list_all2_nthD2:
nipkow@14302
  1332
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1333
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1334
kleing@13863
  1335
lemma list_all2_map1: 
kleing@13863
  1336
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1337
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1338
kleing@13863
  1339
lemma list_all2_map2: 
kleing@13863
  1340
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1341
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1342
kleing@14316
  1343
lemma list_all2_refl [intro?]:
kleing@13863
  1344
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1345
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1346
kleing@13863
  1347
lemma list_all2_update_cong:
kleing@13863
  1348
  "\<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
  1349
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1350
kleing@13863
  1351
lemma list_all2_update_cong2:
kleing@13863
  1352
  "\<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
  1353
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1354
nipkow@14302
  1355
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1356
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1357
  apply (induct xs)
nipkow@14302
  1358
   apply simp
nipkow@14302
  1359
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1360
  apply (case_tac n)
nipkow@14302
  1361
  apply auto
nipkow@14302
  1362
  done
nipkow@14302
  1363
nipkow@14302
  1364
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1365
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1366
  apply (induct as, simp)
kleing@13863
  1367
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1368
  apply (case_tac n, simp, simp)
kleing@13863
  1369
  done
kleing@13863
  1370
kleing@14327
  1371
lemma list_all2_mono [intro?]:
kleing@13863
  1372
  "\<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
  1373
  apply (induct x, simp)
paulson@14208
  1374
  apply (case_tac y, auto)
kleing@13863
  1375
  done
kleing@13863
  1376
wenzelm@13142
  1377
nipkow@14402
  1378
subsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1379
wenzelm@13142
  1380
lemma foldl_append [simp]:
nipkow@13145
  1381
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1382
by (induct xs) auto
wenzelm@13142
  1383
nipkow@14402
  1384
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1385
by (induct xs) auto
nipkow@14402
  1386
nipkow@14402
  1387
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1388
by (induct xs) auto
nipkow@14402
  1389
nipkow@14402
  1390
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1391
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1392
wenzelm@13142
  1393
text {*
nipkow@13145
  1394
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1395
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1396
*}
wenzelm@13142
  1397
wenzelm@13142
  1398
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1399
by (induct ns) auto
wenzelm@13142
  1400
wenzelm@13142
  1401
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1402
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1403
wenzelm@13142
  1404
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1405
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1406
by (induct ns) auto
wenzelm@13114
  1407
wenzelm@13114
  1408
wenzelm@13142
  1409
subsection {* @{text upto} *}
wenzelm@13114
  1410
wenzelm@13142
  1411
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145
  1412
-- {* Does not terminate! *}
nipkow@13145
  1413
by (induct j) auto
wenzelm@13142
  1414
wenzelm@13142
  1415
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145
  1416
by (subst upt_rec) simp
wenzelm@13114
  1417
wenzelm@13142
  1418
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145
  1419
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1420
by simp
wenzelm@13114
  1421
wenzelm@13142
  1422
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145
  1423
apply(rule trans)
nipkow@13145
  1424
apply(subst upt_rec)
paulson@14208
  1425
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1426
done
wenzelm@13114
  1427
wenzelm@13142
  1428
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145
  1429
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1430
by (induct k) auto
wenzelm@13114
  1431
wenzelm@13142
  1432
lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145
  1433
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1434
wenzelm@13142
  1435
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145
  1436
apply (induct j)
nipkow@13145
  1437
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1438
done
wenzelm@13114
  1439
wenzelm@13142
  1440
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208
  1441
apply (induct m, simp)
nipkow@13145
  1442
apply (subst upt_rec)
nipkow@13145
  1443
apply (rule sym)
nipkow@13145
  1444
apply (subst upt_rec)
nipkow@13145
  1445
apply (simp del: upt.simps)
nipkow@13145
  1446
done
nipkow@3507
  1447
wenzelm@13114
  1448
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145
  1449
by (induct n) auto
wenzelm@13114
