author  nipkow 
Mon, 05 Jan 2004 22:43:03 +0100  
changeset 14338  a1add2de7601 
parent 14328  fd063037fdf5 
child 14339  ec575b7bde7a 
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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License: GPL (GNU GENERAL PUBLIC LICENSE) 

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

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

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consts 

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

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

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

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

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fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set" 
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hd:: "'a list => 'a" 
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tl:: "'a list => 'a list" 

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last:: "'a list => 'a" 

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butlast :: "'a list => 'a list" 

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set :: "'a list => 'a set" 

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o2l :: "'a option => 'a list" 
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list_all:: "('a => bool) => ('a list => bool)" 
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" 

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

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mem :: "'a => 'a list => bool" (infixl 55) 

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nth :: "'a list => nat => 'a" (infixl "!" 100) 

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list_update :: "'a list => nat => 'a => 'a list" 

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take:: "nat => 'a list => 'a list" 

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drop:: "nat => 'a list => 'a list" 

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takeWhile :: "('a => bool) => 'a list => 'a list" 

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dropWhile :: "('a => bool) => 'a list => 'a list" 

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rev :: "'a list => 'a list" 

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zip :: "'a list => 'b list => ('a * 'b) list" 

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upt :: "nat => nat => nat list" ("(1[_../_'(])") 

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remdups :: "'a list => 'a list" 

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null:: "'a list => bool" 

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"distinct":: "'a list => bool" 

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replicate :: "nat => 'a => 'a list" 

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postfix :: "'a list => 'a list => bool" 
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syntax (xsymbols) 

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postfix :: "'a list => 'a list => bool" ("(_/ \<sqsupseteq> _)" [51, 51] 50) 

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nonterminals lupdbinds lupdbind 
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syntax 
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 {* list Enumeration *} 
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"@list" :: "args => 'a list" ("[(_)]") 

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 {* Special syntax for filter *} 
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"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") 

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 {* list update *} 
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"_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") 

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"" :: "lupdbind => lupdbinds" ("_") 

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"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") 

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"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) 

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upto:: "nat => nat => nat list" ("(1[_../_])") 
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translations 
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"[x, xs]" == "x#[xs]" 
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"[x]" == "x#[]" 

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"[x:xs . P]"== "filter (%x. P) xs" 

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"_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" 
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"xs[i:=x]" == "list_update xs i x" 

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"[i..j]" == "[i..(Suc j)(]" 
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syntax (xsymbols) 
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") 
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text {* 
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Function @{text size} is overloaded for all datatypes.Users may 
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refer to the list version as @{text length}. *} 

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syntax length :: "'a list => nat" 

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translations "length" => "size :: _ list => nat" 

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typed_print_translation {* 
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let 
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fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = 

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Syntax.const "length" $ t 

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 size_tr' _ _ _ = raise Match; 

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in [("size", size_tr')] end 

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*} 
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primrec 
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"hd(x#xs) = x" 
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primrec 
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"tl([]) = []" 
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"tl(x#xs) = xs" 

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primrec 
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"null([]) = True" 
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"null(x#xs) = False" 

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primrec 
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"last(x#xs) = (if xs=[] then x else last xs)" 
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primrec 
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"butlast []= []" 
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"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" 

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primrec 
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"x mem [] = False" 
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"x mem (y#ys) = (if y=x then True else x mem ys)" 

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primrec 
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"set [] = {}" 
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"set (x#xs) = insert x (set xs)" 

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primrec 
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"o2l None = []" 
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"o2l (Some x) = [x]" 

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primrec 

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list_all_Nil:"list_all P [] = True" 
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list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)" 

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primrec 
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"map f [] = []" 
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"map f (x#xs) = f(x)#map f xs" 

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primrec 
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append_Nil:"[]@ys = ys" 
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append_Cons: "(x#xs)@ys = x#(xs@ys)" 

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primrec 
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"rev([]) = []" 
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"rev(x#xs) = rev(xs) @ [x]" 

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primrec 
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"filter P [] = []" 
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" 

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primrec 
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foldl_Nil:"foldl f a [] = a" 
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" 

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primrec 
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"foldr f [] a = a" 
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"foldr f (x#xs) a = f x (foldr f xs a)" 

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primrec 
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"concat([]) = []" 
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"concat(x#xs) = x @ concat(xs)" 

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primrec 
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drop_Nil:"drop n [] = []" 
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drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs  Suc(m) => drop m xs)" 

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 {* Warning: simpset does not contain this definition *} 

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 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

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primrec 
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take_Nil:"take n [] = []" 
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take_Cons: "take n (x#xs) = (case n of 0 => []  Suc(m) => x # take m xs)" 

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 {* Warning: simpset does not contain this definition *} 

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 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

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primrec 
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nth_Cons:"(x#xs)!n = (case n of 0 => x  (Suc k) => xs!k)" 
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 {* Warning: simpset does not contain this definition *} 

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 {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 

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primrec 
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"[][i:=v] = []" 
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"(x#xs)[i:=v] = 

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(case i of 0 => v # xs 

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 Suc j => x # xs[j:=v])" 

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primrec 
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"takeWhile P [] = []" 
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" 

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primrec 
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"dropWhile P [] = []" 
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" 

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primrec 
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"zip xs [] = []" 
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zip_Cons: "zip xs (y#ys) = (case xs of [] => []  z#zs => (z,y)#zip zs ys)" 

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 {* Warning: simpset does not contain this definition *} 

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 {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} 

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primrec 
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upt_0: "[i..0(] = []" 
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upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])" 

