author  wenzelm 
Tue, 16 Jul 2002 18:26:36 +0200  
changeset 13366  114b7c14084a 
parent 13187  e5434b822a96 
child 13462  56610e2ba220 
permissions  rwrr 
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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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Copyright 1994 TU Muenchen 

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*) 
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header {* The datatype of finite lists *} 
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theory List = PreList: 

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datatype 'a list = 
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Nil ("[]") 
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 Cons 'a "'a list" (infixr "#" 65) 

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consts 

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"@" :: "'a list => 'a list => 'a list" (infixr 65) 
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filter:: "('a => bool) => 'a list => 'a list" 

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concat:: "'a list list => 'a list" 

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foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" 

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foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" 

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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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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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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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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 listforming 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 [rule_format]: 
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"l: lists A ==> l: lists B > l: lists (A Int B)" 
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apply (erule lists.induct) 

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apply blast+ 

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done 

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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" 

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apply (rule mono_Int [THEN equalityI]) 
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apply (simp add: mono_def lists_mono) 

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apply (blast intro!: lists_IntI) 

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done 

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lemma append_in_lists_conv [iff]: 
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" 
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by (induct xs) auto 

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subsection {* @{text length} *} 

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text {* 
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Needs to come before @{text "@"} because of theorem @{text 
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append_eq_append_conv}. 

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*} 
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" 
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by (induct xs) auto 
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lemma length_map [simp]: "length (map f xs) = length xs" 
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by (induct xs) auto 
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lemma length_rev [simp]: "length (rev xs) = length xs" 
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by (induct xs) auto 
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lemma length_tl [simp]: "length (tl xs) = length xs  1" 
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by (cases xs) auto 
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" 
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by (induct xs) auto 
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" 
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by (induct xs) auto 
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lemma length_Suc_conv: 

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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" 
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by (induct xs) auto 

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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 [rule_format, simp]: 
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"\<forall>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_tac xs) 

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apply(rule allI) 

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apply (case_tac ys) 

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apply simp 

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apply force 

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apply (rule allI) 

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apply (case_tac ys) 

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apply force 

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apply simp 

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done 

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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" 

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by simp 
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" 

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by simp 
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" 
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by simp 
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" 
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using append_same_eq [of _ _ "[]"] by auto 
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" 
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using append_same_eq [of "[]"] by auto 
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" 
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by (induct xs) auto 
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" 
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by (induct xs) auto 
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" 
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by (simp add: hd_append split: list.split) 
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys  z#zs => zs @ ys)" 
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by (simp split: list.split) 
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lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" 
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by (simp add: tl_append split: list.split) 
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text {* Trivial rules for solving @{text "@"}equations automatically. *} 
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lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" 

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by simp 
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lemma Cons_eq_appendI: 
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"[ x # xs1 = ys; xs = xs1 @ zs ] ==> x # xs = ys @ zs" 
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by (drule sym) simp 

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lemma append_eq_appendI: 
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"[ xs @ xs1 = zs; ys = xs1 @ us ] ==> xs @ ys = zs @ us" 
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by (drule sym) simp 

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353 

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text {* 
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Simplification procedure for all list equalities. 
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Currently only tries to rearrange @{text "@"} to see if 

357 
 both lists end in a singleton list, 

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 or both lists end in the same list. 

13142  359 
*} 
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361 
ML_setup {* 

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local 
363 

13122  364 
val append_assoc = thm "append_assoc"; 
365 
val append_Nil = thm "append_Nil"; 

366 
val append_Cons = thm "append_Cons"; 

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val append1_eq_conv = thm "append1_eq_conv"; 

368 
val append_same_eq = thm "append_same_eq"; 

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val list_eq_pattern = 
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Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT) 
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373 
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = 

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(case xs of Const("List.list.Nil",_) => cons  _ => last xs) 
375 
 last (Const("List.op @",_) $ _ $ ys) = last ys 

376 
 last t = t 

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378 
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true 

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 list1 _ = false 
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381 
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = 

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(case xs of Const("List.list.Nil",_) => xs  _ => cons $ butlast xs) 
383 
 butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys 

384 
 butlast xs = Const("List.list.Nil",fastype_of xs) 

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386 
val rearr_tac = 

13145  387 
simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]) 
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389 
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = 

13145  390 
let 
391 
val lastl = last lhs and lastr = last rhs 

392 
fun rearr conv = 

393 
let val lhs1 = butlast lhs and rhs1 = butlast rhs 

394 
val Type(_,listT::_) = eqT 

395 
val appT = [listT,listT] > listT 

396 
val app = Const("List.op @",appT) 

397 
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) 

398 
val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) 

399 
val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) 

400 
handle ERROR => 

401 
error("The error(s) above occurred while trying to prove " ^ 

402 
string_of_cterm ct) 