  1450
wenzelm@13114
  1451
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145
  1452
apply (induct n m rule: diff_induct)
nipkow@13145
  1453
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1454
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1455
done
wenzelm@13114
  1456
berghofe@13883
  1457
lemma nth_take_lemma:
berghofe@13883
  1458
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1459
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1460
apply (atomize, induct k)
paulson@14208
  1461
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1462
txt {* Both lists must be non-empty *}
paulson@14208
  1463
apply (case_tac xs, simp)
paulson@14208
  1464
apply (case_tac ys, clarify)
nipkow@13145
  1465
 apply (simp (no_asm_use))
nipkow@13145
  1466
apply clarify
nipkow@13145
  1467
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1468
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1469
apply blast
nipkow@13145
  1470
done
wenzelm@13114
  1471
wenzelm@13114
  1472
lemma nth_equalityI:
wenzelm@13114
  1473
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1474
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1475
apply (simp_all add: take_all)
nipkow@13145
  1476
done
wenzelm@13142
  1477
kleing@13863
  1478
(* needs nth_equalityI *)
kleing@13863
  1479
lemma list_all2_antisym:
kleing@13863
  1480
  "\<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
  1481
  \<Longrightarrow> xs = ys"
kleing@13863
  1482
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1483
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1484
  done
kleing@13863
  1485
wenzelm@13142
  1486
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1487
-- {* The famous take-lemma. *}
nipkow@13145
  1488
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1489
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1490
done
wenzelm@13142
  1491
wenzelm@13142
  1492
wenzelm@13142
  1493
subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1494
wenzelm@13142
  1495
lemma distinct_append [simp]:
nipkow@13145
  1496
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1497
by (induct xs) auto
wenzelm@13142
  1498
wenzelm@13142
  1499
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1500
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1501
wenzelm@13142
  1502
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1503
by (induct xs) auto
wenzelm@13142
  1504
paulson@15072
  1505
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15072
  1506
  by (induct_tac x, auto) 
paulson@15072
  1507
paulson@15072
  1508
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15072
  1509
  by (induct_tac x, auto)
paulson@15072
  1510
nipkow@15245
  1511
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1512
by (induct xs) auto
nipkow@15245
  1513
nipkow@15245
  1514
lemma length_remdups_eq[iff]:
nipkow@15245
  1515
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1516
apply(induct xs)
nipkow@15245
  1517
 apply auto
nipkow@15245
  1518
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1519
 apply arith
nipkow@15245
  1520
apply(rule length_remdups_leq)
nipkow@15245
  1521
done
nipkow@15245
  1522
wenzelm@13142
  1523
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1524
by (induct xs) auto
wenzelm@13114
  1525
wenzelm@13142
  1526
text {*
nipkow@13145
  1527
It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145
  1528
it is useful. *}
wenzelm@13142
  1529
lemma distinct_conv_nth:
nipkow@13145
  1530
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@14208
  1531
apply (induct_tac xs, simp, simp)
paulson@14208
  1532
apply (rule iffI, clarsimp)
nipkow@13145
  1533
 apply (case_tac i)
paulson@14208
  1534
apply (case_tac j, simp)
nipkow@13145
  1535
apply (simp add: set_conv_nth)
nipkow@13145
  1536
 apply (case_tac j)
paulson@14208
  1537
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1538
apply (rule conjI)
nipkow@13145
  1539
 apply (clarsimp simp add: set_conv_nth)
nipkow@13145
  1540
 apply (erule_tac x = 0 in allE)
paulson@14208
  1541
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145
  1542
apply (erule_tac x = "Suc i" in allE)
paulson@14208
  1543
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1544
done
wenzelm@13114
  1545
nipkow@15110
  1546
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1547
  by (induct xs) auto
kleing@14388
  1548
nipkow@15110
  1549
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1550
proof (induct xs)
kleing@14388
  1551
  case Nil thus ?case by simp
kleing@14388
  1552
next
kleing@14388
  1553
  case (Cons x xs)
kleing@14388
  1554
  show ?case
kleing@14388
  1555
  proof (cases "x \<in> set xs")
kleing@14388
  1556
    case False with Cons show ?thesis by simp
kleing@14388
  1557
  next
kleing@14388
  1558
    case True with Cons.prems
kleing@14388
  1559
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1560
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1561
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1562
    ultimately have False by simp
kleing@14388
  1563
    thus ?thesis ..