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primrec 
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"distinct [] = True" 
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)" 

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primrec 
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"remdups [] = []" 
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" 

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primrec 
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replicate_0: "replicate 0 x = []" 
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replicate_Suc: "replicate (Suc n) x = x # replicate n x" 
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defs 
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postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys" 
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defs 

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list_all2_def: 
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"list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" 
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subsection {* Lexicographic orderings on lists *} 
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consts 

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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" 
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primrec 
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"lexn r 0 = {}" 
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"lexn r (Suc n) = 

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(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int 

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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" 

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constdefs 

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lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" 
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"lex r == \<Union>n. lexn r n" 

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lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" 
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" 

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sublist :: "'a list => nat set => 'a list" 
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))" 

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lemma not_Cons_self [simp]: "xs \<noteq> x # xs" 
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by (induct xs) auto 
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] 
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" 
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by (induct xs) auto 
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lemma length_induct: 
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"(!!xs. \<forall>ys. length ys < length xs > P ys ==> P xs) ==> P xs" 
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by (rule measure_induct [of length]) rules 

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subsection {* @{text lists}: the 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: 
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assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l 
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by induct blast+ 
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" 

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

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apply (blast intro!: lists_IntI) 

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done 

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lemma append_in_lists_conv [iff]: 
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" 
251 
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> 

298 
\<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) 

304 
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) 

13145  332 
done 
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" 

13145  335 
by simp 
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" 

13145  338 
by simp 
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13142  340 
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" 
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by simp 
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13142  343 
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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13142  349 
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" 
13145  350 
by (induct xs) auto 
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13142  352 
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" 
13145  353 
by (induct xs) auto 
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13142  355 
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" 
13145  356 
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)" 
13145  359 
by (simp split: list.split) 
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13142  361 
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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364 

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lemma Cons_eq_append_conv: "x#xs = ys@zs = 
366 
(ys = [] & x#xs = zs  (EX ys'. x#ys' = ys & xs = ys'@zs))" 

367 
by(cases ys) auto 

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text {* Trivial rules for solving @{text "@"}equations automatically. *} 
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lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" 

13145  373 
by simp 
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13142  375 
lemma Cons_eq_appendI: 
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"[ x # xs1 = ys; xs = xs1 @ zs ] ==> x # xs = ys @ zs" 
377 
by (drule sym) simp 

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13142  379 
lemma append_eq_appendI: 
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"[ xs @ xs1 = zs; ys = xs1 @ us ] ==> xs @ ys = zs @ us" 
381 
by (drule sym) simp 

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383 

13142  384 
text {* 
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Simplification procedure for all list equalities. 
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Currently only tries to rearrange @{text "@"} to see if 

387 
 both lists end in a singleton list, 

388 
 or both lists end in the same list. 

13142  389 
*} 
390 

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ML_setup {* 

3507  392 
local 
393 

13122  394 
val append_assoc = thm "append_assoc"; 
395 
val append_Nil = thm "append_Nil"; 

396 
val append_Cons = thm "append_Cons"; 

397 
val append1_eq_conv = thm "append1_eq_conv"; 

398 
val append_same_eq = thm "append_same_eq"; 

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fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = 
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(case xs of Const("List.list.Nil",_) => cons  _ => last xs) 
402 
 last (Const("List.op @",_) $ _ $ ys) = last ys 

403 
 last t = t; 

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405 
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true 

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 list1 _ = false; 
13114  407 

408 
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) 
410 
 butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys 

411 
 butlast xs = Const("List.list.Nil",fastype_of xs); 

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413 
val rearr_tac = 

13462  414 
simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); 
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416 
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = 

13462  417 
let 
418 
val lastl = last lhs and lastr = last rhs; 

419 
fun rearr conv = 

420 
let 

421 
val lhs1 = butlast lhs and rhs1 = butlast rhs; 

422 
val Type(_,listT::_) = eqT 

423 
val appT = [listT,listT] > listT 

424 
val app = Const("List.op @",appT) 

425 
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) 

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diff
changeset

426 
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); 
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

427 
val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1)); 
13462  428 
in Some ((conv RS (thm RS trans)) RS eq_reflection) end; 
13114  429 

13462  430 
in 
431 
if list1 lastl andalso list1 lastr then rearr append1_eq_conv 

432 
else if lastl aconv lastr then rearr append_same_eq 

433 
else None 

434 
end; 

435 

13114  436 
in 
13462  437 

438 
val list_eq_simproc = 

439 
Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; 

440 

13114  441 
end; 
442 

443 
Addsimprocs [list_eq_simproc]; 

444 
*} 

445 

446 

13142  447 
subsection {* @{text map} *} 
13114  448 

13142  449 
lemma map_ext: "(!!x. x : set xs > f x = g x) ==> map f xs = map g xs" 
13145  450 
by (induct xs) simp_all 
13114  451 

13142  452 
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" 
13145  453 
by (rule ext, induct_tac xs) auto 
13114  454 

13142  455 
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" 
13145  456 
by (induct xs) auto 
13114  457 

13142  458 
lemma map_compose: "map (f o g) xs = map f (map g xs)" 
13145  459 
by (induct xs) (auto simp add: o_def) 
13114  460 

13142  461 
lemma rev_map: "rev (map f xs) = map f (rev xs)" 
13145  462 
by (induct xs) auto 
13114  463 

13737  464 
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" 
465 
by (induct xs) auto 

466 

13366  467 
lemma map_cong [recdef_cong]: 
13145  468 
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" 
469 
 {* a congruence rule for @{text map} *} 