403 
in Some((conv RS (thm RS trans)) RS eq_reflection) end 

13114  404 

13145  405 
in if list1 lastl andalso list1 lastr 
406 
then rearr append1_eq_conv 

407 
else 

408 
if lastl aconv lastr 

409 
then rearr append_same_eq 

410 
else None 

411 
end 

13114  412 
in 
413 
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq 

414 
end; 

415 

416 
Addsimprocs [list_eq_simproc]; 

417 
*} 

418 

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subsection {* @{text map} *} 
13114  421 

13142  422 
lemma map_ext: "(!!x. x : set xs > f x = g x) ==> map f xs = map g xs" 
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by (induct xs) simp_all 
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lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" 
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by (rule ext, induct_tac xs) auto 
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lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" 
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by (induct xs) auto 
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lemma map_compose: "map (f o g) xs = map f (map g xs)" 
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by (induct xs) (auto simp add: o_def) 
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13142  434 
lemma rev_map: "rev (map f xs) = map f (rev xs)" 
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by (induct xs) auto 
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lemma map_cong [recdef_cong]: 
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"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" 
439 
 {* a congruence rule for @{text map} *} 

440 
by (clarify, induct ys) auto 

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lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" 
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by (cases xs) auto 
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13142  445 
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" 
13145  446 
by (cases xs) auto 
13114  447 

448 
lemma map_eq_Cons: 

13145  449 
"(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)" 
450 
by (cases xs) auto 

13114  451 

452 
lemma map_injective: 

13145  453 
"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y > x = y) ==> xs = ys" 
454 
by (induct ys) (auto simp add: map_eq_Cons) 

13114  455 

456 
lemma inj_mapI: "inj f ==> inj (map f)" 

13145  457 
by (rules dest: map_injective injD intro: injI) 
13114  458 

459 
lemma inj_mapD: "inj (map f) ==> inj f" 

13145  460 
apply (unfold inj_on_def) 
461 
apply clarify 

462 
apply (erule_tac x = "[x]" in ballE) 

463 
apply (erule_tac x = "[y]" in ballE) 

464 
apply simp 

465 
apply blast 

466 
apply blast 

467 
done 

13114  468 

469 
lemma inj_map: "inj (map f) = inj f" 

13145  470 
by (blast dest: inj_mapD intro: inj_mapI) 
13114  471 

472 

13142  473 
subsection {* @{text rev} *} 
13114  474 

13142  475 
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" 
13145  476 
by (induct xs) auto 
13114  477 

13142  478 
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" 
13145  479 
by (induct xs) auto 
13114  480 

13142  481 
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" 
13145  482 
by (induct xs) auto 
13114  483 

13142  484 
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" 
13145  485 
by (induct xs) auto 
13114  486 

13142  487 
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" 
13145  488 
apply (induct xs) 
489 
apply force 

490 
apply (case_tac ys) 

491 
apply simp 

492 
apply force 

493 
done 

13114  494 

13366  495 
lemma rev_induct [case_names Nil snoc]: 
496 
"[ P []; !!x xs. P xs ==> P (xs @ [x]) ] ==> P xs" 

13145  497 
apply(subst rev_rev_ident[symmetric]) 
498 
apply(rule_tac list = "rev xs" in list.induct, simp_all) 

499 
done 

13114  500 

13145  501 
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} "compatibility" 
13114  502 

13366  503 
lemma rev_exhaust [case_names Nil snoc]: 
504 
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" 

13145  505 
by (induct xs rule: rev_induct) auto 
13114  506 

13366  507 
lemmas rev_cases = rev_exhaust 
508 

13114  509 

13142  510 
subsection {* @{text set} *} 
13114  511 

13142  512 
lemma finite_set [iff]: "finite (set xs)" 
13145  513 
by (induct xs) auto 
13114  514 

13142  515 
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" 
13145  516 
by (induct xs) auto 
13114  517 

13142  518 
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" 
13145  519 
by auto 
13114  520 

13142  521 
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" 
13145  522 
by (induct xs) auto 
13114  523 

13142  524 
lemma set_rev [simp]: "set (rev xs) = set xs" 
13145  525 
by (induct xs) auto 
13114  526 

13142  527 
lemma set_map [simp]: "set (map f xs) = f`(set xs)" 
13145  528 
by (induct xs) auto 
13114  529 

13142  530 
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" 
13145  531 
by (induct xs) auto 
13114  532 

13142  533 
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" 
13145  534 
apply (induct j) 
535 
apply simp_all 

536 
apply(erule ssubst) 

537 
apply auto 

538 
done 

13114  539 

13142  540 
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" 
13145  541 
apply (induct xs) 
542 
apply simp 

543 
apply simp 

544 
apply (rule iffI) 

545 
apply (blast intro: eq_Nil_appendI Cons_eq_appendI) 

546 
apply (erule exE)+ 

547 
apply (case_tac ys) 

548 
apply auto 

549 
done 

13142  550 

551 
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" 