kleing@14388
  1564
  qed
kleing@14388
  1565
qed
kleing@14388
  1566
nipkow@15110
  1567
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110
  1568
apply(induct xs)
nipkow@15110
  1569
 apply simp
nipkow@15110
  1570
apply fastsimp
nipkow@15110
  1571
done
nipkow@15110
  1572
nipkow@15110
  1573
lemma inj_on_set_conv:
nipkow@15110
  1574
 "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110
  1575
apply(induct xs)
nipkow@15110
  1576
 apply simp
nipkow@15110
  1577
apply fastsimp
nipkow@15110
  1578
done
nipkow@15110
  1579
nipkow@15110
  1580
nipkow@15110
  1581
subsection {* @{text remove1} *}
nipkow@15110
  1582
nipkow@15110
  1583
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1584
apply(induct xs)
nipkow@15110
  1585
 apply simp
nipkow@15110
  1586
apply simp
nipkow@15110
  1587
apply blast
nipkow@15110
  1588
done
nipkow@15110
  1589
nipkow@15110
  1590
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1591
apply(induct xs)
nipkow@15110
  1592
 apply simp
nipkow@15110
  1593
apply simp
nipkow@15110
  1594
apply blast
nipkow@15110
  1595
done
nipkow@15110
  1596
nipkow@15110
  1597
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1598
apply(insert set_remove1_subset)
nipkow@15110
  1599
apply fast
nipkow@15110
  1600
done
nipkow@15110
  1601
nipkow@15110
  1602
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1603
by (induct xs) simp_all
nipkow@15110
  1604
wenzelm@13114
  1605
wenzelm@13142
  1606
subsection {* @{text replicate} *}
wenzelm@13114
  1607
wenzelm@13142
  1608
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1609
by (induct n) auto
nipkow@13124
  1610
wenzelm@13142
  1611
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1612
by (induct n) auto
wenzelm@13114
  1613
wenzelm@13114
  1614
lemma replicate_app_Cons_same:
nipkow@13145
  1615
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1616
by (induct n) auto
wenzelm@13114
  1617
wenzelm@13142
  1618
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1619
apply (induct n, simp)
nipkow@13145
  1620
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1621
done
wenzelm@13114
  1622
wenzelm@13142
  1623
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1624
by (induct n) auto
wenzelm@13114
  1625
wenzelm@13142
  1626
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1627
by (induct n) auto
wenzelm@13114
  1628
wenzelm@13142
  1629
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1630
by (induct n) auto
wenzelm@13114
  1631
wenzelm@13142
  1632
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1633
by (atomize (full), induct n) auto
wenzelm@13114
  1634
wenzelm@13142
  1635
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1636
apply (induct n, simp)
nipkow@13145
  1637
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1638
done
wenzelm@13114
  1639
wenzelm@13142
  1640
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1641
by (induct n) auto
wenzelm@13114
  1642
wenzelm@13142
  1643
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1644
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1645
wenzelm@13142
  1646
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1647
by auto
wenzelm@13114
  1648
wenzelm@13142
  1649
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1650
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1651
wenzelm@13114
  1652
oheimb@14099
  1653
subsection {* Lexicographic orderings on lists *}
nipkow@3507
  1654
wenzelm@13142
  1655
lemma wf_lexn: "wf r ==> wf (lexn r n)"
paulson@14208
  1656
apply (induct_tac n, simp, simp)
nipkow@13145
  1657
apply(rule wf_subset)
nipkow@13145
  1658
 prefer 2 apply (rule Int_lower1)
nipkow@13145
  1659
apply(rule wf_prod_fun_image)
paulson@14208
  1660
 prefer 2 apply (rule inj_onI, auto)
nipkow@13145
  1661
done
wenzelm@13114
  1662
wenzelm@13114
  1663
lemma lexn_length:
paulson@15168
  1664
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145
  1665
by (induct n) auto
wenzelm@13114
  1666
wenzelm@13142
  1667
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145
  1668
apply (unfold lex_def)
nipkow@13145
  1669
apply (rule wf_UN)
paulson@14208
  1670
apply (blast intro: wf_lexn, clarify)
nipkow@13145
  1671
apply (rename_tac m n)
nipkow@13145
  1672
apply (subgoal_tac "m \<noteq> n")
nipkow@13145
  1673
 prefer 2 apply blast
nipkow@13145
  1674
apply (blast dest: lexn_length not_sym)
nipkow@13145
  1675
done
wenzelm@13114
  1676
wenzelm@13114
  1677
lemma lexn_conv:
nipkow@13145
  1678
"lexn r n =
nipkow@13145
  1679
{(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145
  1680
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
paulson@14208
  1681
apply (induct_tac n, simp, blast)
paulson@14208
  1682
apply (simp add: image_Collect lex_prod_def, safe, blast)