13737  470 
by simp 
13114  471 

13142  472 
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" 
13145  473 
by (cases xs) auto 
13114  474 

13142  475 
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" 
13145  476 
by (cases xs) auto 
13114  477 

14025  478 
lemma map_eq_Cons_conv[iff]: 
479 
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" 

13145  480 
by (cases xs) auto 
13114  481 

14025  482 
lemma Cons_eq_map_conv[iff]: 
483 
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" 

484 
by (cases ys) auto 

485 

14111  486 
lemma ex_map_conv: 
487 
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" 

488 
by(induct ys, auto) 

489 

13114  490 
lemma map_injective: 
14338  491 
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys" 
492 
by (induct ys) (auto dest!:injD) 

13114  493 

494 
lemma inj_mapI: "inj f ==> inj (map f)" 

13585  495 
by (rules dest: map_injective injD intro: inj_onI) 
13114  496 

497 
lemma inj_mapD: "inj (map f) ==> inj f" 

14208  498 
apply (unfold inj_on_def, clarify) 
13145  499 
apply (erule_tac x = "[x]" in ballE) 
14208  500 
apply (erule_tac x = "[y]" in ballE, simp, blast) 
13145  501 
apply blast 
502 
done 

13114  503 

504 
lemma inj_map: "inj (map f) = inj f" 

13145  505 
by (blast dest: inj_mapD intro: inj_mapI) 
13114  506 

507 

13142  508 
subsection {* @{text rev} *} 
13114  509 

13142  510 
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" 
13145  511 
by (induct xs) auto 
13114  512 

13142  513 
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" 
13145  514 
by (induct xs) auto 
13114  515 

13142  516 
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" 
13145  517 
by (induct xs) auto 
13114  518 

13142  519 
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" 
13145  520 
by (induct xs) auto 
13114  521 

13142  522 
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" 
14208  523 
apply (induct xs, force) 
524 
apply (case_tac ys, simp, force) 

13145  525 
done 
13114  526 

13366  527 
lemma rev_induct [case_names Nil snoc]: 
528 
"[ P []; !!x xs. P xs ==> P (xs @ [x]) ] ==> P xs" 

13145  529 
apply(subst rev_rev_ident[symmetric]) 
530 
apply(rule_tac list = "rev xs" in list.induct, simp_all) 

531 
done 

13114  532 

13145  533 
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} "compatibility" 
13114  534 

13366  535 
lemma rev_exhaust [case_names Nil snoc]: 
536 
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" 

13145  537 
by (induct xs rule: rev_induct) auto 
13114  538 

13366  539 
lemmas rev_cases = rev_exhaust 
540 

13114  541 

13142  542 
subsection {* @{text set} *} 
13114  543 

13142  544 
lemma finite_set [iff]: "finite (set xs)" 
13145  545 
by (induct xs) auto 
13114  546 

13142  547 
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" 
13145  548 
by (induct xs) auto 
13114  549 

14099  550 
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l" 
14208  551 
by (case_tac l, auto) 
14099  552 

13142  553 
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" 
13145  554 
by auto 
13114  555 

14099  556 
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
557 
by auto 

558 

13142  559 
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" 
13145  560 
by (induct xs) auto 
13114  561 

13142  562 
lemma set_rev [simp]: "set (rev xs) = set xs" 
13145  563 
by (induct xs) auto 
13114  564 

13142  565 
lemma set_map [simp]: "set (map f xs) = f`(set xs)" 
13145  566 
by (induct xs) auto 
13114  567 

13142  568 
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" 
13145  569 
by (induct xs) auto 
13114  570 

13142  571 
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" 
14208  572 
apply (induct j, simp_all) 
573 
apply (erule ssubst, auto) 

13145  574 
done 
13114  575 

13142  576 
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" 
14208  577 
apply (induct xs, simp, simp) 
13145  578 
apply (rule iffI) 
579 
apply (blast intro: eq_Nil_appendI Cons_eq_appendI) 

580 
apply (erule exE)+ 

14208  581 
apply (case_tac ys, auto) 
13145  582 
done 
13142  583 

584 
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" 

13145  585 
 {* eliminate @{text lists} in favour of @{text set} *} 
586 
by (induct xs) auto 

13142  587 

588 
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" 

13145  589 
by (rule in_lists_conv_set [THEN iffD1]) 
13142  590 

591 
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" 

13145  592 
by (rule in_lists_conv_set [THEN iffD2]) 
13114  593 

13508  594 
lemma finite_list: "finite A ==> EX l. set l = A" 
595 
apply (erule finite_induct, auto) 

596 
apply (rule_tac x="x#l" in exI, auto) 

597 
done 

598 

13114  599 

13142  600 
subsection {* @{text mem} *} 
13114  601 

602 
lemma set_mem_eq: "(x mem xs) = (x : set xs)" 

13145  603 
by (induct xs) auto 
13114  604 

605 

13142  606 
subsection {* @{text list_all} *} 
13114  607 

13142  608 
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" 
13145  609 
by (induct xs) auto 
13114  610 

13142  611 
lemma list_all_append [simp]: 
13145  612 
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" 
613 
by (induct xs) auto 

13114  614 

615 

13142  616 
subsection {* @{text filter} *} 
13114  617 

13142  618 
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" 
13145  619 
by (induct xs) auto 
13114  620 

13142  621 
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" 
13145  622 
by (induct xs) auto 
13114  623 

13142  624 
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" 
13145  625 
by (induct xs) auto 
13114  626 

13142  627 
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" 
13145  628 
by (induct xs) auto 
13114  629 