13145  552 
 {* eliminate @{text lists} in favour of @{text set} *} 
553 
by (induct xs) auto 

13142  554 

555 
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" 

13145  556 
by (rule in_lists_conv_set [THEN iffD1]) 
13142  557 

558 
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" 

13145  559 
by (rule in_lists_conv_set [THEN iffD2]) 
13114  560 

561 

13142  562 
subsection {* @{text mem} *} 
13114  563 

564 
lemma set_mem_eq: "(x mem xs) = (x : set xs)" 

13145  565 
by (induct xs) auto 
13114  566 

567 

13142  568 
subsection {* @{text list_all} *} 
13114  569 

13142  570 
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" 
13145  571 
by (induct xs) auto 
13114  572 

13142  573 
lemma list_all_append [simp]: 
13145  574 
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" 
575 
by (induct xs) auto 

13114  576 

577 

13142  578 
subsection {* @{text filter} *} 
13114  579 

13142  580 
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" 
13145  581 
by (induct xs) auto 
13114  582 

13142  583 
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" 
13145  584 
by (induct xs) auto 
13114  585 

13142  586 
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" 
13145  587 
by (induct xs) auto 
13114  588 

13142  589 
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" 
13145  590 
by (induct xs) auto 
13114  591 

13142  592 
lemma length_filter [simp]: "length (filter P xs) \<le> length xs" 
13145  593 
by (induct xs) (auto simp add: le_SucI) 
13114  594 

13142  595 
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" 
13145  596 
by auto 
13114  597 

598 

13142  599 
subsection {* @{text concat} *} 
13114  600 

13142  601 
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" 
13145  602 
by (induct xs) auto 
13114  603 

13142  604 
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" 
13145  605 
by (induct xss) auto 
13114  606 

13142  607 
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" 
13145  608 
by (induct xss) auto 
13114  609 

13142  610 
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" 
13145  611 
by (induct xs) auto 
13114  612 

13142  613 
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" 
13145  614 
by (induct xs) auto 
13114  615 

13142  616 
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" 
13145  617 
by (induct xs) auto 
13114  618 

13142  619 
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" 
13145  620 
by (induct xs) auto 
13114  621 

622 

13142  623 
subsection {* @{text nth} *} 
13114  624 

13142  625 
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" 
13145  626 
by auto 
13114  627 

13142  628 
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" 
13145  629 
by auto 
13114  630 

13142  631 
declare nth.simps [simp del] 
13114  632 

633 
lemma nth_append: 

13145  634 
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n  length xs))" 
635 
apply(induct "xs") 

636 
apply simp 

637 
apply (case_tac n) 

638 
apply auto 

639 
done 

13114  640 

13142  641 
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" 
13145  642 
apply(induct xs) 
643 
apply simp 

644 
apply (case_tac n) 

645 
apply auto 

646 
done 

13114  647 

13142  648 
lemma set_conv_nth: "set xs = {xs!i  i. i < length xs}" 
13145  649 
apply (induct_tac xs) 
650 
apply simp 

651 
apply simp 

652 
apply safe 

653 
apply (rule_tac x = 0 in exI) 

654 
apply simp 

655 
apply (rule_tac x = "Suc i" in exI) 

656 
apply simp 

657 
apply (case_tac i) 

658 
apply simp 

659 
apply (rename_tac j) 

660 
apply (rule_tac x = j in exI) 

661 
apply simp 

662 
done 

13114  663 

13145  664 
lemma list_ball_nth: "[ n < length xs; !x : set xs. P x] ==> P(xs!n)" 
665 
by (auto simp add: set_conv_nth) 

13114  666 

13142  667 
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" 
13145  668 
by (auto simp add: set_conv_nth) 
13114  669 

670 
lemma all_nth_imp_all_set: 

13145  671 
"[ !i < length xs. P(xs!i); x : set xs] ==> P x" 
672 
by (auto simp add: set_conv_nth) 

13114  673 

674 
lemma all_set_conv_all_nth: 

13145  675 
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs > P (xs ! i))" 
676 
by (auto simp add: set_conv_nth) 

13114  677 

678 

13142  679 
subsection {* @{text list_update} *} 
13114  680 

13142  681 
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" 
13145  682 
by (induct xs) (auto split: nat.split) 
13114  683 

684 
lemma nth_list_update: 

13145  685 
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" 
686 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 

13114  687 

13142  688 
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" 
13145  689 
by (simp add: nth_list_update) 
13114  690 

13142  691 
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" 
13145  692 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 
13114  693 

13142  694 
lemma list_update_overwrite [simp]: 
13145  695 
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" 
696 
by (induct xs) (auto split: nat.split) 

13114  697 

698 
lemma list_update_same_conv: 

13145  699 
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" 
700 
by (induct xs) (auto split: nat.split) 

13114  701 

702 
lemma update_zip: 

13145  703 
"!!i xy xs. length xs = length ys ==> 
704 
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" 