paulson@14208
  1683
 apply (rule_tac x = "ab # xys" in exI, simp)
paulson@14208
  1684
apply (case_tac xys, simp_all, blast)
nipkow@13145
  1685
done
wenzelm@13114
  1686
wenzelm@13114
  1687
lemma lex_conv:
nipkow@13145
  1688
"lex r =
nipkow@13145
  1689
{(xs,ys). length xs = length ys \<and>
nipkow@13145
  1690
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145
  1691
by (force simp add: lex_def lexn_conv)
wenzelm@13114
  1692
wenzelm@13142
  1693
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145
  1694
by (unfold lexico_def) blast
wenzelm@13114
  1695
wenzelm@13114
  1696
lemma lexico_conv:
nipkow@13145
  1697
"lexico r = {(xs,ys). length xs < length ys |
nipkow@13145
  1698
length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145
  1699
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114
  1700
wenzelm@13142
  1701
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145
  1702
by (simp add: lex_conv)
wenzelm@13114
  1703
wenzelm@13142
  1704
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145
  1705
by (simp add:lex_conv)
wenzelm@13114
  1706
wenzelm@13142
  1707
lemma Cons_in_lex [iff]:
nipkow@13145
  1708
"((x # xs, y # ys) : lex r) =
nipkow@13145
  1709
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145
  1710
apply (simp add: lex_conv)
nipkow@13145
  1711
apply (rule iffI)
paulson@14208
  1712
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
paulson@14208
  1713
apply (case_tac xys, simp, simp)
nipkow@13145
  1714
apply blast
nipkow@13145
  1715
done
wenzelm@13114
  1716
wenzelm@13114
  1717
wenzelm@13142
  1718
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  1719
wenzelm@13142
  1720
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  1721
by (auto simp add: sublist_def)
wenzelm@13114
  1722
wenzelm@13142
  1723
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  1724
by (auto simp add: sublist_def)
wenzelm@13114
  1725
wenzelm@13114
  1726
lemma sublist_shift_lemma:
paulson@15168
  1727
     "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
paulson@15168
  1728
      map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145
  1729
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  1730
wenzelm@13114
  1731
lemma sublist_append:
paulson@15168
  1732
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  1733
apply (unfold sublist_def)
paulson@14208
  1734
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  1735
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  1736
apply (simp add: add_commute)
nipkow@13145
  1737
done
wenzelm@13114
  1738
wenzelm@13114
  1739
lemma sublist_Cons:
nipkow@13145
  1740
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  1741
apply (induct l rule: rev_induct)
nipkow@13145
  1742
 apply (simp add: sublist_def)
nipkow@13145
  1743
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  1744
done
wenzelm@13114
  1745
wenzelm@13142
  1746
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  1747
by (simp add: sublist_Cons)
wenzelm@13114
  1748
nipkow@15045
  1749
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  1750
apply (induct l rule: rev_induct, simp)
nipkow@13145
  1751
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  1752
done
wenzelm@13114
  1753
wenzelm@13114
  1754
wenzelm@13142
  1755
lemma take_Cons':
paulson@15168
  1756
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145
  1757
by (cases n) simp_all
wenzelm@13114
  1758
wenzelm@13142
  1759
lemma drop_Cons':
paulson@15168
  1760
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145
  1761
by (cases n) simp_all
wenzelm@13114
  1762
wenzelm@13142
  1763
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145
  1764
by (cases n) simp_all
wenzelm@13142
  1765
nipkow@13145
  1766
lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145
  1767
                drop_Cons'[of "number_of v",standard]
nipkow@13145
  1768
                nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507
  1769
wenzelm@13462
  1770
kleing@14388
  1771
lemma "card (set xs) = size xs \<Longrightarrow> distinct xs"
kleing@14388
  1772
proof (induct xs)
kleing@14388
  1773
  case Nil thus ?case by simp
kleing@14388
  1774
next
kleing@14388
  1775
  case (Cons x xs)
kleing@14388
  1776
  show ?case
kleing@14388
  1777
  proof (cases "x \<in> set xs")
kleing@14388
  1778
    case False with Cons show ?thesis by simp
kleing@14388
  1779
  next
kleing@14388
  1780
    case True with Cons.prems
kleing@14388
  1781
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1782
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1783
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1784
    ultimately have False by simp
kleing@14388
  1785
    thus ?thesis ..