13142  630 
lemma length_filter [simp]: "length (filter P xs) \<le> length xs" 
13145  631 
by (induct xs) (auto simp add: le_SucI) 
13114  632 

13142  633 
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" 
13145  634 
by auto 
13114  635 

636 

13142  637 
subsection {* @{text concat} *} 
13114  638 

13142  639 
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" 
13145  640 
by (induct xs) auto 
13114  641 

13142  642 
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" 
13145  643 
by (induct xss) auto 
13114  644 

13142  645 
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" 
13145  646 
by (induct xss) auto 
13114  647 

13142  648 
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" 
13145  649 
by (induct xs) auto 
13114  650 

13142  651 
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" 
13145  652 
by (induct xs) auto 
13114  653 

13142  654 
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" 
13145  655 
by (induct xs) auto 
13114  656 

13142  657 
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" 
13145  658 
by (induct xs) auto 
13114  659 

660 

13142  661 
subsection {* @{text nth} *} 
13114  662 

13142  663 
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" 
13145  664 
by auto 
13114  665 

13142  666 
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" 
13145  667 
by auto 
13114  668 

13142  669 
declare nth.simps [simp del] 
13114  670 

671 
lemma nth_append: 

13145  672 
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n  length xs))" 
14208  673 
apply (induct "xs", simp) 
674 
apply (case_tac n, auto) 

13145  675 
done 
13114  676 

13142  677 
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" 
14208  678 
apply (induct xs, simp) 
679 
apply (case_tac n, auto) 

13145  680 
done 
13114  681 

13142  682 
lemma set_conv_nth: "set xs = {xs!i  i. i < length xs}" 
14208  683 
apply (induct_tac xs, simp, simp) 
13145  684 
apply safe 
14208  685 
apply (rule_tac x = 0 in exI, simp) 
686 
apply (rule_tac x = "Suc i" in exI, simp) 

687 
apply (case_tac i, simp) 

13145  688 
apply (rename_tac j) 
14208  689 
apply (rule_tac x = j in exI, simp) 
13145  690 
done 
13114  691 

13145  692 
lemma list_ball_nth: "[ n < length xs; !x : set xs. P x] ==> P(xs!n)" 
693 
by (auto simp add: set_conv_nth) 

13114  694 

13142  695 
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" 
13145  696 
by (auto simp add: set_conv_nth) 
13114  697 

698 
lemma all_nth_imp_all_set: 

13145  699 
"[ !i < length xs. P(xs!i); x : set xs] ==> P x" 
700 
by (auto simp add: set_conv_nth) 

13114  701 

702 
lemma all_set_conv_all_nth: 

13145  703 
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs > P (xs ! i))" 
704 
by (auto simp add: set_conv_nth) 

13114  705 

706 

13142  707 
subsection {* @{text list_update} *} 
13114  708 

13142  709 
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" 
13145  710 
by (induct xs) (auto split: nat.split) 
13114  711 

712 
lemma nth_list_update: 

13145  713 
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" 
714 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 

13114  715 

13142  716 
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" 
13145  717 
by (simp add: nth_list_update) 
13114  718 

13142  719 
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" 
13145  720 
by (induct xs) (auto simp add: nth_Cons split: nat.split) 
13114  721 

13142  722 
lemma list_update_overwrite [simp]: 
13145  723 
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" 
724 
by (induct xs) (auto split: nat.split) 

13114  725 

14187  726 
lemma list_update_id[simp]: "!!i. i < length xs \<Longrightarrow> xs[i := xs!i] = xs" 
14208  727 
apply (induct xs, simp) 
14187  728 
apply(simp split:nat.splits) 
729 
done 

730 

13114  731 
lemma list_update_same_conv: 
13145  732 
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" 
733 
by (induct xs) (auto split: nat.split) 

13114  734 

14187  735 
lemma list_update_append1: 
736 
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" 

14208  737 
apply (induct xs, simp) 
14187  738 
apply(simp split:nat.split) 
739 
done 

740 

13114  741 
lemma update_zip: 
13145  742 
"!!i xy xs. length xs = length ys ==> 
743 
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" 

744 
by (induct ys) (auto, case_tac xs, auto split: nat.split) 

13114  745 

746 
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" 

13145  747 
by (induct xs) (auto split: nat.split) 
13114  748 

749 
lemma set_update_subsetI: "[ set xs <= A; x:A ] ==> set(xs[i := x]) <= A" 

13145  750 
by (blast dest!: set_update_subset_insert [THEN subsetD]) 
13114  751 

752 

13142  753 
subsection {* @{text last} and @{text butlast} *} 
13114  754 

13142  755 
lemma last_snoc [simp]: "last (xs @ [x]) = x" 
13145  756 
by (induct xs) auto 
13114  757 

13142  758 
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" 
13145  759 
by (induct xs) auto 
13114  760 

14302  761 
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" 
762 
by(simp add:last.simps) 

763 

764 
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" 

765 
by(simp add:last.simps) 

766 

767 
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" 

768 
by (induct xs) (auto) 

769 

770 
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" 

771 
by(simp add:last_append) 

772 

773 
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" 

774 
by(simp add:last_append) 

775 

776 

777 

13142  778 
lemma length_butlast [simp]: "length (butlast xs) = length xs  1" 
13145  779 
by (induct xs rule: rev_induct) auto 
13114  780 

781 
lemma butlast_append: 

13145  782 
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" 
783 
by (induct xs) auto 

13114  784 

13142  785 
lemma append_butlast_last_id [simp]: 
13145  786 
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" 
787 
by (induct xs) auto 