705 
by (induct ys) (auto, case_tac xs, auto split: nat.split) 

13114  706 

707 
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" 

13145  708 
by (induct xs) (auto split: nat.split) 
13114  709 

710 
lemma set_update_subsetI: "[ set xs <= A; x:A ] ==> set(xs[i := x]) <= A" 

13145  711 
by (blast dest!: set_update_subset_insert [THEN subsetD]) 
13114  712 

713 

13142  714 
subsection {* @{text last} and @{text butlast} *} 
13114  715 

13142  716 
lemma last_snoc [simp]: "last (xs @ [x]) = x" 
13145  717 
by (induct xs) auto 
13114  718 

13142  719 
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" 
13145  720 
by (induct xs) auto 
13114  721 

13142  722 
lemma length_butlast [simp]: "length (butlast xs) = length xs  1" 
13145  723 
by (induct xs rule: rev_induct) auto 
13114  724 

725 
lemma butlast_append: 

13145  726 
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" 
727 
by (induct xs) auto 

13114  728 

13142  729 
lemma append_butlast_last_id [simp]: 
13145  730 
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" 
731 
by (induct xs) auto 

13114  732 

13142  733 
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" 
13145  734 
by (induct xs) (auto split: split_if_asm) 
13114  735 

736 
lemma in_set_butlast_appendI: 

13145  737 
"x : set (butlast xs)  x : set (butlast ys) ==> x : set (butlast (xs @ ys))" 
738 
by (auto dest: in_set_butlastD simp add: butlast_append) 

13114  739 

13142  740 

741 
subsection {* @{text take} and @{text drop} *} 

13114  742 

13142  743 
lemma take_0 [simp]: "take 0 xs = []" 
13145  744 
by (induct xs) auto 
13114  745 

13142  746 
lemma drop_0 [simp]: "drop 0 xs = xs" 
13145  747 
by (induct xs) auto 
13114  748 

13142  749 
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" 
13145  750 
by simp 
13114  751 

13142  752 
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" 
13145  753 
by simp 
13114  754 

13142  755 
declare take_Cons [simp del] and drop_Cons [simp del] 
13114  756 

13142  757 
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" 
13145  758 
by (induct n) (auto, case_tac xs, auto) 
13114  759 

13142  760 
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs  n)" 
13145  761 
by (induct n) (auto, case_tac xs, auto) 
13114  762 

13142  763 
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" 
13145  764 
by (induct n) (auto, case_tac xs, auto) 
13114  765 

13142  766 
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" 
13145  767 
by (induct n) (auto, case_tac xs, auto) 
13114  768 

13142  769 
lemma take_append [simp]: 
13145  770 
"!!xs. take n (xs @ ys) = (take n xs @ take (n  length xs) ys)" 
771 
by (induct n) (auto, case_tac xs, auto) 

13114  772 

13142  773 
lemma drop_append [simp]: 
13145  774 
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n  length xs) ys" 
775 
by (induct n) (auto, case_tac xs, auto) 

13114  776 

13142  777 
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" 
13145  778 
apply (induct m) 
779 
apply auto 

780 
apply (case_tac xs) 

781 
apply auto 

782 
apply (case_tac na) 

783 
apply auto 

784 
done 

13114  785 

13142  786 
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 
13145  787 
apply (induct m) 
788 
apply auto 

789 
apply (case_tac xs) 

790 
apply auto 

791 
done 

13114  792 

793 
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" 

13145  794 
apply (induct m) 
795 
apply auto 

796 
apply (case_tac xs) 

797 
apply auto 

798 
done 

13114  799 

13142  800 
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" 
13145  801 
apply (induct n) 
802 
apply auto 

803 
apply (case_tac xs) 

804 
apply auto 

805 
done 

13114  806 

807 
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" 

13145  808 
apply (induct n) 
809 
apply auto 

810 
apply (case_tac xs) 

811 
apply auto 

812 
done 

13114  813 

13142  814 
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 
13145  815 
apply (induct n) 
816 
apply auto 

817 
apply (case_tac xs) 

818 
apply auto 

819 
done 

13114  820 

821 
lemma rev_take: "!!i. rev (take i xs) = drop (length xs  i) (rev xs)" 

13145  822 
apply (induct xs) 
823 
apply auto 

824 
apply (case_tac i) 

825 
apply auto 

826 
done 

13114  827 

828 
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs  i) (rev xs)" 

13145  829 
apply (induct xs) 
830 
apply auto 

831 
apply (case_tac i) 

832 
apply auto 

833 
done 

13114  834 

13142  835 
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" 
13145  836 
apply (induct xs) 
837 
apply auto 

838 
apply (case_tac n) 

839 
apply(blast ) 

840 
apply (case_tac i) 

841 
apply auto 

842 
done 

13114  843 

13142  844 
lemma nth_drop [simp]: 
13145  845 
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" 
846 
apply (induct n) 