kleing@14388
  1786
  qed
kleing@14388
  1787
qed
kleing@14388
  1788
wenzelm@13366
  1789
subsection {* Characters and strings *}
wenzelm@13366
  1790
wenzelm@13366
  1791
datatype nibble =
wenzelm@13366
  1792
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  1793
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  1794
wenzelm@13366
  1795
datatype char = Char nibble nibble
wenzelm@13366
  1796
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  1797
wenzelm@13366
  1798
types string = "char list"
wenzelm@13366
  1799
wenzelm@13366
  1800
syntax
wenzelm@13366
  1801
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  1802
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  1803
wenzelm@13366
  1804
parse_ast_translation {*
wenzelm@13366
  1805
  let
wenzelm@13366
  1806
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  1807
wenzelm@13366
  1808
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  1809
    fun mk_char c =
wenzelm@13366
  1810
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  1811
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  1812
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  1813
wenzelm@13366
  1814
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  1815
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  1816
wenzelm@13366
  1817
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1818
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1819
          [c] => mk_char c
wenzelm@13366
  1820
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  1821
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  1822
wenzelm@13366
  1823
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  1824
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  1825
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  1826
        | cs => mk_string cs)
wenzelm@13366
  1827
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  1828
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  1829
*}
wenzelm@13366
  1830
berghofe@15064
  1831
ML {*
berghofe@15064
  1832
fun int_of_nibble h =
berghofe@15064
  1833
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  1834
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  1835
  else raise Match;
berghofe@15064
  1836
berghofe@15064
  1837
fun nibble_of_int i =
berghofe@15064
  1838
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064
  1839
*}
berghofe@15064
  1840
wenzelm@13366
  1841
print_ast_translation {*
wenzelm@13366
  1842
  let
wenzelm@13366
  1843
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  1844
        (case explode c of
berghofe@15064
  1845
          ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366
  1846
        | _ => raise Match)
wenzelm@13366
  1847
      | dest_nib _ = raise Match;
wenzelm@13366
  1848
wenzelm@13366
  1849
    fun dest_chr c1 c2 =
wenzelm@13366
  1850
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  1851
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  1852
wenzelm@13366
  1853
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  1854
      | dest_char _ = raise Match;
wenzelm@13366
  1855
wenzelm@13366
  1856
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  1857
wenzelm@13366
  1858
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  1859
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  1860
wenzelm@13366
  1861
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  1862
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  1863
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  1864
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  1865
*}
wenzelm@13366
  1866
berghofe@15064
  1867
subsection {* Code generator setup *}
berghofe@15064
  1868
berghofe@15064
  1869
ML {*
berghofe@15064
  1870
local
berghofe@15064
  1871
berghofe@15064
  1872
fun list_codegen thy gr dep b t =
berghofe@15064
  1873
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064
  1874
    (gr, HOLogic.dest_list t)
berghofe@15064
  1875
  in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064
  1876
berghofe@15064
  1877
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064
  1878
  | dest_nibble _ = raise Match;
berghofe@15064
  1879
berghofe@15064
  1880
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064
  1881
    (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064
  1882
     in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064
  1883
       else None
berghofe@15064
  1884
     end handle LIST _ => None | Match => None)
berghofe@15064
  1885
  | char_codegen thy gr dep b _ = None;
berghofe@15064
  1886
berghofe@15064
  1887
in
berghofe@15064
  1888
berghofe@15064
  1889
val list_codegen_setup =
berghofe@15064
  1890
  [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064
  1891
   Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064
  1892
berghofe@15064
  1893
end;
berghofe@15064
  1894
berghofe@15064
  1895
val term_of_list = HOLogic.mk_list;
berghofe@15064
  1896
berghofe@15064
  1897
fun gen_list' aG i j = frequency
berghofe@15064
  1898
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  1899
and gen_list aG i = gen_list' aG i i;
berghofe@15064
  1900
berghofe@15064
  1901
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  1902
berghofe@15064
  1903
fun term_of_char c =
berghofe@15064
  1904
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064
  1905
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064
  1906
    Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064
  1907
berghofe@15064
  1908
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  1909
*}
berghofe@15064
  1910
berghofe@15064
  1911
types_code
berghofe@15064
  1912
  "list" ("_ list")
berghofe@15064
  1913
  "char" ("string")
berghofe@15064
  1914
berghofe@15064
  1915
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  1916
berghofe@15064
  1917
setup list_codegen_setup
berghofe@15064
  1918
wenzelm@13122
  1919
end