13114  788 

13142  789 
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" 
13145  790 
by (induct xs) (auto split: split_if_asm) 
13114  791 

792 
lemma in_set_butlast_appendI: 

13145  793 
"x : set (butlast xs)  x : set (butlast ys) ==> x : set (butlast (xs @ ys))" 
794 
by (auto dest: in_set_butlastD simp add: butlast_append) 

13114  795 

13142  796 

797 
subsection {* @{text take} and @{text drop} *} 

13114  798 

13142  799 
lemma take_0 [simp]: "take 0 xs = []" 
13145  800 
by (induct xs) auto 
13114  801 

13142  802 
lemma drop_0 [simp]: "drop 0 xs = xs" 
13145  803 
by (induct xs) auto 
13114  804 

13142  805 
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" 
13145  806 
by simp 
13114  807 

13142  808 
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" 
13145  809 
by simp 
13114  810 

13142  811 
declare take_Cons [simp del] and drop_Cons [simp del] 
13114  812 

14187  813 
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" 
814 
by(cases xs, simp_all) 

815 

816 
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" 

817 
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) 

818 

819 
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" 

14208  820 
apply (induct xs, simp) 
14187  821 
apply(simp add:drop_Cons nth_Cons split:nat.splits) 
822 
done 

823 

13913  824 
lemma take_Suc_conv_app_nth: 
825 
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" 

14208  826 
apply (induct xs, simp) 
827 
apply (case_tac i, auto) 

13913  828 
done 
829 

13142  830 
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" 
13145  831 
by (induct n) (auto, case_tac xs, auto) 
13114  832 

13142  833 
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs  n)" 
13145  834 
by (induct n) (auto, case_tac xs, auto) 
13114  835 

13142  836 
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" 
13145  837 
by (induct n) (auto, case_tac xs, auto) 
13114  838 

13142  839 
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" 
13145  840 
by (induct n) (auto, case_tac xs, auto) 
13114  841 

13142  842 
lemma take_append [simp]: 
13145  843 
"!!xs. take n (xs @ ys) = (take n xs @ take (n  length xs) ys)" 
844 
by (induct n) (auto, case_tac xs, auto) 

13114  845 

13142  846 
lemma drop_append [simp]: 
13145  847 
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n  length xs) ys" 
848 
by (induct n) (auto, case_tac xs, auto) 

13114  849 

13142  850 
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" 
14208  851 
apply (induct m, auto) 
852 
apply (case_tac xs, auto) 

853 
apply (case_tac na, auto) 

13145  854 
done 
13114  855 

13142  856 
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" 
14208  857 
apply (induct m, auto) 
858 
apply (case_tac xs, auto) 

13145  859 
done 
13114  860 

861 
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" 

14208  862 
apply (induct m, auto) 
863 
apply (case_tac xs, auto) 

13145  864 
done 
13114  865 

13142  866 
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" 
14208  867 
apply (induct n, auto) 
868 
apply (case_tac xs, auto) 

13145  869 
done 
13114  870 

871 
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" 

14208  872 
apply (induct n, auto) 
873 
apply (case_tac xs, auto) 

13145  874 
done 
13114  875 

13142  876 
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" 
14208  877 
apply (induct n, auto) 
878 
apply (case_tac xs, auto) 

13145  879 
done 
13114  880 

881 
lemma rev_take: "!!i. rev (take i xs) = drop (length xs  i) (rev xs)" 

14208  882 
apply (induct xs, auto) 
883 
apply (case_tac i, auto) 

13145  884 
done 
13114  885 

886 
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs  i) (rev xs)" 

14208  887 
apply (induct xs, auto) 
888 
apply (case_tac i, auto) 

13145  889 
done 
13114  890 

13142  891 
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" 
14208  892 
apply (induct xs, auto) 
893 
apply (case_tac n, blast) 

894 
apply (case_tac i, auto) 

13145  895 
done 
13114  896 

13142  897 
lemma nth_drop [simp]: 
13145  898 
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" 
14208  899 
apply (induct n, auto) 
900 
apply (case_tac xs, auto) 

13145  901 
done 
3507  902 

14025  903 
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" 
904 
by(induct xs)(auto simp:take_Cons split:nat.split) 

905 

906 
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" 

907 
by(induct xs)(auto simp:drop_Cons split:nat.split) 

908 

14187  909 
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" 
910 
using set_take_subset by fast 

911 

912 
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" 

913 
using set_drop_subset by fast 

914 

13114  915 
lemma append_eq_conv_conj: 
13145  916 
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" 
14208  917 
apply (induct xs, simp, clarsimp) 
918 
apply (case_tac zs, auto) 

13145  919 
done 
13142  920 

14050  921 
lemma take_add [rule_format]: 
922 
"\<forall>i. i+j \<le> length(xs) > take (i+j) xs = take i xs @ take j (drop i xs)" 

923 
apply (induct xs, auto) 

924 
apply (case_tac i, simp_all) 

925 
done 

926 

14300  927 
lemma append_eq_append_conv_if: 
928 
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = 

929 
(if size xs\<^isub>1 \<le> size ys\<^isub>1 

930 
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 

931 
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)" 

932 
apply(induct xs\<^isub>1) 

933 
apply simp 

934 
apply(case_tac ys\<^isub>1) 

935 
apply simp_all 

936 
done 

937 

13114  938 

13142  939 
subsection {* @{text takeWhile} and @{text dropWhile} *} 
13114  940 

13142  941 
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" 
13145  942 
by (induct xs) auto 
13114  943 