847 
apply auto 

848 
apply (case_tac xs) 

849 
apply auto 

850 
done 

3507  851 

13114  852 
lemma append_eq_conv_conj: 
13145  853 
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" 
854 
apply(induct xs) 

855 
apply simp 

856 
apply clarsimp 

857 
apply (case_tac zs) 

858 
apply auto 

859 
done 

13142  860 

13114  861 

13142  862 
subsection {* @{text takeWhile} and @{text dropWhile} *} 
13114  863 

13142  864 
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" 
13145  865 
by (induct xs) auto 
13114  866 

13142  867 
lemma takeWhile_append1 [simp]: 
13145  868 
"[ x:set xs; ~P(x)] ==> takeWhile P (xs @ ys) = takeWhile P xs" 
869 
by (induct xs) auto 

13114  870 

13142  871 
lemma takeWhile_append2 [simp]: 
13145  872 
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" 
873 
by (induct xs) auto 

13114  874 

13142  875 
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" 
13145  876 
by (induct xs) auto 
13114  877 

13142  878 
lemma dropWhile_append1 [simp]: 
13145  879 
"[ x : set xs; ~P(x)] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" 
880 
by (induct xs) auto 

13114  881 

13142  882 
lemma dropWhile_append2 [simp]: 
13145  883 
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" 
884 
by (induct xs) auto 

13114  885 

13142  886 
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" 
13145  887 
by (induct xs) (auto split: split_if_asm) 
13114  888 

889 

13142  890 
subsection {* @{text zip} *} 
13114  891 

13142  892 
lemma zip_Nil [simp]: "zip [] ys = []" 
13145  893 
by (induct ys) auto 
13114  894 

13142  895 
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" 
13145  896 
by simp 
13114  897 

13142  898 
declare zip_Cons [simp del] 
13114  899 

13142  900 
lemma length_zip [simp]: 
13145  901 
"!!xs. length (zip xs ys) = min (length xs) (length ys)" 
902 
apply(induct ys) 

903 
apply simp 

904 
apply (case_tac xs) 

905 
apply auto 

906 
done 

13114  907 

908 
lemma zip_append1: 

13145  909 
"!!xs. zip (xs @ ys) zs = 
910 
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" 

911 
apply (induct zs) 

912 
apply simp 

913 
apply (case_tac xs) 

914 
apply simp_all 

915 
done 

13114  916 

917 
lemma zip_append2: 

13145  918 
"!!ys. zip xs (ys @ zs) = 
919 
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" 

920 
apply (induct xs) 

921 
apply simp 

922 
apply (case_tac ys) 

923 
apply simp_all 

924 
done 

13114  925 

13142  926 
lemma zip_append [simp]: 
927 
"[ length xs = length us; length ys = length vs ] ==> 

13145  928 
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" 
929 
by (simp add: zip_append1) 

13114  930 

931 
lemma zip_rev: 

13145  932 
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" 
933 
apply(induct ys) 

934 
apply simp 

935 
apply (case_tac xs) 

936 
apply simp_all 

937 
done 

13114  938 

13142  939 
lemma nth_zip [simp]: 
13145  940 
"!!i xs. [ i < length xs; i < length ys] ==> (zip xs ys)!i = (xs!i, ys!i)" 
941 
apply (induct ys) 

942 
apply simp 

943 
apply (case_tac xs) 

944 
apply (simp_all add: nth.simps split: nat.split) 

945 
done 

13114  946 

947 
lemma set_zip: 

13145  948 
"set (zip xs ys) = {(xs!i, ys!i)  i. i < min (length xs) (length ys)}" 
949 
by (simp add: set_conv_nth cong: rev_conj_cong) 

13114  950 

951 
lemma zip_update: 

13145  952 
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" 
953 
by (rule sym, simp add: update_zip) 

13114  954 

13142  955 
lemma zip_replicate [simp]: 
13145  956 
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" 
957 
apply (induct i) 

958 
apply auto 

959 
apply (case_tac j) 

960 
apply auto 

961 
done 

13114  962 

13142  963 

964 
subsection {* @{text list_all2} *} 

13114  965 

966 
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys" 

13145  967 
by (simp add: list_all2_def) 
13114  968 

13142  969 
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" 
13145  970 
by (simp add: list_all2_def) 
13114  971 

13142  972 
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" 
13145  973 
by (simp add: list_all2_def) 
13114  974 

13142  975 
lemma list_all2_Cons [iff]: 
13145  976 
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" 
977 
by (auto simp add: list_all2_def) 

13114  978 

979 
lemma list_all2_Cons1: 

13145  980 
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" 
981 
by (cases ys) auto 

13114  982 

983 
lemma list_all2_Cons2: 

13145  984 
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" 
985 
by (cases xs) auto 

13114  986 

13142  987 
lemma list_all2_rev [iff]: 
13145  988 
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" 
989 
by (simp add: list_all2_def zip_rev cong: conj_cong) 

13114  990 

991 
lemma list_all2_append1: 

13145  992 
"list_all2 P (xs @ ys) zs = 
993 
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> 

994 
list_all2 P xs us \<and> list_all2 P ys vs)" 

995 
apply (simp add: list_all2_def zip_append1) 

996 
apply (rule iffI) 

997 
apply (rule_tac x = "take (length xs) zs" in exI) 

998 
apply (rule_tac x = "drop (length xs) zs" in exI) 

999 
apply (force split: nat_diff_split simp add: min_def) 

1000 
apply clarify 

1001 
apply (simp add: ball_Un) 

1002 
done 

13114  1003 

1004 
lemma list_all2_append2: 

13145  1005 
"list_all2 P xs (ys @ zs) = 
1006 
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> 

1007 
list_all2 P us ys \<and> list_all2 P vs zs)" 

1008 
apply (simp add: list_all2_def zip_append2) 

1009 
apply (rule iffI) 

1010 
apply (rule_tac x = "take (length ys) xs" in exI) 

1011 
apply (rule_tac x = "drop (length ys) xs" in exI) 

1012 
apply (force split: nat_diff_split simp add: min_def) 

1013 
apply clarify 

1014 
apply (simp add: ball_Un) 

1015 
done 

13114  1016 

1017 
lemma list_all2_conv_all_nth: 

13145  1018 
"list_all2 P xs ys = 
1019 
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" 

1020 
by (force simp add: list_all2_def set_zip) 

13114  1021 

1022 
lemma list_all2_trans[rule_format]: 

13145  1023 
"\<forall>a b c. P1 a b > P2 b c > P3 a c ==> 
1024 
\<forall>bs cs. list_all2 P1 as bs > list_all2 P2 bs cs > list_all2 P3 as cs" 

1025 
apply(induct_tac as) 

1026 
apply simp 

1027 
apply(rule allI) 

1028 
apply(induct_tac bs) 

1029 
apply simp 

1030 
apply(rule allI) 

1031 
apply(induct_tac cs) 

1032 
apply auto 

1033 
done 

13142  1034 

1035 

1036 
subsection {* @{text foldl} *} 

1037 

1038 
lemma foldl_append [simp]: 

13145  1039 
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" 
1040 
by (induct xs) auto 

13142  1041 

1042 
text {* 

13145  1043 
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more 
1044 
difficult to use because it requires an additional transitivity step. 

13142  1045 
*} 
1046 

1047 
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" 

13145  1048 
by (induct ns) auto 
13142  1049 

1050 
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" 

13145  1051 
by (force intro: start_le_sum simp add: in_set_conv_decomp) 
13142  1052 

1053 
lemma sum_eq_0_conv [iff]: 

13145  1054 
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" 
1055 
by (induct ns) auto 

13114  1056 

1057 

13142  1058 
subsection {* @{text upto} *} 
13114  1059 

13142  1060 
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" 
13145  1061 
 {* Does not terminate! *} 
1062 
by (induct j) auto 

13142  1063 

1064 
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" 

13145  1065 
by (subst upt_rec) simp 
13114  1066 

13142  1067 
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" 
13145  1068 
 {* Only needed if @{text upt_Suc} is deleted from the simpset. *} 
1069 
by simp 

13114  1070 

13142  1071 
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" 
13145  1072 
apply(rule trans) 
1073 
apply(subst upt_rec) 

1074 
prefer 2 apply(rule refl) 

1075 
apply simp 

1076 
done 

13114  1077 

13142  1078 
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" 
13145  1079 
 {* LOOPS as a simprule, since @{text "j <= j"}. *} 
1080 
by (induct k) auto 

13114  1081 

13142  1082 
lemma length_upt [simp]: "length [i..j(] = j  i" 
13145  1083 
by (induct j) (auto simp add: Suc_diff_le) 
13114  1084 

13142  1085 
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" 
13145  1086 
apply (induct j) 
1087 
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) 

1088 
done 

13114  1089 

13142  1090 
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" 
13145  1091 
apply (induct m) 
1092 
apply simp 

1093 
apply (subst upt_rec) 

1094 
apply (rule sym) 

1095 
apply (subst upt_rec) 

1096 
apply (simp del: upt.simps) 

1097 
done 

3507  1098 

13114  1099 
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" 
13145  1100 
by (induct n) auto 
13114  1101 

1102 
lemma nth_map_upt: "!!i. i < nm ==> (map f [m..n(]) ! i = f(m+i)" 

13145  1103 
apply (induct n m rule: diff_induct) 
1104 
prefer 3 apply (subst map_Suc_upt[symmetric]) 

1105 
apply (auto simp add: less_diff_conv nth_upt) 

1106 
done 

13114  1107 

13142  1108 
lemma nth_take_lemma [rule_format]: 
13145  1109 
"ALL xs ys. k <= length xs > k <= length ys 
1110 
> (ALL i. i < k > xs!i = ys!i) 