13142  944 
lemma takeWhile_append1 [simp]: 
13145  945 
"[ x:set xs; ~P(x)] ==> takeWhile P (xs @ ys) = takeWhile P xs" 
946 
by (induct xs) auto 

13114  947 

13142  948 
lemma takeWhile_append2 [simp]: 
13145  949 
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" 
950 
by (induct xs) auto 

13114  951 

13142  952 
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" 
13145  953 
by (induct xs) auto 
13114  954 

13142  955 
lemma dropWhile_append1 [simp]: 
13145  956 
"[ x : set xs; ~P(x)] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" 
957 
by (induct xs) auto 

13114  958 

13142  959 
lemma dropWhile_append2 [simp]: 
13145  960 
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" 
961 
by (induct xs) auto 

13114  962 

13142  963 
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" 
13145  964 
by (induct xs) (auto split: split_if_asm) 
13114  965 

13913  966 
lemma takeWhile_eq_all_conv[simp]: 
967 
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" 

968 
by(induct xs, auto) 

969 

970 
lemma dropWhile_eq_Nil_conv[simp]: 

971 
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" 

972 
by(induct xs, auto) 

973 

974 
lemma dropWhile_eq_Cons_conv: 

975 
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" 

976 
by(induct xs, auto) 

977 

13114  978 

13142  979 
subsection {* @{text zip} *} 
13114  980 

13142  981 
lemma zip_Nil [simp]: "zip [] ys = []" 
13145  982 
by (induct ys) auto 
13114  983 

13142  984 
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" 
13145  985 
by simp 
13114  986 

13142  987 
declare zip_Cons [simp del] 
13114  988 

13142  989 
lemma length_zip [simp]: 
13145  990 
"!!xs. length (zip xs ys) = min (length xs) (length ys)" 
14208  991 
apply (induct ys, simp) 
992 
apply (case_tac xs, auto) 

13145  993 
done 
13114  994 

995 
lemma zip_append1: 

13145  996 
"!!xs. zip (xs @ ys) zs = 
997 
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" 

14208  998 
apply (induct zs, simp) 
999 
apply (case_tac xs, simp_all) 

13145  1000 
done 
13114  1001 

1002 
lemma zip_append2: 

13145  1003 
"!!ys. zip xs (ys @ zs) = 
1004 
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" 

14208  1005 
apply (induct xs, simp) 
1006 
apply (case_tac ys, simp_all) 

13145  1007 
done 
13114  1008 

13142  1009 
lemma zip_append [simp]: 
1010 
"[ length xs = length us; length ys = length vs ] ==> 

13145  1011 
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" 
1012 
by (simp add: zip_append1) 

13114  1013 

1014 
lemma zip_rev: 

14247  1015 
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" 
1016 
by (induct rule:list_induct2, simp_all) 

13114  1017 

13142  1018 
lemma nth_zip [simp]: 
13145  1019 
"!!i xs. [ i < length xs; i < length ys] ==> (zip xs ys)!i = (xs!i, ys!i)" 
14208  1020 
apply (induct ys, simp) 
13145  1021 
apply (case_tac xs) 
1022 
apply (simp_all add: nth.simps split: nat.split) 

1023 
done 

13114  1024 

1025 
lemma set_zip: 

13145  1026 
"set (zip xs ys) = {(xs!i, ys!i)  i. i < min (length xs) (length ys)}" 
1027 
by (simp add: set_conv_nth cong: rev_conj_cong) 

13114  1028 

1029 
lemma zip_update: 

13145  1030 
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" 
1031 
by (rule sym, simp add: update_zip) 

13114  1032 

13142  1033 
lemma zip_replicate [simp]: 
13145  1034 
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" 
14208  1035 
apply (induct i, auto) 
1036 
apply (case_tac j, auto) 

13145  1037 
done 
13114  1038 

13142  1039 

1040 
subsection {* @{text list_all2} *} 

13114  1041 

14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset

1042 
lemma list_all2_lengthD [intro?]: 
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset

1043 
"list_all2 P xs ys ==> length xs = length ys" 
13145  1044 
by (simp add: list_all2_def) 
13114  1045 

13142  1046 
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" 
13145  1047 
by (simp add: list_all2_def) 
13114  1048 

13142  1049 
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" 
13145  1050 
by (simp add: list_all2_def) 
13114  1051 

13142  1052 
lemma list_all2_Cons [iff]: 
13145  1053 
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" 
1054 
by (auto simp add: list_all2_def) 

13114  1055 

1056 
lemma list_all2_Cons1: 

13145  1057 
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" 
1058 
by (cases ys) auto 

13114  1059 

1060 
lemma list_all2_Cons2: 

13145  1061 
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" 
1062 
by (cases xs) auto 

13114  1063 

13142  1064 
lemma list_all2_rev [iff]: 
13145  1065 
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" 
1066 
by (simp add: list_all2_def zip_rev cong: conj_cong) 

13114  1067 

13863  1068 
lemma list_all2_rev1: 
1069 
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" 

1070 
by (subst list_all2_rev [symmetric]) simp 

1071 

13114  1072 
lemma list_all2_append1: 
13145  1073 
"list_all2 P (xs @ ys) zs = 
1074 
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> 

1075 
list_all2 P xs us \<and> list_all2 P ys vs)" 

1076 
apply (simp add: list_all2_def zip_append1) 

1077 
apply (rule iffI) 

1078 
apply (rule_tac x = "take (length xs) zs" in exI) 

1079 
apply (rule_tac x = "drop (length xs) zs" in exI) 