1111 
> take k xs = take k ys" 

1112 
apply (induct k) 

1113 
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib) 

1114 
apply clarify 

1115 
txt {* Both lists must be nonempty *} 

1116 
apply (case_tac xs) 

1117 
apply simp 

1118 
apply (case_tac ys) 

1119 
apply clarify 

1120 
apply (simp (no_asm_use)) 

1121 
apply clarify 

1122 
txt {* prenexing's needed, not miniscoping *} 

1123 
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) 

1124 
apply blast 

1125 
done 

13114  1126 

1127 
lemma nth_equalityI: 

1128 
"[ length xs = length ys; ALL i < length xs. xs!i = ys!i ] ==> xs = ys" 

13145  1129 
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) 
1130 
apply (simp_all add: take_all) 

1131 
done 

13142  1132 

1133 
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" 

13145  1134 
 {* The famous takelemma. *} 
1135 
apply (drule_tac x = "max (length xs) (length ys)" in spec) 

1136 
apply (simp add: le_max_iff_disj take_all) 

1137 
done 

13142  1138 

1139 

1140 
subsection {* @{text "distinct"} and @{text remdups} *} 

1141 

1142 
lemma distinct_append [simp]: 

13145  1143 
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" 
1144 
by (induct xs) auto 

13142  1145 

1146 
lemma set_remdups [simp]: "set (remdups xs) = set xs" 

13145  1147 
by (induct xs) (auto simp add: insert_absorb) 
13142  1148 

1149 
lemma distinct_remdups [iff]: "distinct (remdups xs)" 

13145  1150 
by (induct xs) auto 
13142  1151 

1152 
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" 

13145  1153 
by (induct xs) auto 
13114  1154 

13142  1155 
text {* 
13145  1156 
It is best to avoid this indexed version of distinct, but sometimes 
1157 
it is useful. *} 

13142  1158 
lemma distinct_conv_nth: 
13145  1159 
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j > xs!i \<noteq> xs!j)" 
1160 
apply (induct_tac xs) 

1161 
apply simp 

1162 
apply simp 

1163 
apply (rule iffI) 

1164 
apply clarsimp 

1165 
apply (case_tac i) 

1166 
apply (case_tac j) 

1167 
apply simp 

1168 
apply (simp add: set_conv_nth) 

1169 
apply (case_tac j) 

1170 
apply (clarsimp simp add: set_conv_nth) 

1171 
apply simp 

1172 
apply (rule conjI) 

1173 
apply (clarsimp simp add: set_conv_nth) 

1174 
apply (erule_tac x = 0 in allE) 

1175 
apply (erule_tac x = "Suc i" in allE) 

1176 
apply simp 

1177 
apply clarsimp 

1178 
apply (erule_tac x = "Suc i" in allE) 

1179 
apply (erule_tac x = "Suc j" in allE) 

1180 
apply simp 

1181 
done 

13114  1182 

1183 

13142  1184 
subsection {* @{text replicate} *} 
13114  1185 

13142  1186 
lemma length_replicate [simp]: "length (replicate n x) = n" 
13145  1187 
by (induct n) auto 
13124  1188 

13142  1189 
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" 
13145  1190 
by (induct n) auto 
13114  1191 

1192 
lemma replicate_app_Cons_same: 

13145  1193 
"(replicate n x) @ (x # xs) = x # replicate n x @ xs" 
1194 
by (induct n) auto 

13114  1195 

13142  1196 
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" 
13145  1197 
apply(induct n) 
1198 
apply simp 

1199 
apply (simp add: replicate_app_Cons_same) 

1200 
done 

13114  1201 

13142  1202 
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" 
13145  1203 
by (induct n) auto 
13114  1204 

13142  1205 
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" 
13145  1206 
by (induct n) auto 
13114  1207 

13142  1208 
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n  1) x" 
13145  1209 
by (induct n) auto 
13114  1210 

13142  1211 
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" 
13145  1212 
by (atomize (full), induct n) auto 
13114  1213 

13142  1214 
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" 
13145  1215 
apply(induct n) 
1216 
apply simp 

1217 
apply (simp add: nth_Cons split: nat.split) 

1218 
done 

13114  1219 

13142  1220 
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" 
13145  1221 
by (induct n) auto 
13114  1222 

13142  1223 
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" 
13145  1224 
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) 
13114  1225 

13142  1226 
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" 
13145  1227 
by auto 
13114  1228 

13142  1229 
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" 
13145  1230 
by (simp add: set_replicate_conv_if split: split_if_asm) 
13114  1231 

1232 

13142  1233 
subsection {* Lexcicographic orderings on lists *} 
3507  1234 

13142  1235 
lemma wf_lexn: "wf r ==> wf (lexn r n)" 
13145  1236 
apply (induct_tac n) 
1237 
apply simp 