14208  1080 
apply (force split: nat_diff_split simp add: min_def, clarify) 
13145  1081 
apply (simp add: ball_Un) 
1082 
done 

13114  1083 

1084 
lemma list_all2_append2: 

13145  1085 
"list_all2 P xs (ys @ zs) = 
1086 
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> 

1087 
list_all2 P us ys \<and> list_all2 P vs zs)" 

1088 
apply (simp add: list_all2_def zip_append2) 

1089 
apply (rule iffI) 

1090 
apply (rule_tac x = "take (length ys) xs" in exI) 

1091 
apply (rule_tac x = "drop (length ys) xs" in exI) 

14208  1092 
apply (force split: nat_diff_split simp add: min_def, clarify) 
13145  1093 
apply (simp add: ball_Un) 
1094 
done 

13114  1095 

13863  1096 
lemma list_all2_append: 
14247  1097 
"length xs = length ys \<Longrightarrow> 
1098 
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" 

1099 
by (induct rule:list_induct2, simp_all) 

13863  1100 

1101 
lemma list_all2_appendI [intro?, trans]: 

1102 
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" 

1103 
by (simp add: list_all2_append list_all2_lengthD) 

1104 

13114  1105 
lemma list_all2_conv_all_nth: 
13145  1106 
"list_all2 P xs ys = 
1107 
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" 

1108 
by (force simp add: list_all2_def set_zip) 

13114  1109 

13883
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1110 
lemma list_all2_trans: 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1111 
assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1112 
shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1113 
(is "!!bs cs. PROP ?Q as bs cs") 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1114 
proof (induct as) 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1115 
fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1116 
show "!!cs. PROP ?Q (x # xs) bs cs" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1117 
proof (induct bs) 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1118 
fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1119 
show "PROP ?Q (x # xs) (y # ys) cs" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1120 
by (induct cs) (auto intro: tr I1 I2) 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1121 
qed simp 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1122 
qed simp 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1123 

13863  1124 
lemma list_all2_all_nthI [intro?]: 
1125 
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" 

1126 
by (simp add: list_all2_conv_all_nth) 

1127 

14328  1128 
lemma list_all2_nthD: 
13863  1129 
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" 
1130 
by (simp add: list_all2_conv_all_nth) 

1131 

14302  1132 
lemma list_all2_nthD2: 
1133 
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" 

1134 
by (frule list_all2_lengthD) (auto intro: list_all2_nthD) 

1135 

13863  1136 
lemma list_all2_map1: 
1137 
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" 

1138 
by (simp add: list_all2_conv_all_nth) 

1139 

1140 
lemma list_all2_map2: 

1141 
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" 

1142 
by (auto simp add: list_all2_conv_all_nth) 

1143 

14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset

1144 
lemma list_all2_refl [intro?]: 
13863  1145 
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" 
1146 
by (simp add: list_all2_conv_all_nth) 

1147 

1148 
lemma list_all2_update_cong: 

1149 
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" 

1150 
by (simp add: list_all2_conv_all_nth nth_list_update) 

1151 

1152 
lemma list_all2_update_cong2: 

1153 
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" 

1154 
by (simp add: list_all2_lengthD list_all2_update_cong) 

1155 

14302  1156 
lemma list_all2_takeI [simp,intro?]: 
1157 
"\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" 

1158 
apply (induct xs) 

1159 
apply simp 

1160 
apply (clarsimp simp add: list_all2_Cons1) 

1161 
apply (case_tac n) 

1162 
apply auto 

1163 
done 

1164 

1165 
lemma list_all2_dropI [simp,intro?]: 

13863  1166 
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" 
14208  1167 
apply (induct as, simp) 
13863  1168 
apply (clarsimp simp add: list_all2_Cons1) 
14208  1169 
apply (case_tac n, simp, simp) 
13863  1170 
done 
1171 

14327  1172 
lemma list_all2_mono [intro?]: 
13863  1173 
"\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y" 
14208  1174 
apply (induct x, simp) 
1175 
apply (case_tac y, auto) 

13863  1176 
done 
1177 

13142  1178 

1179 
subsection {* @{text foldl} *} 

1180 

1181 
lemma foldl_append [simp]: 

13145  1182 
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" 
1183 
by (induct xs) auto 

13142  1184 

1185 
text {* 

13145  1186 
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more 
1187 
difficult to use because it requires an additional transitivity step. 

13142  1188 
*} 
1189 

1190 
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" 

13145  1191 
by (induct ns) auto 
13142  1192 

1193 
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" 

13145  1194 
by (force intro: start_le_sum simp add: in_set_conv_decomp) 
13142  1195 

1196 
lemma sum_eq_0_conv [iff]: 

13145  1197 
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" 
1198 
by (induct ns) auto 

13114  1199 

1200 

14099  1201 
subsection {* folding a relation over a list *} 
1202 

1203 
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*) 

1204 
inductive "fold_rel R" intros 

1205 
Nil: "(a, [],a) : fold_rel R" 

1206 
Cons: "[(a,x,b) : R; (b,xs,c) : fold_rel R] ==> (a,x#xs,c) : fold_rel R" 

1207 
inductive_cases fold_rel_elim_case [elim!]: 

14208  1208 
"(a, [] , b) : fold_rel R" 
14099  1209 
"(a, x#xs, b) : fold_rel R" 
1210 

1211 
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" 

1212 
by (simp add: fold_rel.Nil) 

1213 
declare fold_rel.Cons [intro!] 

1214 

1215 

13142  1216 
subsection {* @{text upto} *} 
13114  1217 

13142  1218 
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" 
13145  1219 
 {* Does not terminate! *} 
1220 
by (induct j) auto 

13142  1221 

1222 
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" 

13145  1223 
by (subst upt_rec) simp 
13114  1224 

13142  1225 
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" 
13145  1226 
 {* Only needed if @{text upt_Suc} is deleted from the simpset. *} 
1227 
by simp 

13114  1228 

13142  1229 
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" 
13145  1230 
apply(rule trans) 
1231 
apply(subst upt_rec) 

14208  1232 
prefer 2 apply (rule refl, simp) 
13145  1233 
done 
13114  1234 

13142  1235 
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" 
13145  1236 
 {* LOOPS as a simprule, since @{text "j <= j"}. *} 
1237 
by (induct k) auto 

13114  1238 

13142  1239 
lemma length_upt [simp]: "length [i..j(] = j  i" 
13145  1240 
by (induct j) (auto simp add: Suc_diff_le) 
13114  1241 

13142  1242 
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" 
13145  1243 
apply (induct j) 
1244 
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) 

1245 
done 

13114  1246 

13142  1247 
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" 
14208  1248 
apply (induct m, simp) 
13145  1249 
apply (subst upt_rec) 
1250 
apply (rule sym) 

1251 
apply (subst upt_rec) 

1252 
apply (simp del: upt.simps) 

1253 
done 

3507  1254 

13114  1255 
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" 
13145  1256 
by (induct n) auto 
13114  1257 

1258 
lemma nth_map_upt: "!!i. i < nm ==> (map f [m..n(]) ! i = f(m+i)" 

13145  1259 
apply (induct n m rule: diff_induct) 
1260 
prefer 3 apply (subst map_Suc_upt[symmetric]) 

1261 
apply (auto simp add: less_diff_conv nth_upt) 

1262 
done 

13114  1263 

13883
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1264 
lemma nth_take_lemma: 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1265 
"!!xs ys. k <= length xs ==> k <= length ys ==> 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1266 
(!!i. i < k > xs!i = ys!i) ==> take k xs = take k ys" 
0451e0fb3f22
Restructured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset

1267 
apply (atomize, induct k) 
14208  1268 
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) 
13145  1269 
txt {* Both lists must be nonempty *} 
14208  1270 
apply (case_tac xs, simp) 
1271 
apply (case_tac ys, clarify) 

13145  1272 
apply (simp (no_asm_use)) 
1273 
apply clarify 

1274 
txt {* prenexing's needed, not miniscoping *} 

1275 
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) 

1276 
apply blast 

1277 
done 

13114  1278 

1279 
lemma nth_equalityI: 

1280 
"[ length xs = length ys; ALL i < length xs. xs!i = ys!i ] ==> xs = ys" 

13145  1281 
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) 
1282 
apply (simp_all add: take_all) 

1283 
done 

13142  1284 

13863  1285 
(* needs nth_equalityI *) 
1286 
lemma list_all2_antisym: 

1287 
"\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 

1288 
\<Longrightarrow> xs = ys" 

1289 
apply (simp add: list_all2_conv_all_nth) 

14208  1290 
apply (rule nth_equalityI, blast, simp) 
13863  1291 
done 
1292 

13142  1293 
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" 
13145  1294 
 {* The famous takelemma. *} 
1295 
apply (drule_tac x = "max (length xs) (length ys)" in spec) 

1296 
apply (simp add: le_max_iff_disj take_all) 

1297 
done 

13142  1298 

1299 

1300 
subsection {* @{text "distinct"} and @{text remdups} *} 

1301 

1302 
lemma distinct_append [simp]: 

13145  1303 
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" 
1304 
by (induct xs) auto 

13142  1305 

1306 
lemma set_remdups [simp]: "set (remdups xs) = set xs" 

13145  1307 
by (induct xs) (auto simp add: insert_absorb) 
13142  1308 

1309 
lemma distinct_remdups [iff]: "distinct (remdups xs)" 

13145  1310 
by (induct xs) auto 
13142  1311 

1312 
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" 

13145  1313 
by (induct xs) auto 
13114  1314 

13142  1315 
text {* 
13145  1316 
It is best to avoid this indexed version of distinct, but sometimes 
1317 
it is useful. *} 

13142  1318 
lemma distinct_conv_nth: 
13145  1319 
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j > xs!i \<noteq> xs!j)" 
14208  1320 
apply (induct_tac xs, simp, simp) 
1321 
apply (rule iffI, clarsimp) 

13145  1322 
apply (case_tac i) 
14208  1323 
apply (case_tac j, simp) 
13145  1324 
apply (simp add: set_conv_nth) 
1325 
apply (case_tac j) 

14208  1326 
apply (clarsimp simp add: set_conv_nth, simp) 
13145  1327 
apply (rule conjI) 
1328 
apply (clarsimp simp add: set_conv_nth) 

1329 
apply (erule_tac x = 0 in allE) 

14208  1330 
apply (erule_tac x = "Suc i" in allE, simp, clarsimp) 
13145  1331 
apply (erule_tac x = "Suc i" in allE) 
14208  1332 
apply (erule_tac x = "Suc j" in allE, simp) 
13145  1333 
done 
13114  1334 

1335 

13142  1336 
subsection {* @{text replicate} *} 
13114  1337 

13142  1338 
lemma length_replicate [simp]: "length (replicate n x) = n" 
13145  1339 
by (induct n) auto 
13124  1340 

13142  1341 
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" 
13145  1342 
by (induct n) auto 
13114 