1238 
apply simp 

1239 
apply(rule wf_subset) 

1240 
prefer 2 apply (rule Int_lower1) 

1241 
apply(rule wf_prod_fun_image) 

1242 
prefer 2 apply (rule injI) 

1243 
apply auto 

1244 
done 

13114  1245 

1246 
lemma lexn_length: 

13145  1247 
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" 
1248 
by (induct n) auto 

13114  1249 

13142  1250 
lemma wf_lex [intro!]: "wf r ==> wf (lex r)" 
13145  1251 
apply (unfold lex_def) 
1252 
apply (rule wf_UN) 

1253 
apply (blast intro: wf_lexn) 

1254 
apply clarify 

1255 
apply (rename_tac m n) 

1256 
apply (subgoal_tac "m \<noteq> n") 

1257 
prefer 2 apply blast 

1258 
apply (blast dest: lexn_length not_sym) 

1259 
done 

13114  1260 

1261 
lemma lexn_conv: 

13145  1262 
"lexn r n = 
1263 
{(xs,ys). length xs = n \<and> length ys = n \<and> 

1264 
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" 

1265 
apply (induct_tac n) 

1266 
apply simp 

1267 
apply blast 

1268 
apply (simp add: image_Collect lex_prod_def) 

1269 
apply auto 

1270 
apply blast 

1271 
apply (rename_tac a xys x xs' y ys') 

1272 
apply (rule_tac x = "a # xys" in exI) 

1273 
apply simp 

1274 
apply (case_tac xys) 

1275 
apply simp_all 

1276 
apply blast 

1277 
done 

13114  1278 

1279 
lemma lex_conv: 

13145  1280 
"lex r = 
1281 
{(xs,ys). length xs = length ys \<and> 

1282 
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" 

1283 
by (force simp add: lex_def lexn_conv) 

13114  1284 

13142  1285 
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)" 
13145  1286 
by (unfold lexico_def) blast 
13114  1287 

1288 
lemma lexico_conv: 

13145  1289 
"lexico r = {(xs,ys). length xs < length ys  
1290 
length xs = length ys \<and> (xs, ys) : lex r}" 

1291 
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) 

13114  1292 

13142  1293 
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" 
13145  1294 
by (simp add: lex_conv) 
13114  1295 

13142  1296 
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" 
13145  1297 
by (simp add:lex_conv) 
13114  1298 

13142  1299 
lemma Cons_in_lex [iff]: 
13145  1300 
"((x # xs, y # ys) : lex r) = 
1301 
((x, y) : r \<and> length xs = length ys  x = y \<and> (xs, ys) : lex r)" 

1302 
apply (simp add: lex_conv) 

1303 
apply (rule iffI) 

1304 
prefer 2 apply (blast intro: Cons_eq_appendI) 

1305 
apply clarify 

1306 
apply (case_tac xys) 

1307 
apply simp 

1308 
apply simp 

1309 
apply blast 

1310 
done 

13114  1311 

1312 

13142  1313 
subsection {* @{text sublist}  a generalization of @{text nth} to sets *} 
13114  1314 

13142  1315 
lemma sublist_empty [simp]: "sublist xs {} = []" 
13145  1316 
by (auto simp add: sublist_def) 
13114  1317 

13142  1318 
lemma sublist_nil [simp]: "sublist [] A = []" 
13145  1319 
by (auto simp add: sublist_def) 
13114  1320 

1321 
lemma sublist_shift_lemma: 

13145  1322 
"map fst [p:zip xs [i..i + length xs(] . snd p : A] = 
1323 
map fst [p:zip xs [0..length xs(] . snd p + i : A]" 

1324 
by (induct xs rule: rev_induct) (simp_all add: add_commute) 

13114  1325 

1326 
lemma sublist_append: 

13145  1327 
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" 
1328 
apply (unfold sublist_def) 

1329 
apply (induct l' rule: rev_induct) 

1330 
apply simp 

1331 
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) 

1332 
apply (simp add: add_commute) 

1333 
done 

13114  1334 

1335 
lemma sublist_Cons: 

13145  1336 
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" 
1337 
apply (induct l rule: rev_induct) 

1338 
apply (simp add: sublist_def) 

1339 
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) 

1340 
done 

13114  1341 

13142  1342 
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" 
13145  1343 
by (simp add: sublist_Cons) 
13114  1344 

13142  1345 
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l" 
13145  1346 
apply (induct l rule: rev_induct) 
1347 
apply simp 

1348 
apply (simp split: nat_diff_split add: sublist_append) 

1349 
done 

13114  1350 

1351 

13142  1352 
lemma take_Cons': 
13145  1353 
"take n (x # xs) = (if n = 0 then [] else x # take (n  1) xs)" 
1354 
by (cases n) simp_all 

13114  1355 

13142  1356 
lemma drop_Cons': 
13145
59bc43b51aa2
*** empty log me 